A staff puts a note in pitch space, and position is cheap to draw. Duration is not a position but a relationship: a note is short only next to a longer one, and a page of polyphony has to state that relationship for every voice at once. European music had a workable pitch notation by the eleventh century and no workable rhythmic notation until the fourteenth, and the system that finally arrived, mensural notation, is the direct ancestor of the note values you read today.
\nThe earliest Western notation was not a staff. Ninth-century scribes wrote neumes, small signs above a text that recorded the shape of a melodic gesture well enough for a singer who already knew the melody. The next step was to give the signs vertical position: heighted neumes, then a single line drawn through them, then several lines. Guido of Arezzo, working in the eleventh century, is conventionally credited with the four-line staff and with the syllables ut, re, mi, fa, sol, la taken from the hymn Ut queant laxis, which gave singers a way to name and find the intervals.
\nThat system solves pitch and ignores duration, because chant has only one voice. The rhythm comes from the words, from the syllable lengths, and from a performing tradition no notation needed to record. The moment a second voice is added, the notation fails. Two singers in different ranges still have to agree on when, and nothing on the page says when.
\nThirteenth-century scribes began to carry duration in the shape of the note, and the fourteenth century completed the system by adding the semibrevis and the minima to the older brevis and longa. What makes mensural notation look alien is that the value of a shape was not fixed. It depended on the mensuration in force, that is, on how many of the next smaller value the shape contained.
\nFour relationships did the work:
\nThree of something was called perfect, two imperfect. Fourteenth-century theorists attached Trinitarian weight to the number three, which is part of why perfect mensuration carried prestige, but the practical effect was arithmetic: the same written note meant different lengths in different pieces, and the mensuration sign at the start told you which.
\n| Sign | Tempus | Prolatio | Semibreves per breve | Minimae per semibrevis |
|---|---|---|---|---|
| O | perfect | minor | 3 | 2 |
| O with a dot | perfect | major | 3 | 3 |
| C | imperfect | minor | 2 | 2 |
| C with a dot | imperfect | major | 2 | 3 |
The note shapes, written in black ink before the fifteenth century:
\n| Shape | Name | Contains | Conventional modern value |
|---|---|---|---|
| filled rectangle | maxima | 2 or 3 longae | none |
| rectangle with a stem | longa | 2 or 3 breves | none |
| filled square | brevis | 2 or 3 semibreves | double whole note |
| filled lozenge | semibrevis | 2 or 3 minimae | whole note |
| lozenge with a stem | minima | the smallest value | half note |
| stem with one flag | semiminima | half a minima | quarter note |
The modern equivalents in the right-hand column are conventions of transcription, not facts about the fourteenth century; how long a semibrevis lasted depended on tempo, which was not notated at all. The shapes also interacted. In perfect mensuration a note could be imperfected by a following smaller value, losing a third of its length, or altered by a following note of the same value, which doubled it. Readers were trained for years to resolve these contexts at sight.
\nThe phrase ars nova comes from a treatise usually dated to the 1320s and attributed to Philippe de Vitry. Later writers used ars antiqua for the older style, sometimes with contempt, and music historians turned both labels into period names for French music of roughly 1310 to 1375.
\nThe change the label marks is not a new emotion or a new genre. It is the acceptance of duple division as a legitimate, fully notated basis for music. Earlier thirteenth-century notation had treated triple division as the norm and duple as a tolerated exception; after the Ars Nova both were structural, and the semibrevis and minima became the working units of the rhythmic surface. That is what made the notation of Guillaume de Machaut’s songs and motets possible, and his Messe de Nostre Dame is usually described as the earliest complete polyphonic setting of the Mass Ordinary by a single named composer.
\nAt the end of the century the system was pushed further still. The repertory called Ars subtilior, associated with the papal court at Avignon and preserved in manuscripts such as the Chantilly Codex, uses proportional signs, coloured notes, and notated canons that make the page itself the puzzle. It is notation written for specialists, and it is a useful reminder that complexity in a system is not the same thing as expressive depth.
\nThe fifteenth century changed the ink before it changed the logic. Scribes moved to hollow shapes for the longer values and kept filled shapes for the shortest ones, a system modern editors call white notation. Hollow rectangles and lozenges took less ink, wore the paper less, and read more clearly at a distance than a page of solid black. The underlying mensural arithmetic stayed exactly where it was.
\nProportional signs handled the cases where the surface had to change without a new mensuration. A sign such as sesquialtera told the performer to sing three of a value in the time previously given to two, which is the notated form of the hemiola a Renaissance singer feels at a cadence.
\n\nSix equal written notes. Whether they fall as three plus three or as an even six is a decision the mensuration and proportion signs make, not the note shapes, and the same page read under a different sign produces a different piece.
\nPrinting then fixed the layout. Ottaviano Petrucci’s Harmonice Musices Odhecaton, printed in Venice in 1501, is the first printed collection of polyphonic music, and like most polyphonic prints of the period it appeared in partbooks: one voice per book, four singers around a table, no score anywhere. That format is why so much Renaissance music survives only as separate lines, and why a modern editor’s first job is often to work out what the vertical sonorities actually were.
\nThree things a modern score gives you were left to the singer.
\nMusica ficta. Accidentals were written only when they were unusual. Everything else fell under conventions: the cadential leading tone was raised even when the signature did not say so, and the note a fifth above the final tended to be flattened to keep the tritone out of the mode. Fourteenth- and fifteenth-century scribes and theorists compressed this into mnemonics, one of which runs una nota super la semper est canendum fa, a note above la is always sung fa.
\n\nA cadence on D. The C is sharpened by convention to make a leading tone, and nothing on the page marks it.
\n\nThe same figure with the signature left alone: the modal sixth step, no leading tone, a weaker and older close. Both readings exist in the sources, and which one a modern performer chooses is an editorial position.
\nText underlay. Manuscripts frequently give text only to the top voice. Where the syllables fall in the lower voices was decided by convention and by the singers, so editions differ on which note carries which syllable.
\nTempo. Not notated. The proportional signs give relative tempo, and a vertical stroke through the C sign, later called cut time or diminution, indicated that the same values went by at roughly double speed. Nothing gives an absolute rate, which is why the same motet can be recorded at four and a half minutes and at seven.
\nIf you sing or play from a modern edition of medieval and Renaissance music, the clean rhythmic surface is a reconstruction. Bracketed accidentals are the editor’s musica ficta; a bracketed tempo is an inference from notation and from genre; underlay decisions are arguments. None of that makes the edition wrong, but it changes what you are judging. Two editions that disagree about a C sharp are disagreeing about a convention the original never wrote down, so the useful move is to look at the critical report rather than to assume one of them made a mistake.
\nFor practice, take a short passage in a reliable edition and count the editorial accidents on the page: the sharps in brackets, the tempo, the text distribution. The count tells you how much of the sound you are hearing is fifteenth-century and how much is twentieth-century scholarship, and it is the fastest way to stop reading early music as if it were written by a composer with modern notation habits.
", "content_text": "A staff puts a note in pitch space, and position is cheap to draw. Duration is not a position but a relationship: a note is short only next to a longer one, and a page of polyphony has to state that relationship for every voice at once. European music had a workable pitch notation by the eleventh century and no workable rhythmic notation until the fourteenth, and the system that finally arrived, mensural notation, is the direct ancestor of the note values you read today.\n#Pitch was the easy problem\nThe earliest Western notation was not a staff. Ninth-century scribes wrote neumes, small signs above a text that recorded the shape of a melodic gesture well enough for a singer who already knew the melody. The next step was to give the signs vertical position: heighted neumes, then a single line drawn through them, then several lines. Guido of Arezzo, working in the eleventh century, is conventionally credited with the four-line staff and with the syllables ut, re, mi, fa, sol, la taken from the hymn Ut queant laxis, which gave singers a way to name and find the intervals.\nThat system solves pitch and ignores duration, because chant has only one voice. The rhythm comes from the words, from the syllable lengths, and from a performing tradition no notation needed to record. The moment a second voice is added, the notation fails. Two singers in different ranges still have to agree on when, and nothing on the page says when.\n#Mensural notation: value as a ratio\nThirteenth-century scribes began to carry duration in the shape of the note, and the fourteenth century completed the system by adding the semibrevis and the minima to the older brevis and longa. What makes mensural notation look alien is that the value of a shape was not fixed. It depended on the mensuration in force, that is, on how many of the next smaller value the shape contained.\nFour relationships did the work:\nmodus maior: how many longae in a maxima, two or threemodus minor: how many breves in a longatempus: how many semibreves in a brevisprolatio: how many minimae in a semibrevis\nThree of something was called perfect, two imperfect. Fourteenth-century theorists attached Trinitarian weight to the number three, which is part of why perfect mensuration carried prestige, but the practical effect was arithmetic: the same written note meant different lengths in different pieces, and the mensuration sign at the start told you which.\nSignTempusProlatioSemibreves per breveMinimae per semibrevisOperfectminor32O with a dotperfectmajor33Cimperfectminor22C with a dotimperfectmajor23\nThe note shapes, written in black ink before the fifteenth century:\nShapeNameContainsConventional modern valuefilled rectanglemaxima2 or 3 longaenonerectangle with a stemlonga2 or 3 brevesnonefilled squarebrevis2 or 3 semibrevesdouble whole notefilled lozengesemibrevis2 or 3 minimaewhole notelozenge with a stemminimathe smallest valuehalf notestem with one flagsemiminimahalf a minimaquarter note\nThe modern equivalents in the right-hand column are conventions of transcription, not facts about the fourteenth century; how long a semibrevis lasted depended on tempo, which was not notated at all. The shapes also interacted. In perfect mensuration a note could be imperfected by a following smaller value, losing a third of its length, or altered by a following note of the same value, which doubled it. Readers were trained for years to resolve these contexts at sight.\n#Ars Nova: duple time gets equal standing\nThe phrase ars nova comes from a treatise usually dated to the 1320s and attributed to Philippe de Vitry. Later writers used ars antiqua for the older style, sometimes with contempt, and music historians turned both labels into period names for French music of roughly 1310 to 1375.\nThe change the label marks is not a new emotion or a new genre. It is the acceptance of duple division as a legitimate, fully notated basis for music. Earlier thirteenth-century notation had treated triple division as the norm and duple as a tolerated exception; after the Ars Nova both were structural, and the semibrevis and minima became the working units of the rhythmic surface. That is what made the notation of Guillaume de Machaut’s songs and motets possible, and his Messe de Nostre Dame is usually described as the earliest complete polyphonic setting of the Mass Ordinary by a single named composer.\nAt the end of the century the system was pushed further still. The repertory called Ars subtilior, associated with the papal court at Avignon and preserved in manuscripts such as the Chantilly Codex, uses proportional signs, coloured notes, and notated canons that make the page itself the puzzle. It is notation written for specialists, and it is a useful reminder that complexity in a system is not the same thing as expressive depth.\n#White notation and the printed partbook\nThe fifteenth century changed the ink before it changed the logic. Scribes moved to hollow shapes for the longer values and kept filled shapes for the shortest ones, a system modern editors call white notation. Hollow rectangles and lozenges took less ink, wore the paper less, and read more clearly at a distance than a page of solid black. The underlying mensural arithmetic stayed exactly where it was.\nProportional signs handled the cases where the surface had to change without a new mensuration. A sign such as sesquialtera told the performer to sing three of a value in the time previously given to two, which is the notated form of the hemiola a Renaissance singer feels at a cadence.\nC4+E4 C4+E4 C4+E4 G3+D4 G3+D4 G3+D4\nSix equal written notes. Whether they fall as three plus three or as an even six is a decision the mensuration and proportion signs make, not the note shapes, and the same page read under a different sign produces a different piece.\nPrinting then fixed the layout. Ottaviano Petrucci’s Harmonice Musices Odhecaton, printed in Venice in 1501, is the first printed collection of polyphonic music, and like most polyphonic prints of the period it appeared in partbooks: one voice per book, four singers around a table, no score anywhere. That format is why so much Renaissance music survives only as separate lines, and why a modern editor’s first job is often to work out what the vertical sonorities actually were.\n#What the performer supplied\nThree things a modern score gives you were left to the singer.\nMusica ficta. Accidentals were written only when they were unusual. Everything else fell under conventions: the cadential leading tone was raised even when the signature did not say so, and the note a fifth above the final tended to be flattened to keep the tritone out of the mode. Fourteenth- and fifteenth-century scribes and theorists compressed this into mnemonics, one of which runs una nota super la semper est canendum fa, a note above la is always sung fa.\nD4 C#4 D4\nA cadence on D. The C is sharpened by convention to make a leading tone, and nothing on the page marks it.\nD4 C4 D4\nThe same figure with the signature left alone: the modal sixth step, no leading tone, a weaker and older close. Both readings exist in the sources, and which one a modern performer chooses is an editorial position.\nText underlay. Manuscripts frequently give text only to the top voice. Where the syllables fall in the lower voices was decided by convention and by the singers, so editions differ on which note carries which syllable.\nTempo. Not notated. The proportional signs give relative tempo, and a vertical stroke through the C sign, later called cut time or diminution, indicated that the same values went by at roughly double speed. Nothing gives an absolute rate, which is why the same motet can be recorded at four and a half minutes and at seven.\n#Reading a transcription with the right expectations\nIf you sing or play from a modern edition of medieval and Renaissance music, the clean rhythmic surface is a reconstruction. Bracketed accidentals are the editor’s musica ficta; a bracketed tempo is an inference from notation and from genre; underlay decisions are arguments. None of that makes the edition wrong, but it changes what you are judging. Two editions that disagree about a C sharp are disagreeing about a convention the original never wrote down, so the useful move is to look at the critical report rather than to assume one of them made a mistake.\nFor practice, take a short passage in a reliable edition and count the editorial accidents on the page: the sharps in brackets, the tempo, the text distribution. The count tells you how much of the sound you are hearing is fifteenth-century and how much is twentieth-century scholarship, and it is the fastest way to stop reading early music as if it were written by a composer with modern notation habits.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "notation", "mensural notation", "renaissance" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/ars-nova-and-renaissance-notation.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/borrowed-chords-and-modal-mixture/", "url": "https://musics.name/articles/borrowed-chords-and-modal-mixture/", "title": "Borrowed Chords: Modal Mixture in Practice", "summary": "Mixture borrows chords from the parallel mode. What the common borrowed chords do to a phrase, how they resolve, and where labelling fails.", "content_html": "An F minor triad in C major requires an A flat, which the key signature does not give. Unlike a secondary dominant, the chord is not pointing at another key: F minor is the subdominant of C minor, and C minor is C major’s parallel mode. The tonic has not moved. Only the mode has leaned.
\nEvery major key has a parallel minor key with the same tonic, and the two share a set of chords that differ only in the third, sixth and seventh degrees. Mixture, also called modal mixture or modal interchange, is the practice of taking a chord from the parallel mode while the passage continues in the original one.
\nThis is what separates mixture from the chromatic devices that surround it. A secondary dominant imports the dominant of a different tonic, so it points somewhere. Mixture imports a chord with a different colour but the same tonic, so it points nowhere in particular. It can sit in the middle of a phrase without any obligation to confirm anything, because nothing was claimed.
\nThe borrowed chord does not need to be a chord that exists in the parallel minor. The third, sixth and seventh degrees are the variables, and any combination of them produces a usable chord, which is why the list below runs to half a dozen members rather than three.
\n| Chord in C major | Notes | Where it comes from | Usual function | Typical resolution |
|---|---|---|---|---|
| iv | F–A flat–C | C minor subdominant | pre-dominant | to V, or plagally to I |
| flat VI | A flat–C–E flat | C minor submediant | pre-dominant | to V, or to flat VII |
| flat III | E flat–G–B flat | C minor mediant | tonic substitute | to IV, flat VI or V |
| flat VII | B flat–D–F | C minor subtonic | dominant substitute | to I, or to IV |
| ii half-diminished seventh | D–F–A flat–C | C minor supertonic | pre-dominant | to V |
| flat II in first inversion | D flat–F–A flat, F in bass | Neapolitan, borrowed | pre-dominant | to V, often through a cadential six-four |
| v | G–B flat–D | C minor dominant | modal cadence | to I, with a lowered seventh degree |
The lowered sixth degree is the note that most of these chords have in common, and it is the note that does most of the expressive work. A flat VI, an iv, a ii half-diminished seventh and a Neapolitan all contain A flat in C major, and each of them approaches the dominant from a semitone above its bass note or its fifth. That is why mixture is so often a pre-dominant device: it strengthens the arrival on V by making the ear wait for it.
\n\nThe borrowed chords that behave like pre-dominants are easy to place, because they resolve the way their diatonic equivalents do. The iv chord resolves to V, and to I when it wants a plagal close. The ii half-diminished seventh resolves to V with the same downward pull by fifth that a ii seventh has. The Neapolitan resolves to V, and in the classical repertoire it very often passes first through a cadential six-four, which puts the dominant in the bass a beat early and delays the leading tone.
\nTwo of the chords on the list do not behave functionally in the traditional sense, and that is their value. The flat VII is a subtonic rather than a leading-tone chord, so it can move directly to the tonic without any dominant preparation at all, which is the motion behind an enormous amount of rock and folk repertoire. The flat VI can precede the flat VII and produce the same arrival in two steps: A flat, B flat, C. Both progressions are diatonic in the Aeolian mode, which raises a question of labelling worth settling before the analysis gets carried away.
\n\nIf the whole passage is Aeolian, with a natural minor scale and no raised leading tone, then a flat VII is not borrowed at all. It is simply the dominant of the prevailing mode. Mixture presupposes a major or minor tonal centre with a functioning dominant, and a chord that contradicts that assumption is evidence that the centre is not what the analyst assumed. Deciding which of the two is happening is a matter of the cadence: a piece that ends V–i in the minor mode and also uses flat VII in passing is mixing modes, while a piece that ends flat VII–I throughout is in a mode.
\nThe augmented sixth chords are the most chromatic members of the borrowed set, and they are grouped by the interval rather than by the chord. All three are built on the lowered sixth degree, which sits in the bass, and all three contain an augmented sixth above that bass note. In C minor the bass is A flat and the augmented sixth is F sharp.
\n| Name | Notes in C minor | Intervals above A flat | Resolution |
|---|---|---|---|
| Italian sixth | A flat–C–F sharp | major third, augmented sixth | the sixth expands to the octave on G |
| French sixth | A flat–C–D–F sharp | major third, augmented fourth, augmented sixth | same, with the fourth resolving to the fifth of V |
| German sixth | A flat–C–E flat–F sharp | major third, perfect fifth, augmented sixth | same, with the fifth held into the cadential six-four |
The gap between A flat and F sharp is the point. The interval is an augmented sixth: ten semitones, one more than a major sixth, which is why the two notes are under pressure to move outward to the octave, and in practice they do: A flat descends to G and F sharp ascends to G. The result is a dominant, usually reached through a cadential six-four. In a major key the chord requires the borrowed lowered sixth degree, which is why the augmented sixth belongs in a discussion of mixture rather than in a chapter of its own.
\nThe German sixth has a second identity. Respell F sharp as G flat and the chord A flat, C, E flat, G flat is an A flat dominant seventh, which resolves not to C but to D flat. The same four pitches are a pre-dominant in one key and a dominant in another, and later composers used that overlap deliberately. The modulation article follows that thread; here it is enough to note that a chord can belong to two keys at once and that the notation is what tells the player which one is intended.
\n\nThree errors recur often enough to be worth naming.
\nThe first is calling every chromatic chord mixture. A chromatic passing tone between two diatonic notes is not mixture, and neither is a secondary dominant, whose raised third points at a different tonic. The test is the third of the chord. In C major, an E major triad contains G sharp and behaves as V/vi. An E flat major triad contains the borrowed lowered third degree and is mixture. The two chords differ by one scale degree and by their entire function.
\nThe second is labelling mixture in a modal context. A song that stays on a flat VII chord and never raises the seventh degree is not borrowing from anywhere. Retrospective labels that call the modal chords of a rock song “borrowed from the parallel minor” describe the surface and miss the structure.
\nThe third is stacking borrowed chords faster than they can resolve. Mixture is a colouring device, and a passage in four parts in which every chord carries a lowered degree gives the ear no diatonic frame to measure against. Two borrowed chords in a phrase are usually enough for the effect they produce; four in a row tend to sound like a mistake in the parts.
\nCompetent use of mixture is mostly a question of placement.
\nPut a borrowed iv where a diatonic IV would go and the phrase turns inward for a beat, then resolves as usual. Put a flat VI before the dominant and the arrival is delayed and darkened. Put a flat VII directly before the tonic and the cadence becomes modal and open, which is why it appears so often at the end of a chorus. The Neapolitan earns its keep in minor keys, where it supplies the strongest pre-dominant available, and in major keys where a sudden flat II in first inversion announces a serious moment.
\nThe practical discipline is to check the resolutions. Every lowered sixth degree wants to descend to the dominant, and every raised fourth degree wants to ascend to it. When a borrowed chord arrives and neither of those happens, the ear hears the chromaticism as unexplained, and the next phrase has to work harder than it should. Play the chord, resolve it as the table suggests, and the difference between a good borrow and an accidental one is audible in a single hearing.
\nThe virtual piano is enough to test that difference: hold a plain IV, then an iv, and listen to which one makes the following dominant feel inevitable.
", "content_text": "An F minor triad in C major requires an A flat, which the key signature does not give. Unlike a secondary dominant, the chord is not pointing at another key: F minor is the subdominant of C minor, and C minor is C major’s parallel mode. The tonic has not moved. Only the mode has leaned.\n#Mixture as a mode change, not a key change\nEvery major key has a parallel minor key with the same tonic, and the two share a set of chords that differ only in the third, sixth and seventh degrees. Mixture, also called modal mixture or modal interchange, is the practice of taking a chord from the parallel mode while the passage continues in the original one.\nThis is what separates mixture from the chromatic devices that surround it. A secondary dominant imports the dominant of a different tonic, so it points somewhere. Mixture imports a chord with a different colour but the same tonic, so it points nowhere in particular. It can sit in the middle of a phrase without any obligation to confirm anything, because nothing was claimed.\nThe borrowed chord does not need to be a chord that exists in the parallel minor. The third, sixth and seventh degrees are the variables, and any combination of them produces a usable chord, which is why the list below runs to half a dozen members rather than three.\n#The chords you will meet most\nChord in C majorNotesWhere it comes fromUsual functionTypical resolutionivF–A flat–CC minor subdominantpre-dominantto V, or plagally to Iflat VIA flat–C–E flatC minor submediantpre-dominantto V, or to flat VIIflat IIIE flat–G–B flatC minor medianttonic substituteto IV, flat VI or Vflat VIIB flat–D–FC minor subtonicdominant substituteto I, or to IVii half-diminished seventhD–F–A flat–CC minor supertonicpre-dominantto Vflat II in first inversionD flat–F–A flat, F in bassNeapolitan, borrowedpre-dominantto V, often through a cadential six-fourvG–B flat–DC minor dominantmodal cadenceto I, with a lowered seventh degree\nThe lowered sixth degree is the note that most of these chords have in common, and it is the note that does most of the expressive work. A flat VI, an iv, a ii half-diminished seventh and a Neapolitan all contain A flat in C major, and each of them approaches the dominant from a semitone above its bass note or its fifth. That is why mixture is so often a pre-dominant device: it strengthens the arrival on V by making the ear wait for it.\nC4+E4+G4 F3+Ab3+C4 G3+B3+D4 C4+E4+G4\n#How borrowed chords resolve\nThe borrowed chords that behave like pre-dominants are easy to place, because they resolve the way their diatonic equivalents do. The iv chord resolves to V, and to I when it wants a plagal close. The ii half-diminished seventh resolves to V with the same downward pull by fifth that a ii seventh has. The Neapolitan resolves to V, and in the classical repertoire it very often passes first through a cadential six-four, which puts the dominant in the bass a beat early and delays the leading tone.\nTwo of the chords on the list do not behave functionally in the traditional sense, and that is their value. The flat VII is a subtonic rather than a leading-tone chord, so it can move directly to the tonic without any dominant preparation at all, which is the motion behind an enormous amount of rock and folk repertoire. The flat VI can precede the flat VII and produce the same arrival in two steps: A flat, B flat, C. Both progressions are diatonic in the Aeolian mode, which raises a question of labelling worth settling before the analysis gets carried away.\nAb3+C4+Eb4 Bb3+D4+F4 C4+E4+G4\nIf the whole passage is Aeolian, with a natural minor scale and no raised leading tone, then a flat VII is not borrowed at all. It is simply the dominant of the prevailing mode. Mixture presupposes a major or minor tonal centre with a functioning dominant, and a chord that contradicts that assumption is evidence that the centre is not what the analyst assumed. Deciding which of the two is happening is a matter of the cadence: a piece that ends V–i in the minor mode and also uses flat VII in passing is mixing modes, while a piece that ends flat VII–I throughout is in a mode.\n#The augmented sixth family\nThe augmented sixth chords are the most chromatic members of the borrowed set, and they are grouped by the interval rather than by the chord. All three are built on the lowered sixth degree, which sits in the bass, and all three contain an augmented sixth above that bass note. In C minor the bass is A flat and the augmented sixth is F sharp.\nNameNotes in C minorIntervals above A flatResolutionItalian sixthA flat–C–F sharpmajor third, augmented sixththe sixth expands to the octave on GFrench sixthA flat–C–D–F sharpmajor third, augmented fourth, augmented sixthsame, with the fourth resolving to the fifth of VGerman sixthA flat–C–E flat–F sharpmajor third, perfect fifth, augmented sixthsame, with the fifth held into the cadential six-four\nThe gap between A flat and F sharp is the point. The interval is an augmented sixth: ten semitones, one more than a major sixth, which is why the two notes are under pressure to move outward to the octave, and in practice they do: A flat descends to G and F sharp ascends to G. The result is a dominant, usually reached through a cadential six-four. In a major key the chord requires the borrowed lowered sixth degree, which is why the augmented sixth belongs in a discussion of mixture rather than in a chapter of its own.\nThe German sixth has a second identity. Respell F sharp as G flat and the chord A flat, C, E flat, G flat is an A flat dominant seventh, which resolves not to C but to D flat. The same four pitches are a pre-dominant in one key and a dominant in another, and later composers used that overlap deliberately. The modulation article follows that thread; here it is enough to note that a chord can belong to two keys at once and that the notation is what tells the player which one is intended.\nAb3+C4+Eb4+F#4 G3+C4+Eb4 G3+B3+D4 C4+Eb4+G4\n#Where the labelling goes wrong\nThree errors recur often enough to be worth naming.\nThe first is calling every chromatic chord mixture. A chromatic passing tone between two diatonic notes is not mixture, and neither is a secondary dominant, whose raised third points at a different tonic. The test is the third of the chord. In C major, an E major triad contains G sharp and behaves as V/vi. An E flat major triad contains the borrowed lowered third degree and is mixture. The two chords differ by one scale degree and by their entire function.\nThe second is labelling mixture in a modal context. A song that stays on a flat VII chord and never raises the seventh degree is not borrowing from anywhere. Retrospective labels that call the modal chords of a rock song “borrowed from the parallel minor” describe the surface and miss the structure.\nThe third is stacking borrowed chords faster than they can resolve. Mixture is a colouring device, and a passage in four parts in which every chord carries a lowered degree gives the ear no diatonic frame to measure against. Two borrowed chords in a phrase are usually enough for the effect they produce; four in a row tend to sound like a mistake in the parts.\n#Using mixture deliberately\nCompetent use of mixture is mostly a question of placement.\nPut a borrowed iv where a diatonic IV would go and the phrase turns inward for a beat, then resolves as usual. Put a flat VI before the dominant and the arrival is delayed and darkened. Put a flat VII directly before the tonic and the cadence becomes modal and open, which is why it appears so often at the end of a chorus. The Neapolitan earns its keep in minor keys, where it supplies the strongest pre-dominant available, and in major keys where a sudden flat II in first inversion announces a serious moment.\nThe practical discipline is to check the resolutions. Every lowered sixth degree wants to descend to the dominant, and every raised fourth degree wants to ascend to it. When a borrowed chord arrives and neither of those happens, the ear hears the chromaticism as unexplained, and the next phrase has to work harder than it should. Play the chord, resolve it as the table suggests, and the difference between a good borrow and an accidental one is audible in a single hearing.\nThe virtual piano is enough to test that difference: hold a plain IV, then an iv, and listen to which one makes the following dominant feel inevitable.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "modal mixture", "borrowed chords", "chromatic harmony" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/borrowed-chords-and-modal-mixture.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/cadences-and-phrase-endings/", "url": "https://musics.name/articles/cadences-and-phrase-endings/", "title": "Cadences: How Phrases End in Tonal Music", "summary": "Authentic, half, plagal and deceptive cadences, and the real reason a close sounds strong: what happens in the bass, the top voice and the metre.", "content_html": "Music has punctuation, and it is made of harmony. A phrase ends when the harmony, the bass, the metre and the melody all agree that something has finished. That agreement is a cadence, and its strength is not a matter of taste: it comes from four or five identifiable conditions, each of which can be weakened or strengthened on purpose.
\nAuthentic is the arrival everybody knows: a dominant chord resolving to the tonic. When both chords are in root position and the soprano lands on the tonic, it is a perfect authentic cadence, and it is the strongest close tonal music has. Change either condition, inverting a chord or ending with scale degree 3 or 5 on top, and it becomes an imperfect authentic cadence: the same V–I, less final.
\nHalf ends on the dominant. Nothing resolves; the phrase simply stops asking. In minor there is a distinctive version, the Phrygian half cadence, where a first-inversion subdominant moves to the dominant while the bass descends a semitone from scale degree 6 to 5. In A minor that is a D minor triad over F moving to E major, and the semitone in the bass gives the pause its characteristic colour.
\nPlagal is IV to I, the “Amen” motion. It confirms a key more than it closes a phrase, which is why it appears so often after an authentic cadence rather than instead of one. In minor the same motion with the lowered sixth scale degree, iv to i, is darker and equally common.
\nDeceptive replaces the expected tonic with another chord, usually vi, or ♭VI in minor. The dominant has done its work and the arrival is withheld. Deceptive cadences almost always extend a phrase rather than end it.
\n| Cadence | Formula in major | Effect |
|---|---|---|
| Perfect authentic | V(7)–I, root position, tonic in the soprano | full close, ends the phrase |
| Imperfect authentic | V–I with an inverted chord or 3/5 on top | close, weaker |
| Half | ends on V, commonly preceded by I or IV | pause, keeps the phrase open |
| Phrygian half | iv6–V in minor, bass 6 falling to 5 | pause with a semitone pull |
| Plagal | IV–I | confirmation rather than closure |
| Deceptive | V–vi | delayed arrival, phrase continues |
Here is a perfect authentic cadence, then a deceptive one, with the same dominant preparing both.
\n\n\nThe chord names matter less than the conditions around them.
\nThat last point is where the cadence connects to the rest of tonal syntax. The dominant does the closing, but the predominant does the travelling, and a phrase whose dominant arrives without preparation tends to sound abrupt. The reverse error is just as common in student writing: a heavily prepared predominant followed by a cadence so soft that the phrase does not feel finished.
\nA tonic triad in second inversion sitting on scale degree 5 in the bass is one of the most common things in tonal music and one of the hardest to label. Its notes belong to the tonic; its bass and its behaviour belong to the dominant.
\nTwo analyses compete. One treats the six-four as a tonic chord with two notes held over as suspensions, which resolve into the dominant. The other treats the whole figure as an expanded dominant, with scale degree 5 in the bass and the sixth and fourth above it heard as accented dissonances. The disagreement is real and does not affect what you do with the chord: it cannot function as a point of rest, and it must resolve to V. Its practical value is that it puts dominant weight at the end of a phrase while the upper voices still sound like the tonic triad, which is why it so often carries the melody’s final descent.
\nNot every pause is a cadence, and not every cadence stops the music. A useful test is to ask what the bass is doing and whether a conventional pair of chords is present. A long tonic chord in the middle of a phrase is a resting point; a dominant at the end of a phrase is a half cadence, whether or not it is followed by a bar of silence.
\nPhrase structure follows from this. An antecedent phrase typically ends on a half cadence and a consequent answers it with an authentic one, which is the smallest complete musical sentence the tonal repertoire uses. Phrases can also overlap, where the resolution of one phrase is the beginning of the next, and they can be extended by a deceptive cadence or by a cadential six-four that is repeated. When you analyse a longer piece, the cadences are the load-bearing walls: mark them first, and the phrase boundaries, the keys and the sections mostly fall into place. Sonata form is the clearest example, since its large sections are defined by which cadences they avoid and which they reach.
\n\nThat is the smallest open-ended phrase there is: a tonic followed by a half cadence, with the bass stepping up to scale degree 5 and nothing resolving.
\nPre-tonal music cadences by different means. In modal polyphony the arrival is made by two voices moving in contrary motion into an octave or a fifth, with the leading tone often supplied by an editorial accidental rather than by the mode. The effect is a cadence, but the mechanism is intervallic and contrapuntal rather than harmonic.
\nPopular and blues-based music often avoids cadential closure entirely. A twelve-bar blues turns around through the dominant and starts again rather than resolving; a modal rock progression can loop indefinitely with no chord asked to function as a dominant. In jazz, the ii–V–I does the work of a cadence, and the phrase ends where the tonic arrives, even when the tonic chord is a seventh chord that a common-practice writer would hear as unstable.
\nThat last case is the useful warning: cadence is a behaviour, not a chord symbol. A seventh chord on the tonic can close a jazz phrase because the style treats it as home. The same chord in a Bach chorale would sound like a chord on the way somewhere else.
\nTake a piece of tonal repertoire and mark only the cadences, with no other analysis. Then label each one by type and ask what would have happened if the composer had chosen the other option: authentic instead of deceptive, half instead of authentic. Most of the interesting harmonic decisions in tonal music are visible from this angle, and the ear training studio lets you hear authentic, plagal and deceptive cadences back to back, which is how the difference stops being a definition and becomes a sound you recognise before you analyse it.
", "content_text": "Music has punctuation, and it is made of harmony. A phrase ends when the harmony, the bass, the metre and the melody all agree that something has finished. That agreement is a cadence, and its strength is not a matter of taste: it comes from four or five identifiable conditions, each of which can be weakened or strengthened on purpose.\n#The four cadences you meet first\nAuthentic is the arrival everybody knows: a dominant chord resolving to the tonic. When both chords are in root position and the soprano lands on the tonic, it is a perfect authentic cadence, and it is the strongest close tonal music has. Change either condition, inverting a chord or ending with scale degree 3 or 5 on top, and it becomes an imperfect authentic cadence: the same V–I, less final.\nHalf ends on the dominant. Nothing resolves; the phrase simply stops asking. In minor there is a distinctive version, the Phrygian half cadence, where a first-inversion subdominant moves to the dominant while the bass descends a semitone from scale degree 6 to 5. In A minor that is a D minor triad over F moving to E major, and the semitone in the bass gives the pause its characteristic colour.\nPlagal is IV to I, the “Amen” motion. It confirms a key more than it closes a phrase, which is why it appears so often after an authentic cadence rather than instead of one. In minor the same motion with the lowered sixth scale degree, iv to i, is darker and equally common.\nDeceptive replaces the expected tonic with another chord, usually vi, or ♭VI in minor. The dominant has done its work and the arrival is withheld. Deceptive cadences almost always extend a phrase rather than end it.\nCadenceFormula in majorEffectPerfect authenticV(7)–I, root position, tonic in the sopranofull close, ends the phraseImperfect authenticV–I with an inverted chord or 3/5 on topclose, weakerHalfends on V, commonly preceded by I or IVpause, keeps the phrase openPhrygian halfiv6–V in minor, bass 6 falling to 5pause with a semitone pullPlagalIV–Iconfirmation rather than closureDeceptiveV–videlayed arrival, phrase continues\nHere is a perfect authentic cadence, then a deceptive one, with the same dominant preparing both.\nG3+B3+D4+B4 C4+E4+G4+C5\nG3+B3+D4+B4 A3+C4+E4+C5\n#What actually makes a close strong\nThe chord names matter less than the conditions around them.\nThe bass. A cadence needs the bass to move 5 to 1. Root-position dominant and tonic give that motion; inversions blur it.The top voice. The tonic in the soprano is what makes the arrival audible as an arrival rather than a continuation. A close with scale degree 3 on top sounds like an arrival facing sideways.The metre. A cadence that lands on a strong beat closes harder than the same chords landing off the beat, because the metric accent and the harmonic resolution agree.Directness. V7 to I with nothing in between presses harder than a resolution interrupted by a passing chord.Duration. A final chord held longer than the rhythm of the phrase establishes closure; a tonic that flashes past sounds like a passing chord.Preparation. A predominant in front of the dominant, especially a first-inversion ii or the cadential six-four, makes the dominant feel earned.\nThat last point is where the cadence connects to the rest of tonal syntax. The dominant does the closing, but the predominant does the travelling, and a phrase whose dominant arrives without preparation tends to sound abrupt. The reverse error is just as common in student writing: a heavily prepared predominant followed by a cadence so soft that the phrase does not feel finished.\n#The cadential six-four\nA tonic triad in second inversion sitting on scale degree 5 in the bass is one of the most common things in tonal music and one of the hardest to label. Its notes belong to the tonic; its bass and its behaviour belong to the dominant.\nTwo analyses compete. One treats the six-four as a tonic chord with two notes held over as suspensions, which resolve into the dominant. The other treats the whole figure as an expanded dominant, with scale degree 5 in the bass and the sixth and fourth above it heard as accented dissonances. The disagreement is real and does not affect what you do with the chord: it cannot function as a point of rest, and it must resolve to V. Its practical value is that it puts dominant weight at the end of a phrase while the upper voices still sound like the tonic triad, which is why it so often carries the melody’s final descent.\n#Telling a cadence from a resting point\nNot every pause is a cadence, and not every cadence stops the music. A useful test is to ask what the bass is doing and whether a conventional pair of chords is present. A long tonic chord in the middle of a phrase is a resting point; a dominant at the end of a phrase is a half cadence, whether or not it is followed by a bar of silence.\nPhrase structure follows from this. An antecedent phrase typically ends on a half cadence and a consequent answers it with an authentic one, which is the smallest complete musical sentence the tonal repertoire uses. Phrases can also overlap, where the resolution of one phrase is the beginning of the next, and they can be extended by a deceptive cadence or by a cadential six-four that is repeated. When you analyse a longer piece, the cadences are the load-bearing walls: mark them first, and the phrase boundaries, the keys and the sections mostly fall into place. Sonata form is the clearest example, since its large sections are defined by which cadences they avoid and which they reach.\nC4+E4+G4 G3+B3+D4\nThat is the smallest open-ended phrase there is: a tonic followed by a half cadence, with the bass stepping up to scale degree 5 and nothing resolving.\n#Where cadences are not the point\nPre-tonal music cadences by different means. In modal polyphony the arrival is made by two voices moving in contrary motion into an octave or a fifth, with the leading tone often supplied by an editorial accidental rather than by the mode. The effect is a cadence, but the mechanism is intervallic and contrapuntal rather than harmonic.\nPopular and blues-based music often avoids cadential closure entirely. A twelve-bar blues turns around through the dominant and starts again rather than resolving; a modal rock progression can loop indefinitely with no chord asked to function as a dominant. In jazz, the ii–V–I does the work of a cadence, and the phrase ends where the tonic arrives, even when the tonic chord is a seventh chord that a common-practice writer would hear as unstable.\nThat last case is the useful warning: cadence is a behaviour, not a chord symbol. A seventh chord on the tonic can close a jazz phrase because the style treats it as home. The same chord in a Bach chorale would sound like a chord on the way somewhere else.\n#Practising with cadences\nTake a piece of tonal repertoire and mark only the cadences, with no other analysis. Then label each one by type and ask what would have happened if the composer had chosen the other option: authentic instead of deceptive, half instead of authentic. Most of the interesting harmonic decisions in tonal music are visible from this angle, and the ear training studio lets you hear authentic, plagal and deceptive cadences back to back, which is how the difference stops being a definition and becomes a sound you recognise before you analyse it.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "cadence", "phrase structure", "tonality" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/cadences-and-phrase-endings.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/clefs-and-transposition/", "url": "https://musics.name/articles/clefs-and-transposition/", "title": "Clefs and Transposition: What Players Actually Read", "summary": "A clef fixes one pitch and the staff follows. Transposing instruments move it again, so the same written note reaches the audience at different pitches.", "content_html": "A staff is a grid without absolute meaning. A clef gives one line or space a fixed pitch, and every other position follows from it by step. That single decision determines what a player reads, and two further conventions determine what the audience hears: clefs that displace an octave, and instruments whose written pitch is not their sounding pitch.
\nThree clef families cover almost all modern notation.
\nThe G clef, the treble clef of everyday use, curls around the line that is G4, which is the second line from the bottom of the staff. Positions above and below are read from there.
\nThe F clef places its two dots on either side of the line that is F3, the fourth line from the bottom in the bass clef.
\nThe C clef marks whichever line it sits on as middle C, meaning C4. The same clef shape produces different staves depending on its placement: on the bottom line it is the soprano clef, on the second line the mezzo-soprano clef, on the third line the alto clef, on the fourth line the tenor clef, and on the fifth line the baritone clef.
\nAll of these clefs are movable in principle, and the G clef placed on the bottom line gives the French violin clef. The historically unusual placements survive mostly in facsimiles and older editions, but the principle matters more than the inventory: a clef names positions, not staves. A note is whatever the clef says it is, on whatever staff it is printed.
\nThe viola reads alto clef, and it is the only instrument in the standard orchestra that does so as a matter of course. The middle line is middle C, so the viola’s central register sits in the middle of the staff instead of hanging off the bottom of a treble staff with ledger lines.
\nCello, bassoon and trombone use the tenor clef for high passages. A cello line that would run four ledger lines above the bass staff, awkward to read at speed, fits comfortably in tenor clef, where the fourth line is middle C. The instrument switched clef, not register; the pitches are the same.
\nReading these clefs has one reliable method: locate middle C and read intervals from it. In alto clef it is the middle line, in tenor clef the fourth line, and once that anchor is set, every other position is a third, a fifth or a step away. The alternative method, converting from a clef you already know, works but is arithmetic and therefore slow: the same staff position reads as a perfect fifth lower in bass clef than it does in tenor clef, so a cellist using that shortcut is transposing by a fifth on every note. Players who read these clefs daily do not convert at all; they read the anchor and the interval.
\nSome notation moves pitches by an octave without changing the letter names or the clef shape. The convention is marked by a small 8 below or above the clef, though it is printed inconsistently.
\nThe practical rule for a reader is to distrust an unmarked treble staff in an orchestral part until the instrument is identified, and to distrust an unmarked bass staff in the same way. The composer’s octave displacement may appear as a printed 8, as a footnote in the instrumentation list, or nowhere.
\nAn instrument’s name states the pitch that sounds when the player reads and plays a written C. That one sentence generates every case.
\nWhy the convention exists is easiest to see in the instrument families. Clarinets are built in several sizes, saxophones in several sizes, and the notation is transposed so that the same written note corresponds to the same fingering across the family. A saxophonist moving from alto to tenor keeps the same written scale patterns and only has to adjust tone and register, because the written notes already encode the fingering. The historical reason in the horn section is different: natural horns changed key by adding crooks, and players read notation in C and transposed at sight depending on which crook was fitted, which is why horn players still develop the skill and why horn transposition appears in several keys in older repertoire.
\nTwo consequences follow for anyone writing or reading a score. First, the key signature is transposed with the part. A work in concert D major, two sharps, appears with four sharps for a B♭ instrument, because the written key is a major second above the sounding key. Second, the octave displacement is separate from the key transposition and has to be checked on its own. The B♭ tenor saxophone sounds a major ninth below written pitch, so a written C4 emerges as B♭2, while the B♭ clarinet’s written C4 emerges as B♭3. The letter name of the transposition is the same; the octave is not.
\n| Instrument | Written clef | Written C4 sounds | Interval |
|---|---|---|---|
| Piccolo | treble | C5 | octave higher |
| Flute | treble | C4 | as written |
| Oboe | treble | C4 | as written |
| B♭ clarinet | treble | B♭3 | major second lower |
| A clarinet | treble | A3 | minor third lower |
| E♭ alto saxophone | treble | E♭3 | major sixth lower |
| B♭ tenor saxophone | treble | B♭2 | major ninth lower |
| F horn | treble and bass | F3 | perfect fifth lower |
| B♭ trumpet | treble | B♭3 | major second lower |
| Viola | alto | C4 | as written |
| Cello | bass and tenor | C4 | as written |
| Double bass | bass | C3 | octave lower |
| Guitar | treble | C3 | octave lower |
Written pitches are what the player sees and fingers; sounding pitches are what the audience hears. Both are needed, and a range chart that omits one of them is unusable.
\nFull scores from most publishers are transposed, which means the page shows what each player reads rather than what sounds. A conductor studying a transposed score without accounting for the transpositions will hear intervals that do not match the page, and a harmonic analysis done straight from the page will be wrong in every key except C.
\nThree habits keep a mixed score readable. Identify each instrument’s transposition once, at the top of the page, ideally as a pencilled note that states the sounding pitch of that instrument’s middle C. Work from a concert-pitch score for analysis and keep the transposed score for rehearsal. And when checking a chord across a section, convert outward from the bass rather than upward from the top, since the bass instruments are usually non-transposing and give a reliable anchor.
\nA piano cannot retune between two chords, so its tuning has to work for every key in advance. Equal temperament is the arrangement that does that, and it earns the ability by making every fifth slightly narrow and every major third noticeably wide. The compromise is easy to state and worth understanding in numbers, because the numbers explain what a keyboard sounds like and why ensembles that can adjust pitch do something else.
\nEqual temperament divides the octave into twelve steps of identical size. A step is the frequency ratio 2 to the twelfth root of two, about 1.0595, and the system is usually described in cents, where an octave is 1200 and a semitone is therefore exactly 100.
\nThe arithmetic is compact enough to do on the back of an envelope. If A4 is 440 Hz, the note a semitone above it sits at 440 × 1.0595, about 466.2 Hz, and three semitones above A4, which is C5, sits at 440 × 2^(3/12), about 523.3 Hz. An octave above A4 is 880 Hz, which is exactly twice the frequency and exactly 1200 cents, since cents are defined by 1200 × log₂ of the frequency ratio.
\nTwo properties follow from the equality of the steps. Transposition is exact: the same fingering, fret or valve combination produces the same interval in every key. And enharmonic pairs collapse, so F sharp and G flat are the same key on a keyboard and the same fret on a guitar, which means a piece can move through remote keys without the instrument changing shape underneath the player.
\nThe reason a compromise is needed at all is that pure intervals do not add up.
\nTake the fifth, the ratio 3:2. Stack twelve of them and the result is 129.746 times the starting frequency. Seven octaves is 128 times. Twelve pure fifths overshoot seven octaves by the ratio 531441 to 524288, roughly 1.0136, which is about 23.5 cents. That discrepancy is the Pythagorean comma, and it means a keyboard tuned in pure fifths cannot close its own circle: one fifth has to absorb the error, and that fifth is the one players call the wolf.
\nThe second problem is inside the third. Four stacked pure fifths produce a major third of 81:64, which measures about 407.8 cents. The pure major third, the one the ear hears as a 5:4, measures about 386.3 cents. The gap between them is the syntonic comma, roughly 21.5 cents. Pure fifths and pure thirds cannot both be had, because the fifths generate thirds that are too wide by exactly that amount. Tuning a chain of pure fifths therefore produces screaming thirds, and tuning pure thirds requires fifths that are narrowed. The temperaments built on that trade-off before equal temperament are the subject of meantone and well temperament.
\nEqual temperament resolves both problems by refusing purity everywhere. Each fifth is narrowed by about two cents, which is enough to make twelve of them fit seven octaves exactly, and the accumulated slack is distributed across all twelve fifths instead of being dumped into one.
\nThe errors are not equally audible, and the table shows where the cost is concentrated. The just column gives the pure five-limit value, the size the interval has when it is built from small whole-number ratios.
\n| Interval | Just ratio | Just size | Equal-tempered size | Error |
|---|---|---|---|---|
| Octave | 2:1 | 1200.0 cents | 1200 cents | 0 |
| Fifth | 3:2 | 701.96 cents | 700 cents | −1.96 |
| Fourth | 4:3 | 498.04 cents | 500 cents | +1.96 |
| Major third | 5:4 | 386.31 cents | 400 cents | +13.69 |
| Minor third | 6:5 | 315.64 cents | 300 cents | −15.64 |
| Major sixth | 5:3 | 884.36 cents | 900 cents | +15.64 |
| Minor sixth | 8:5 | 813.69 cents | 800 cents | −13.69 |
| Major second | 9:8 | 203.91 cents | 200 cents | −3.91 |
| Minor seventh | 9:5 | 1017.60 cents | 1000 cents | −17.60 |
The fifth’s error is small, which is why a string quartet can tune its open strings in fifths and still play with a piano without much trouble. The third’s error is seven times larger and in the opposite direction, and it has an audible consequence: the partials that a pure third would align are left slightly apart, and they beat. Near middle C the numbers are easy to check. C4 at 261.6 Hz has its fifth partial at 1308.1 Hz, and E4 at 329.6 Hz has its fourth partial at 1318.5 Hz, so the two partials sit about 10 Hz apart and produce roughly ten beats per second. That is too fast to hear as a wobble and slow enough to be heard as a roughness, which is part of why an equal-tempered piano chord has a shimmer that a just-tuned chord does not.
\nSemitones lose their two sizes as well. In just intonation a diatonic semitone such as C sharp to D is 16:15, about 111.7 cents, and a chromatic semitone such as C to C sharp is 25:24, about 70.7 cents. Equal temperament gives both 100, so the small step that makes a leading tone lean and the step that merely colours a note become the same distance.
\n\nThe octave, the one interval equal temperament gets exactly right, and the only one where the just and tempered sizes are identical.
\n\n\nAn equal-tempered major third and an equal-tempered fifth. The fifth is two cents away from pure, which is why it sounds settled; the third is about fourteen cents wide, which is why it carries the beating.
\n\nTwo equal steps of 100 cents. In just intonation the first of these steps, the chromatic semitone, is about 71 cents and the second, the diatonic semitone, about 112, so the two moves are audibly different distances rather than the same one repeated.
\nFour things, and they are the reasons the compromise won.
\nThe first is that fixed-pitch instruments work in every key. A keyboard, a fretted instrument, a xylophone and a set of tuned percussion have no way to adjust individual notes while playing, so any tuning that privileges some keys makes the others unusable. Equal temperament privileges none and therefore permits all of them.
\nThe second is exact transposition. Every interval has the same size in every key, so a shape learned in C major transfers unchanged to F sharp major. For players of instruments with a fixed layout this is not a convenience but the basis of technique.
\nThe third is enharmonic equivalence. Treating F sharp and G flat as one pitch makes chromatic harmony navigable and makes remote modulations cheap, since the new key does not need new notes, only different ones. Tonal music after roughly 1800 exploits this constantly, and much of the harmonic freedom of the nineteenth century depends on an instrument on which every key is equally available.
\nThe fourth is that the compromise is uniform. A meantone tuning with pure thirds leaves one fifth so wide that the keys using it are avoided or retuned, while equal temperament spreads the same total error thinly enough that no interval is unusable. What the system gives up is the differentiation between keys: when every key is tuned identically, no key has a distinct character, and the historical claims about the sound of particular keys lose their acoustic footing.
\nThe piano is equal-tempered, and most of the music played by flexible-pitch instruments is not. String players tune their open strings in fifths and then adjust continuously, playing leading tones high and thirds narrow when the texture allows. Wind players lip notes into place, and brass players work with an instrument whose natural resonance is the harmonic series, which is a just system rather than a tempered one. Singers in an unaccompanied ensemble drift towards pure intervals, and one consequence is measurable in performance: an ensemble that tunes pure thirds and pure fifths does not return to the same pitch after a long passage, because the intervals do not add up to the arithmetic of the octave.
\nWhere a piano is present it becomes the reference, and the flexible instruments move to it, which is why a string quartet with piano sounds subtly different from the same quartet alone. The tuning a listener hears in a concert hall is therefore a negotiation, and the equal-tempered grid is one participant in it rather than the definition of being in tune. The practical use of the cents figures above is diagnostic: a tuner reading of two cents sharp is inside the noise of an ensemble, and a reading of fifteen cents sharp on a third is the difference between a tempered interval and a pure one, which players can hear even when they cannot name it.
", "content_text": "A piano cannot retune between two chords, so its tuning has to work for every key in advance. Equal temperament is the arrangement that does that, and it earns the ability by making every fifth slightly narrow and every major third noticeably wide. The compromise is easy to state and worth understanding in numbers, because the numbers explain what a keyboard sounds like and why ensembles that can adjust pitch do something else.\n#Twelve equal steps\nEqual temperament divides the octave into twelve steps of identical size. A step is the frequency ratio 2 to the twelfth root of two, about 1.0595, and the system is usually described in cents, where an octave is 1200 and a semitone is therefore exactly 100.\nThe arithmetic is compact enough to do on the back of an envelope. If A4 is 440 Hz, the note a semitone above it sits at 440 × 1.0595, about 466.2 Hz, and three semitones above A4, which is C5, sits at 440 × 2^(3/12), about 523.3 Hz. An octave above A4 is 880 Hz, which is exactly twice the frequency and exactly 1200 cents, since cents are defined by 1200 × log₂ of the frequency ratio.\nTwo properties follow from the equality of the steps. Transposition is exact: the same fingering, fret or valve combination produces the same interval in every key. And enharmonic pairs collapse, so F sharp and G flat are the same key on a keyboard and the same fret on a guitar, which means a piece can move through remote keys without the instrument changing shape underneath the player.\n#Two commas, one problem\nThe reason a compromise is needed at all is that pure intervals do not add up.\nTake the fifth, the ratio 3:2. Stack twelve of them and the result is 129.746 times the starting frequency. Seven octaves is 128 times. Twelve pure fifths overshoot seven octaves by the ratio 531441 to 524288, roughly 1.0136, which is about 23.5 cents. That discrepancy is the Pythagorean comma, and it means a keyboard tuned in pure fifths cannot close its own circle: one fifth has to absorb the error, and that fifth is the one players call the wolf.\nThe second problem is inside the third. Four stacked pure fifths produce a major third of 81:64, which measures about 407.8 cents. The pure major third, the one the ear hears as a 5:4, measures about 386.3 cents. The gap between them is the syntonic comma, roughly 21.5 cents. Pure fifths and pure thirds cannot both be had, because the fifths generate thirds that are too wide by exactly that amount. Tuning a chain of pure fifths therefore produces screaming thirds, and tuning pure thirds requires fifths that are narrowed. The temperaments built on that trade-off before equal temperament are the subject of meantone and well temperament.\nEqual temperament resolves both problems by refusing purity everywhere. Each fifth is narrowed by about two cents, which is enough to make twelve of them fit seven octaves exactly, and the accumulated slack is distributed across all twelve fifths instead of being dumped into one.\n#What the tuning distorts\nThe errors are not equally audible, and the table shows where the cost is concentrated. The just column gives the pure five-limit value, the size the interval has when it is built from small whole-number ratios.\nIntervalJust ratioJust sizeEqual-tempered sizeErrorOctave2:11200.0 cents1200 cents0Fifth3:2701.96 cents700 cents−1.96Fourth4:3498.04 cents500 cents+1.96Major third5:4386.31 cents400 cents+13.69Minor third6:5315.64 cents300 cents−15.64Major sixth5:3884.36 cents900 cents+15.64Minor sixth8:5813.69 cents800 cents−13.69Major second9:8203.91 cents200 cents−3.91Minor seventh9:51017.60 cents1000 cents−17.60\nThe fifth’s error is small, which is why a string quartet can tune its open strings in fifths and still play with a piano without much trouble. The third’s error is seven times larger and in the opposite direction, and it has an audible consequence: the partials that a pure third would align are left slightly apart, and they beat. Near middle C the numbers are easy to check. C4 at 261.6 Hz has its fifth partial at 1308.1 Hz, and E4 at 329.6 Hz has its fourth partial at 1318.5 Hz, so the two partials sit about 10 Hz apart and produce roughly ten beats per second. That is too fast to hear as a wobble and slow enough to be heard as a roughness, which is part of why an equal-tempered piano chord has a shimmer that a just-tuned chord does not.\nSemitones lose their two sizes as well. In just intonation a diatonic semitone such as C sharp to D is 16:15, about 111.7 cents, and a chromatic semitone such as C to C sharp is 25:24, about 70.7 cents. Equal temperament gives both 100, so the small step that makes a leading tone lean and the step that merely colours a note become the same distance.\nA4 A5\nThe octave, the one interval equal temperament gets exactly right, and the only one where the just and tempered sizes are identical.\nC4 E4\nC4 G4\nAn equal-tempered major third and an equal-tempered fifth. The fifth is two cents away from pure, which is why it sounds settled; the third is about fourteen cents wide, which is why it carries the beating.\nC4 C#4 D4\nTwo equal steps of 100 cents. In just intonation the first of these steps, the chromatic semitone, is about 71 cents and the second, the diatonic semitone, about 112, so the two moves are audibly different distances rather than the same one repeated.\n#What it buys\nFour things, and they are the reasons the compromise won.\nThe first is that fixed-pitch instruments work in every key. A keyboard, a fretted instrument, a xylophone and a set of tuned percussion have no way to adjust individual notes while playing, so any tuning that privileges some keys makes the others unusable. Equal temperament privileges none and therefore permits all of them.\nThe second is exact transposition. Every interval has the same size in every key, so a shape learned in C major transfers unchanged to F sharp major. For players of instruments with a fixed layout this is not a convenience but the basis of technique.\nThe third is enharmonic equivalence. Treating F sharp and G flat as one pitch makes chromatic harmony navigable and makes remote modulations cheap, since the new key does not need new notes, only different ones. Tonal music after roughly 1800 exploits this constantly, and much of the harmonic freedom of the nineteenth century depends on an instrument on which every key is equally available.\nThe fourth is that the compromise is uniform. A meantone tuning with pure thirds leaves one fifth so wide that the keys using it are avoided or retuned, while equal temperament spreads the same total error thinly enough that no interval is unusable. What the system gives up is the differentiation between keys: when every key is tuned identically, no key has a distinct character, and the historical claims about the sound of particular keys lose their acoustic footing.\n#Why ensembles still play unequal\nThe piano is equal-tempered, and most of the music played by flexible-pitch instruments is not. String players tune their open strings in fifths and then adjust continuously, playing leading tones high and thirds narrow when the texture allows. Wind players lip notes into place, and brass players work with an instrument whose natural resonance is the harmonic series, which is a just system rather than a tempered one. Singers in an unaccompanied ensemble drift towards pure intervals, and one consequence is measurable in performance: an ensemble that tunes pure thirds and pure fifths does not return to the same pitch after a long passage, because the intervals do not add up to the arithmetic of the octave.\nWhere a piano is present it becomes the reference, and the flexible instruments move to it, which is why a string quartet with piano sounds subtly different from the same quartet alone. The tuning a listener hears in a concert hall is therefore a negotiation, and the equal-tempered grid is one participant in it rather than the definition of being in tune. The practical use of the cents figures above is diagnostic: a tuner reading of two cents sharp is inside the noise of an ensemble, and a reading of fifteen cents sharp on a third is the difference between a tempered interval and a pure one, which players can hear even when they cannot name it.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "tuning", "temperament", "intervals" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/equal-temperament.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/figured-bass/", "url": "https://musics.name/articles/figured-bass/", "title": "Figured Bass: Reading Intervals Above the Bass Line", "summary": "Figures name intervals above the bass, not chords. What each shorthand means, how accidentals override the key, and what the player decides.", "content_html": "A figured bass is a bass line with numerals underneath it, and the numerals are not chord symbols. Each one names an interval above the note it sits under, counted from that note. The player supplies the rest. Understanding what is written and what is deliberately left unwritten is most of the skill.
\nA 6 under a bass note means that a sixth above that note sounds in the chord. Put the figure under E in C major and the sixth above E is C, so the chord is E, G, C — C major in first inversion. Put a 6/4 under G in the same key and the intervals above the bass are E and C, so the chord is G, C, E — the same C major triad, this time in second inversion. The figure follows the interval above the bass, and the bass note decides which inversion the ear hears.
\nThat relativity is the point of the notation. A chord symbol names an entire simultaneity, including its root and its quality. A figure names one or two intervals above a given note, and leaves the quality, the root and the doubling to be inferred from the key and the idiom. What the notation records with great precision is inversion, which is exactly the information a chord symbol struggles to express.
\n\nBoth figures describe the same triad in different inversions. A bare 6 marks C major over E, its first inversion; a 6/4 marks the same chord over G, its second inversion. The note named by the 6 is C in the first case and E in the second, because the numeral always counts upwards from whatever the bass is doing.
\nThe default is a root-position triad, intervals of a fifth and a third above the bass, and it is left unfigured. Figures appear only where the intervals differ from that default, which is why a continuo part has long stretches of empty space under the staff.
\n| Figure | Intervals above the bass | Chord | Example in C major |
|---|---|---|---|
| 5/3, often unfigured | fifth, third | root-position triad | C: C–E–G |
| 6 or 6/3 | sixth, third | first-inversion triad | C/E: E–G–C |
| 6/4 | sixth, fourth | second-inversion triad | C/G: G–C–E |
| 7 | seventh, fifth, third | root-position seventh chord | G7: G–B–D–F |
| 6/5 | sixth, fifth, third | first-inversion seventh chord | G7/B: B–D–F–G |
| 4/3 | fourth, third, sixth implied | second-inversion seventh chord | G7/D: D–F–G–B |
| 4/2 or 2 | fourth, second, sixth implied | third-inversion seventh chord | G7/F: F–G–B–D |
The abbreviations are conventions rather than abbreviations of arithmetic. A bare 6 means 6/3 with the third taken for granted. A bare 2 means 4/2. A bare 4 in certain cadential contexts is understood as 6/4. The player who reads numerals literally, as instructions to place a second above the bass and nothing else, will produce four-note chords where two were intended and thin chords where a full triad was wanted.
\nA sharp, flat or natural attached to a numeral alters that interval above the bass, and it does so regardless of the key signature. The figure 6 with a sharp under a bass note in C major raises the sixth above that note, which is how a continuo part asks for a chromatic note the key does not contain. An accidental with no numeral applies to the third, so a lone sharp alone means a raised third.
\nTwo subtleties follow. The first is that the figures override the key signature rather than adding to it, so a natural sign can be as informative as a sharp: it cancels an accidental that the key signature would otherwise supply. The second is that a slash through a sharp or flat raises that note by a further semitone, which is how the notation writes a raised leading tone in minor, a chromatic passing tone, or any other inflection that the key signature does not already supply.
\nWhat the figures never state is register. A 6 above the bass can be placed in any octave, and a realisation in which every sixth sits immediately above the bass will sound cramped in a way the notation did not ask for.
\nEverything about the upper voices that is not an interval is a decision. Spacing, doubling, which voice moves where, whether the chord is played in three parts or four, whether the keyboard doubles the bass at the octave, and how long each chord is held: none of that is written, and all of it is heard.
\nThis is why a realisation is a small act of composition rather than an act of decoding. The figures give the harmonic skeleton, and the player builds the texture. A continuo part played by one hand in the middle of the keyboard and one in the bass produces a very different result from four independent voices, and both are correct readings of the same figures.
\nThree conventions narrow the field. A bass player and a keyboard player divide the labour: the melodic instrument states the line as written, and the keyboard fills in above it, usually keeping the right hand below the melody so that the texture does not double. The marking tasto solo instructs the keyboard player to lay out for a passage and leave the bass line unaccompanied, which is used where the texture needs thinning or where the harmony is deliberately bare. And where upper parts exist, the keyboard avoids doubling the notated melody, since two instruments on the same line at the same octave is a colour, not a texture.
\n\nTake a short progression in C major: bass A, then F, then G, then C, with the figures 5/3, 6, 7 and 5/3.
\n| Bass | Figure | Intervals above | Chord | Function |
|---|---|---|---|---|
| A | unfigured, 5/3 | C, E | A minor | vi |
| F | 6 | A, D | D minor in first inversion | ii6 |
| G | 7 | B, D, F | G dominant seventh | V7 |
| C | unfigured, 5/3 | E, G | C major | I |
The line descends by step from A to F to G, and the harmony is a tonic expansion followed by a cadence. Nothing in the figures is unusual: the bare 6 marks the first inversion, the 7 marks the seventh chord, and the two unfigured notes are the plain triads at the ends. A player would add a third above the bass in the first chord, place the sixth above the F in the second, resolve the F of the seventh chord down to the E of the final triad, and lead the leading tone B up to C. All of that is idiomatic knowledge, and none of it is in the figures.
\nFigured bass survived as a working notation for something like two centuries after it appeared with the basso continuo in the early seventeenth century, and it is still taught for reasons that have little to do with performing old music.
\nIt records inversion and function in the same symbol, which no chord symbol does as clearly. It forces the student to think in intervals above a given note rather than in memorised shapes on an instrument. And it makes voice leading unavoidable, because the player has to decide what the inner voices do while the bass moves, which is exactly the decision that part-writing exercises formalise and that performance demands in real time.
\nThe voice-leading article covers the constraints a realisation has to satisfy. Those constraints are what turned a notation for accompanists into a training method, and they are why a continuo exercise is still one of the fastest ways to find out whether the harmony in a passage has been understood or merely labelled.
\nThe virtual piano is a practical place to work through a realisation at the keyboard: play the bass line with the left hand as written, and add the intervals above it with the right.
", "content_text": "A figured bass is a bass line with numerals underneath it, and the numerals are not chord symbols. Each one names an interval above the note it sits under, counted from that note. The player supplies the rest. Understanding what is written and what is deliberately left unwritten is most of the skill.\n#The figures are intervals, not chords\nA 6 under a bass note means that a sixth above that note sounds in the chord. Put the figure under E in C major and the sixth above E is C, so the chord is E, G, C — C major in first inversion. Put a 6/4 under G in the same key and the intervals above the bass are E and C, so the chord is G, C, E — the same C major triad, this time in second inversion. The figure follows the interval above the bass, and the bass note decides which inversion the ear hears.\nThat relativity is the point of the notation. A chord symbol names an entire simultaneity, including its root and its quality. A figure names one or two intervals above a given note, and leaves the quality, the root and the doubling to be inferred from the key and the idiom. What the notation records with great precision is inversion, which is exactly the information a chord symbol struggles to express.\nE3+G3+C4 G3+C4+E4\nBoth figures describe the same triad in different inversions. A bare 6 marks C major over E, its first inversion; a 6/4 marks the same chord over G, its second inversion. The note named by the 6 is C in the first case and E in the second, because the numeral always counts upwards from whatever the bass is doing.\n#The standard shorthand\nThe default is a root-position triad, intervals of a fifth and a third above the bass, and it is left unfigured. Figures appear only where the intervals differ from that default, which is why a continuo part has long stretches of empty space under the staff.\nFigureIntervals above the bassChordExample in C major5/3, often unfiguredfifth, thirdroot-position triadC: C–E–G6 or 6/3sixth, thirdfirst-inversion triadC/E: E–G–C6/4sixth, fourthsecond-inversion triadC/G: G–C–E7seventh, fifth, thirdroot-position seventh chordG7: G–B–D–F6/5sixth, fifth, thirdfirst-inversion seventh chordG7/B: B–D–F–G4/3fourth, third, sixth impliedsecond-inversion seventh chordG7/D: D–F–G–B4/2 or 2fourth, second, sixth impliedthird-inversion seventh chordG7/F: F–G–B–D\nThe abbreviations are conventions rather than abbreviations of arithmetic. A bare 6 means 6/3 with the third taken for granted. A bare 2 means 4/2. A bare 4 in certain cadential contexts is understood as 6/4. The player who reads numerals literally, as instructions to place a second above the bass and nothing else, will produce four-note chords where two were intended and thin chords where a full triad was wanted.\n#Accidentals in the figures\nA sharp, flat or natural attached to a numeral alters that interval above the bass, and it does so regardless of the key signature. The figure 6 with a sharp under a bass note in C major raises the sixth above that note, which is how a continuo part asks for a chromatic note the key does not contain. An accidental with no numeral applies to the third, so a lone sharp alone means a raised third.\nTwo subtleties follow. The first is that the figures override the key signature rather than adding to it, so a natural sign can be as informative as a sharp: it cancels an accidental that the key signature would otherwise supply. The second is that a slash through a sharp or flat raises that note by a further semitone, which is how the notation writes a raised leading tone in minor, a chromatic passing tone, or any other inflection that the key signature does not already supply.\nWhat the figures never state is register. A 6 above the bass can be placed in any octave, and a realisation in which every sixth sits immediately above the bass will sound cramped in a way the notation did not ask for.\n#What the notation leaves to the player\nEverything about the upper voices that is not an interval is a decision. Spacing, doubling, which voice moves where, whether the chord is played in three parts or four, whether the keyboard doubles the bass at the octave, and how long each chord is held: none of that is written, and all of it is heard.\nThis is why a realisation is a small act of composition rather than an act of decoding. The figures give the harmonic skeleton, and the player builds the texture. A continuo part played by one hand in the middle of the keyboard and one in the bass produces a very different result from four independent voices, and both are correct readings of the same figures.\nThree conventions narrow the field. A bass player and a keyboard player divide the labour: the melodic instrument states the line as written, and the keyboard fills in above it, usually keeping the right hand below the melody so that the texture does not double. The marking tasto solo instructs the keyboard player to lay out for a passage and leave the bass line unaccompanied, which is used where the texture needs thinning or where the harmony is deliberately bare. And where upper parts exist, the keyboard avoids doubling the notated melody, since two instruments on the same line at the same octave is a colour, not a texture.\nA3+C4+E4 F3+A3+D4 G3+B3+D4+F4 C4+E4+G4\n#Reading a line in practice\nTake a short progression in C major: bass A, then F, then G, then C, with the figures 5/3, 6, 7 and 5/3.\nBassFigureIntervals aboveChordFunctionAunfigured, 5/3C, EA minorviF6A, DD minor in first inversionii6G7B, D, FG dominant seventhV7Cunfigured, 5/3E, GC majorI\nThe line descends by step from A to F to G, and the harmony is a tonic expansion followed by a cadence. Nothing in the figures is unusual: the bare 6 marks the first inversion, the 7 marks the seventh chord, and the two unfigured notes are the plain triads at the ends. A player would add a third above the bass in the first chord, place the sixth above the F in the second, resolve the F of the seventh chord down to the E of the final triad, and lead the leading tone B up to C. All of that is idiomatic knowledge, and none of it is in the figures.\n#Why the notation is still taught\nFigured bass survived as a working notation for something like two centuries after it appeared with the basso continuo in the early seventeenth century, and it is still taught for reasons that have little to do with performing old music.\nIt records inversion and function in the same symbol, which no chord symbol does as clearly. It forces the student to think in intervals above a given note rather than in memorised shapes on an instrument. And it makes voice leading unavoidable, because the player has to decide what the inner voices do while the bass moves, which is exactly the decision that part-writing exercises formalise and that performance demands in real time.\nThe voice-leading article covers the constraints a realisation has to satisfy. Those constraints are what turned a notation for accompanists into a training method, and they are why a continuo exercise is still one of the fastest ways to find out whether the harmony in a passage has been understood or merely labelled.\nThe virtual piano is a practical place to work through a realisation at the keyboard: play the bass line with the left hand as written, and add the intervals above it with the right.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "figured bass", "basso continuo", "inversion" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/figured-bass.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/first-species-counterpoint/", "url": "https://musics.name/articles/first-species-counterpoint/", "title": "First Species Counterpoint, Rule by Rule", "summary": "Species counterpoint isolates one compositional decision at a time. First species isolates the hardest one: how to make two voices sound like two voices.", "content_html": "A cantus firmus is a tune with no accompaniment, eight to sixteen notes long, that moves mostly by step and comes to rest on its final. First species asks you to add exactly one note against each of those notes: two voices, both in whole notes, moving together through the whole exercise. Nothing about the task sounds difficult until the fourth or fifth note, when the obvious moves have run out and the remaining options are all wrong for reasons you have to learn to hear.
\nWhat makes the exercise worth doing is that it removes almost everything else. There is no rhythm to hide behind, no accompaniment to thicken the texture, no register change to reset the ear. Every vertical interval is exposed for a full bar, and the quality of the counterpoint is decided by which intervals you choose and how you travel between them.
\nSpecies counterpoint reaches its canonical form in Johann Joseph Fux’s Gradus ad Parnassum (1725), which distilled the practice of sixteenth-century vocal polyphony into a graded method: note against note, then two notes against one, then four, then syncopated, then florid. Fux was teaching a style, not an abstract system, and the style had a specific sound: massed consonant intervals, each line shaped like a singable melody, cadences that come to rest on the final.
\nThe order matters. In later species the added voice has rhythmic motion, which lets consonance and dissonance alternate quickly and lets melodic momentum cover weak moments. In first species everything is held. Two voices that are consonant but inert will sound inert, and two voices that are individually beautiful but collide will sound wrong immediately, and you will not be able to explain why until you look at the intervals.
\nSo first species is really testing three separate skills at once: choosing consonant intervals, moving between them in a way that keeps the voices independent, and shaping each line as a melody. Keep them separate while you work. Most beginner errors come from trying to satisfy all three in one pass.
\nConsonances are divided into two classes, and the division carries real information. Imperfect consonances (thirds and sixths) can be used freely, including in parallel motion. Perfect consonances (fifths, octaves, unisons) are structurally stronger and therefore rationed: they open the piece, close it, and mark important points in between.
\n| Interval against the cantus | Status in first species | Conditions |
|---|---|---|
| Third, sixth (imperfect) | freely usable | the substance of the middle of the exercise |
| Fifth, octave (perfect) | usable, sparingly | approach by contrary or oblique motion |
| Unison | opening and close only | a mid-exercise unison erases the two-voice texture |
| Fourth | not usable | a dissonance in two-voice writing against the lower voice |
| Second, seventh, any altered interval | not usable | dissonant |
| Parallel fifths or octaves | forbidden | both voices move in the same direction by the same perfect interval |
| Hidden (direct) fifths or octaves | forbidden | similar motion into a perfect consonance |
The unison rule surprises people. A perfect unison is the most stable interval available, so it seems like a natural resting point. In two voices it is the opposite: at a unison the two lines have merged into one pitch, the texture momentarily has one voice instead of two, and the ear loses the independence that the whole exercise exists to demonstrate.
\nThe rule against parallel perfect intervals has nothing to do with ugliness. Fifths and octaves are so stable that a pair of them in a row fuses the two voices into a single reinforced line; the independence that makes counterpoint counterpoint disappears. Fifths and octaves arrived at by similar motion produce a milder version of the same fusion, and in two-voice writing the damage is audible: the upper voice seems to slip into the lower voice’s track.
\nContrary motion is the fix, and it is worth thinking of it as the engine of the style. When one voice rises while the other falls, both lines are audible as lines, and any perfect interval they arrive at sounds like a convergence rather than an accident. Oblique motion, where one voice holds while the other moves, is nearly as useful.
\n\nPlay that pair of chords and listen to how quickly the interest drains out of it. Both voices step up, the fifth stays a fifth, and the result is not two voices but one thick one. Now the same bass with a contrary upper voice:
\n\nThe interval changes from a fifth to a third, the upper line has its own direction, and the ear registers two parts. Nothing in the pitches changed except one note.
\nA counterpoint that satisfies every interval rule can still be a bad counterpoint, because the melodic constraints are doing half the work. The added voice should be singable by a person with an ordinary range, which means staying inside roughly a tenth and avoiding anything that would make a singer reposition.
\nThe practical melodic rules are these. Stepwise motion should predominate; the exercise is not the place for a chain of leaps. Leaps are permitted up to the fifth and to the minor sixth, plus the octave, and the major sixth leap is generally avoided in strict two-voice writing. Any leap larger than a third should be recovered by step in the opposite direction, which is what makes a leap sound like an excursion rather than a series of disconnected pitches. Repeated notes are not part of the strict style at this species. Outlining a tritone, meaning the augmented fourth or its inversion, is forbidden, both by direct leap and by two leaps that trace the interval between them.
\nThen there is the shape rule. The added voice should have one high point, reached deliberately, usually by step, and not returned to. A line with two climaxes has no climax. A line whose highest note is also its first note has nowhere to go.
\n\nThat example is legal almost everywhere and dull: a chain of parallel thirds, then a contrary arrival at the octave. Parallel thirds are permitted precisely because moving between imperfect consonances does not fuse the voices, but a whole exercise of them has no structural shape. Compare a version that spends its perfect intervals where they count, opens on the octave, uses thirds and sixths through the middle, and returns to the octave at the close.
\nThe cadence deserves special attention because it is the only place where the rules relax in a specific direction. The final interval should be an octave or a unison, approached so that the finality is unmistakable, and in first species the standard close is a stepwise approach into the final in one or both voices. That is the ancestor of every cadence you will meet later in tonal music, and it is worth noticing here where it comes from: the counterpoint’s ending is a melodic event before it is a harmonic one.
\nThe most common way to do this badly is to write the added voice note by note from left to right, choosing the best available interval at each step. That method has no view of the whole line, so it produces the equivalent of a walk that never leaves the block.
\nWrite it in two passes instead. First pass: decide the skeleton. Place the opening interval, the closing interval, any perfect intervals you intend to use, and the climactic note of the added voice. Write those five or six decisions down. Second pass: fill the gaps with stepwise motion, favoring thirds and sixths, and let the melodic requirement that leaps recover by step do the work of keeping the line coherent.
\nThen check it the other way around. Read the added voice alone and ask whether it is a melody you would want to sing. Read the cantus alone, which you should already know is one. Then read the intervals from bottom to top, note by note, and name each one out loud. Errors in first species are almost never subtle; they survive precisely because the student stopped checking.
\nThe rules are a residue, not a foundation. They describe what sixteenth-century polyphony does most of the time, and Fux’s codification of that practice served a pedagogical purpose in a later era. Treating them as laws of music leads to the conclusion that parallel fifths are intrinsically wrong, which is false in most of the world’s music and a great deal of European music after 1890.
\nFirst species also teaches nothing about dissonance treatment, because there is no dissonance to treat. The interesting half of counterpoint begins in second species, where a passing dissonance on a weak beat is legal and the same note on a strong beat is not. If you do only first species, you have learned interval grammar without learning syntax.
\nAnd the exercise cannot measure quality, only legality. A counterpoint with no faults can be lifeless, and a slightly irregular one can be memorable. The rules exist to protect the independence of two lines; independence is what you are actually after.
\nSet yourself a cantus firmus, sing it until it is memorized, then write the counterpoint in the opposite direction. If you write below the cantus, the melodic shape you naturally favor will be wrong, and you will have to think about register. Sing each line separately before playing them together; the voice finds parallel fifths that the eye and the keyboard both forgive. Play the finished exercise on the virtual piano with the two voices an octave apart, which makes the intervals visible as well as audible.
\nThen invert the exercise. Rewrite the counterpoint below the cantus instead of above, keeping the pitch classes unchanged, and see which of your intervals survive the flip. Most first attempts do not survive it, and that failure is the entrance to the next subject.
", "content_text": "A cantus firmus is a tune with no accompaniment, eight to sixteen notes long, that moves mostly by step and comes to rest on its final. First species asks you to add exactly one note against each of those notes: two voices, both in whole notes, moving together through the whole exercise. Nothing about the task sounds difficult until the fourth or fifth note, when the obvious moves have run out and the remaining options are all wrong for reasons you have to learn to hear.\nWhat makes the exercise worth doing is that it removes almost everything else. There is no rhythm to hide behind, no accompaniment to thicken the texture, no register change to reset the ear. Every vertical interval is exposed for a full bar, and the quality of the counterpoint is decided by which intervals you choose and how you travel between them.\n#What first species is actually testing\nSpecies counterpoint reaches its canonical form in Johann Joseph Fux’s Gradus ad Parnassum (1725), which distilled the practice of sixteenth-century vocal polyphony into a graded method: note against note, then two notes against one, then four, then syncopated, then florid. Fux was teaching a style, not an abstract system, and the style had a specific sound: massed consonant intervals, each line shaped like a singable melody, cadences that come to rest on the final.\nThe order matters. In later species the added voice has rhythmic motion, which lets consonance and dissonance alternate quickly and lets melodic momentum cover weak moments. In first species everything is held. Two voices that are consonant but inert will sound inert, and two voices that are individually beautiful but collide will sound wrong immediately, and you will not be able to explain why until you look at the intervals.\nSo first species is really testing three separate skills at once: choosing consonant intervals, moving between them in a way that keeps the voices independent, and shaping each line as a melody. Keep them separate while you work. Most beginner errors come from trying to satisfy all three in one pass.\n#The intervallic framework\nConsonances are divided into two classes, and the division carries real information. Imperfect consonances (thirds and sixths) can be used freely, including in parallel motion. Perfect consonances (fifths, octaves, unisons) are structurally stronger and therefore rationed: they open the piece, close it, and mark important points in between.\nInterval against the cantusStatus in first speciesConditionsThird, sixth (imperfect)freely usablethe substance of the middle of the exerciseFifth, octave (perfect)usable, sparinglyapproach by contrary or oblique motionUnisonopening and close onlya mid-exercise unison erases the two-voice textureFourthnot usablea dissonance in two-voice writing against the lower voiceSecond, seventh, any altered intervalnot usabledissonantParallel fifths or octavesforbiddenboth voices move in the same direction by the same perfect intervalHidden (direct) fifths or octavesforbiddensimilar motion into a perfect consonance\nThe unison rule surprises people. A perfect unison is the most stable interval available, so it seems like a natural resting point. In two voices it is the opposite: at a unison the two lines have merged into one pitch, the texture momentarily has one voice instead of two, and the ear loses the independence that the whole exercise exists to demonstrate.\nThe rule against parallel perfect intervals has nothing to do with ugliness. Fifths and octaves are so stable that a pair of them in a row fuses the two voices into a single reinforced line; the independence that makes counterpoint counterpoint disappears. Fifths and octaves arrived at by similar motion produce a milder version of the same fusion, and in two-voice writing the damage is audible: the upper voice seems to slip into the lower voice’s track.\nContrary motion is the fix, and it is worth thinking of it as the engine of the style. When one voice rises while the other falls, both lines are audible as lines, and any perfect interval they arrive at sounds like a convergence rather than an accident. Oblique motion, where one voice holds while the other moves, is nearly as useful.\nC4+G4 D4+A4\nPlay that pair of chords and listen to how quickly the interest drains out of it. Both voices step up, the fifth stays a fifth, and the result is not two voices but one thick one. Now the same bass with a contrary upper voice:\nC4+G4 D4+F4\nThe interval changes from a fifth to a third, the upper line has its own direction, and the ear registers two parts. Nothing in the pitches changed except one note.\n#The added voice still has to be a melody\nA counterpoint that satisfies every interval rule can still be a bad counterpoint, because the melodic constraints are doing half the work. The added voice should be singable by a person with an ordinary range, which means staying inside roughly a tenth and avoiding anything that would make a singer reposition.\nThe practical melodic rules are these. Stepwise motion should predominate; the exercise is not the place for a chain of leaps. Leaps are permitted up to the fifth and to the minor sixth, plus the octave, and the major sixth leap is generally avoided in strict two-voice writing. Any leap larger than a third should be recovered by step in the opposite direction, which is what makes a leap sound like an excursion rather than a series of disconnected pitches. Repeated notes are not part of the strict style at this species. Outlining a tritone, meaning the augmented fourth or its inversion, is forbidden, both by direct leap and by two leaps that trace the interval between them.\nThen there is the shape rule. The added voice should have one high point, reached deliberately, usually by step, and not returned to. A line with two climaxes has no climax. A line whose highest note is also its first note has nowhere to go.\nC4+E4 D4+F4 E4+G4 F4+A4 G4+B4 A4+C5 G4+B4 F4+A4 C4+C5\nThat example is legal almost everywhere and dull: a chain of parallel thirds, then a contrary arrival at the octave. Parallel thirds are permitted precisely because moving between imperfect consonances does not fuse the voices, but a whole exercise of them has no structural shape. Compare a version that spends its perfect intervals where they count, opens on the octave, uses thirds and sixths through the middle, and returns to the octave at the close.\nThe cadence deserves special attention because it is the only place where the rules relax in a specific direction. The final interval should be an octave or a unison, approached so that the finality is unmistakable, and in first species the standard close is a stepwise approach into the final in one or both voices. That is the ancestor of every cadence you will meet later in tonal music, and it is worth noticing here where it comes from: the counterpoint’s ending is a melodic event before it is a harmonic one.\n#Two passes, not one\nThe most common way to do this badly is to write the added voice note by note from left to right, choosing the best available interval at each step. That method has no view of the whole line, so it produces the equivalent of a walk that never leaves the block.\nWrite it in two passes instead. First pass: decide the skeleton. Place the opening interval, the closing interval, any perfect intervals you intend to use, and the climactic note of the added voice. Write those five or six decisions down. Second pass: fill the gaps with stepwise motion, favoring thirds and sixths, and let the melodic requirement that leaps recover by step do the work of keeping the line coherent.\nThen check it the other way around. Read the added voice alone and ask whether it is a melody you would want to sing. Read the cantus alone, which you should already know is one. Then read the intervals from bottom to top, note by note, and name each one out loud. Errors in first species are almost never subtle; they survive precisely because the student stopped checking.\n#Where the exercise misleads people\nThe rules are a residue, not a foundation. They describe what sixteenth-century polyphony does most of the time, and Fux’s codification of that practice served a pedagogical purpose in a later era. Treating them as laws of music leads to the conclusion that parallel fifths are intrinsically wrong, which is false in most of the world’s music and a great deal of European music after 1890.\nFirst species also teaches nothing about dissonance treatment, because there is no dissonance to treat. The interesting half of counterpoint begins in second species, where a passing dissonance on a weak beat is legal and the same note on a strong beat is not. If you do only first species, you have learned interval grammar without learning syntax.\nAnd the exercise cannot measure quality, only legality. A counterpoint with no faults can be lifeless, and a slightly irregular one can be memorable. The rules exist to protect the independence of two lines; independence is what you are actually after.\n#How to work with it\nSet yourself a cantus firmus, sing it until it is memorized, then write the counterpoint in the opposite direction. If you write below the cantus, the melodic shape you naturally favor will be wrong, and you will have to think about register. Sing each line separately before playing them together; the voice finds parallel fifths that the eye and the keyboard both forgive. Play the finished exercise on the virtual piano with the two voices an octave apart, which makes the intervals visible as well as audible.\nThen invert the exercise. Rewrite the counterpoint below the cantus instead of above, keeping the pitch classes unchanged, and see which of your intervals survive the flip. Most first attempts do not survive it, and that failure is the entrance to the next subject.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "species counterpoint", "voice leading", "two voices" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/first-species-counterpoint.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/fugue-subject-and-answer/", "url": "https://musics.name/articles/fugue-subject-and-answer/", "title": "Fugue Subjects and Answers", "summary": "A fugue's opening notes commit the piece to more than a tune. The subject's shape and the answer's adjustment decide what can happen next.", "content_html": "Almost every description of fugue begins by warning that fugue is not a form. The warning is necessary and usually wasted, because the terms that follow (subject, answer, countersubject, exposition, episode) get used as if they named fixed compartments. They are better understood as the vocabulary a composer uses to solve a specific problem: how to state an idea, leave it, return to it, and make every return sound like a consequence rather than a repeat.
\nThe subject is where that problem is decided. Before the second voice enters, the first eight notes have already fixed the key, the harmonic direction, the rhythmic character and the contours that later entries can be inverted against.
\nThe subject is the idea itself, stated alone by one voice. The answer is its reappearance in another voice, normally at the interval of a fifth or an octave. The countersubject is a line that accompanies the answer and is designed to combine with the subject thereafter. The exposition is the section in which each voice enters with the answer or subject in turn. An episode is a passage of transition and development built from figures extracted from the subject, usually moving to a new key and preparing the next entry. Later passages repeat the subject in related keys and are called middle entries, and a stretto compresses the entries so that each voice enters before the previous one has finished.
\nOnly two of those terms have an unambiguous referent in the score: the subject and the exposition. “Countersubject” is an analyst’s convenience, applied after the fact to whichever line happened to accompany the answer and proved useful later. Many fugues have no counted countersubject at all. And a fugal passage inside a larger movement, a fugato, uses the same texture without committing to a full exposition.
\nThe one structural rule with real teeth is that the exposition must state each voice’s entry in a clear order and establish the key. Everything after that is variation, and the variation is the point.
\nThe simplest answer is the real answer: the subject transposed literally, usually up a fifth. If the subject is the five notes C–D–E–F–G, the real answer is G–A–B–C–D, and nothing has been adjusted.
\nThe complication arrives with subjects that emphasize the dominant in their opening. Take a subject that begins by leaping from the tonic up to the dominant, then steps down: C–G–F. Transposed literally up a fifth, the answer begins G–D–C, and the second note, D, is the supertonic of the key, which means the answer’s opening outlines the dominant’s own dominant rather than a stable dominant. The relationship between the two entries is wrong: instead of hearing the tonic answered by the dominant, the ear hears a dominant answered by something that wants to leave it.
\nThe traditional correction is the tonal answer, and its rule is worth stating exactly: when the subject’s opening emphasizes the dominant by leap, the answer mirrors the intervals, so that a leap of a fifth is answered by a leap of a fourth. The subject C–G–F becomes the answer G–C–B. The second note is now the tonic, the tonic-dominant axis is preserved, and the rest of the subject proceeds as before.
\n\nThe button plays three three-note heads in order: the subject, then a literal real answer, then the corrected tonal answer. The first and second differ only in the size of the opening leap; the second and third differ in the second note, and it is that single note that keeps the answer inside the key.
\nReal answers are not simply the lazy choice. A subject that begins on the tonic and moves by step, or that leaps up a fourth, is usually answered literally, because a literal transposition keeps the harmonic meaning intact. The decision is made by the subject’s opening intervals, not by its length, and treatises have always disagreed about the hardest cases. In Bach’s Well-Tempered Clavier both kinds appear throughout, sometimes within the same pair of pieces.
\nThe practical guide is short enough to memorize:
\n| The subject’s opening | Typical answer | Reason |
|---|---|---|
| Steps away from the tonic (1–2–3) | real | A literal transposition keeps the tonic-dominant relationship intact |
| Leaps from 1 up to 5 | tonal | A fifth answered by a fourth keeps the dominant answered by the tonic |
| Arpeggiates the dominant triad | tonal | A literal answer would push the music toward the dominant’s own dominant |
| Begins on the dominant itself | tonal | The answer enters on the tonic, a fourth above |
None of the four rows is a law. They describe which choice preserves the relationship the ear is tracking, and the ear is the final authority in the cases where a subject sits between two categories.
\nA subject is a melody and a harmonic plan at once. Its first three or four notes imply a chord, and that implication is the fugue’s engine: entries are placed where the harmony they imply can be prepared, and episodes exist to move from one entry’s key to the next entry’s key.
\nSeveral properties matter more than the tune itself. The head motive, the first few intervals and their rhythm, is what listeners recognize when the subject returns after an episode, so it needs something distinctive: an unusual rhythm, a characteristic leap, a repeated note. A subject that opens with four stepwise notes at the same speed gives the ear nothing to hold, and the whole fugue will depend on its countersubject to be memorable.
\nThe harmonic profile determines how much freedom the composer has. A subject that arpeggiates the tonic triad is easy to place and hard to make interesting; a subject whose opening implies a dominant or a seventh chord creates tension that later entries can resolve differently. Most well-made subjects sit between those extremes, stating the tonic clearly but ending in a way that leaves the phrase open.
\nThe range is a practical constraint that beginners overlook. The answer is a fifth above the subject, so a subject that climbs to the top of a voice’s range will push its answer out of it. This is why so many subjects begin in the lower middle of the voice and rise late.
\nFinally, the tail of the subject is often designed to become the countersubject. If the last few notes state a rhythm that can continue in counterpoint against the head, the fugue has a ready-made second idea without needing a separate invention.
\nA countersubject is judged by what it can do when the subject returns, not by how it sounds on its own. It accompanies the answer in the exposition, and it must then work when the two lines trade places: the subject below with the countersubject above, and later the reverse, at whatever inversion interval the composer has chosen.
\nThat requirement is exactly the interval arithmetic of invertible counterpoint. A countersubject invertible at the octave cannot use perfect fifths against the subject, because a fifth inverts to a fourth; invertible at the twelfth, it can use fifths but not sixths, since a sixth inverts to a seventh. Most fugue countersubjects are written as note-against-note or in deliberately complementary rhythm, so that the two lines never compete for the same metrical position.
\n\nThat is the shape to aim for: the upper line moving mostly in steps, the lower line turning against it, and every vertical interval a third or a sixth. Nothing in the example is a fifth or an octave, which is what makes it survive being turned over.
\nThe deeper obligation is negative. A countersubject that merely fills out the harmony of the subject will survive inversion technically and sound like an accompaniment in both versions. The test is whether the line, played alone, is a line.
\nTwo habits cause most of the confusion around fugue. The first is assuming that every fugue has all the parts named above. Plenty of fugues begin with a subject and a countersubject that is never used again, or build their middle sections from new material rather than episodes, or dispense with stretto entirely. The vocabulary describes what a fugue is likely to contain, and the piece is the authority.
\nThe second habit is treating the answer’s type as a fact about the piece rather than about the subject. A tonal answer is not a lesser version of a real answer, and a real answer is not a sign of a less sophisticated subject. The question that decides it is narrow: does the subject’s opening need to have its dominant emphasized, or would a literal transposition preserve the tonic-dominant relationship? When a subject begins on the dominant, or arpeggiates it, the answer usually employs the same adjustment in the other direction.
\nThere is also a naming problem specific to the countersubject. Because it is identified retrospectively, two analysts can disagree about which line deserves the name, and both can be right about the counterpoint while differing about the label. This does not matter for analysis and matters a great deal for writing: if you are composing, you choose which line will return with the subject, and you write it accordingly.
\nStart with the head, not the whole subject. Compose three or four notes with a rhythm and an interval you would recognize if you heard them again after thirty seconds of other music, and check what chord they imply. Then extend to the length of a normal phrase, ending on a note that leaves the phrase open, and check that the answer at the fifth stays inside the voice’s range.
\nThen answer it twice: literally, and with the opening interval adjusted if the head emphasizes the dominant. Sing or play both, and decide which one preserves the tonic-dominant relationship rather than merely transposing the notes. The ear training studio is useful at this stage, because hearing a fourth where you expected a fifth is the whole skill in miniature.
\nOnly then write the countersubject, and write it with the inversion in mind from the first note. Play the subject and countersubject together on the virtual piano in both arrangements before you decide which one you prefer. If the second arrangement collapses, the countersubject is not finished, however good the first arrangement sounded.
", "content_text": "Almost every description of fugue begins by warning that fugue is not a form. The warning is necessary and usually wasted, because the terms that follow (subject, answer, countersubject, exposition, episode) get used as if they named fixed compartments. They are better understood as the vocabulary a composer uses to solve a specific problem: how to state an idea, leave it, return to it, and make every return sound like a consequence rather than a repeat.\nThe subject is where that problem is decided. Before the second voice enters, the first eight notes have already fixed the key, the harmonic direction, the rhythmic character and the contours that later entries can be inverted against.\n#The anatomy, as working vocabulary\nThe subject is the idea itself, stated alone by one voice. The answer is its reappearance in another voice, normally at the interval of a fifth or an octave. The countersubject is a line that accompanies the answer and is designed to combine with the subject thereafter. The exposition is the section in which each voice enters with the answer or subject in turn. An episode is a passage of transition and development built from figures extracted from the subject, usually moving to a new key and preparing the next entry. Later passages repeat the subject in related keys and are called middle entries, and a stretto compresses the entries so that each voice enters before the previous one has finished.\nOnly two of those terms have an unambiguous referent in the score: the subject and the exposition. “Countersubject” is an analyst’s convenience, applied after the fact to whichever line happened to accompany the answer and proved useful later. Many fugues have no counted countersubject at all. And a fugal passage inside a larger movement, a fugato, uses the same texture without committing to a full exposition.\nThe one structural rule with real teeth is that the exposition must state each voice’s entry in a clear order and establish the key. Everything after that is variation, and the variation is the point.\n#Two answers, and why only one of them can be literal\nThe simplest answer is the real answer: the subject transposed literally, usually up a fifth. If the subject is the five notes C–D–E–F–G, the real answer is G–A–B–C–D, and nothing has been adjusted.\nThe complication arrives with subjects that emphasize the dominant in their opening. Take a subject that begins by leaping from the tonic up to the dominant, then steps down: C–G–F. Transposed literally up a fifth, the answer begins G–D–C, and the second note, D, is the supertonic of the key, which means the answer’s opening outlines the dominant’s own dominant rather than a stable dominant. The relationship between the two entries is wrong: instead of hearing the tonic answered by the dominant, the ear hears a dominant answered by something that wants to leave it.\nThe traditional correction is the tonal answer, and its rule is worth stating exactly: when the subject’s opening emphasizes the dominant by leap, the answer mirrors the intervals, so that a leap of a fifth is answered by a leap of a fourth. The subject C–G–F becomes the answer G–C–B. The second note is now the tonic, the tonic-dominant axis is preserved, and the rest of the subject proceeds as before.\nC4 G4 F4 G4 D5 C5 G4 C5 B4\nThe button plays three three-note heads in order: the subject, then a literal real answer, then the corrected tonal answer. The first and second differ only in the size of the opening leap; the second and third differ in the second note, and it is that single note that keeps the answer inside the key.\nReal answers are not simply the lazy choice. A subject that begins on the tonic and moves by step, or that leaps up a fourth, is usually answered literally, because a literal transposition keeps the harmonic meaning intact. The decision is made by the subject’s opening intervals, not by its length, and treatises have always disagreed about the hardest cases. In Bach’s Well-Tempered Clavier both kinds appear throughout, sometimes within the same pair of pieces.\nThe practical guide is short enough to memorize:\nThe subject’s openingTypical answerReasonSteps away from the tonic (1–2–3)realA literal transposition keeps the tonic-dominant relationship intactLeaps from 1 up to 5tonalA fifth answered by a fourth keeps the dominant answered by the tonicArpeggiates the dominant triadtonalA literal answer would push the music toward the dominant’s own dominantBegins on the dominant itselftonalThe answer enters on the tonic, a fourth above\nNone of the four rows is a law. They describe which choice preserves the relationship the ear is tracking, and the ear is the final authority in the cases where a subject sits between two categories.\n#What the subject has already decided\nA subject is a melody and a harmonic plan at once. Its first three or four notes imply a chord, and that implication is the fugue’s engine: entries are placed where the harmony they imply can be prepared, and episodes exist to move from one entry’s key to the next entry’s key.\nSeveral properties matter more than the tune itself. The head motive, the first few intervals and their rhythm, is what listeners recognize when the subject returns after an episode, so it needs something distinctive: an unusual rhythm, a characteristic leap, a repeated note. A subject that opens with four stepwise notes at the same speed gives the ear nothing to hold, and the whole fugue will depend on its countersubject to be memorable.\nThe harmonic profile determines how much freedom the composer has. A subject that arpeggiates the tonic triad is easy to place and hard to make interesting; a subject whose opening implies a dominant or a seventh chord creates tension that later entries can resolve differently. Most well-made subjects sit between those extremes, stating the tonic clearly but ending in a way that leaves the phrase open.\nThe range is a practical constraint that beginners overlook. The answer is a fifth above the subject, so a subject that climbs to the top of a voice’s range will push its answer out of it. This is why so many subjects begin in the lower middle of the voice and rise late.\nFinally, the tail of the subject is often designed to become the countersubject. If the last few notes state a rhythm that can continue in counterpoint against the head, the fugue has a ready-made second idea without needing a separate invention.\n#The countersubject’s obligations\nA countersubject is judged by what it can do when the subject returns, not by how it sounds on its own. It accompanies the answer in the exposition, and it must then work when the two lines trade places: the subject below with the countersubject above, and later the reverse, at whatever inversion interval the composer has chosen.\nThat requirement is exactly the interval arithmetic of invertible counterpoint. A countersubject invertible at the octave cannot use perfect fifths against the subject, because a fifth inverts to a fourth; invertible at the twelfth, it can use fifths but not sixths, since a sixth inverts to a seventh. Most fugue countersubjects are written as note-against-note or in deliberately complementary rhythm, so that the two lines never compete for the same metrical position.\nC4+E4 D4+F4 E4+C5 D4+B4\nThat is the shape to aim for: the upper line moving mostly in steps, the lower line turning against it, and every vertical interval a third or a sixth. Nothing in the example is a fifth or an octave, which is what makes it survive being turned over.\nThe deeper obligation is negative. A countersubject that merely fills out the harmony of the subject will survive inversion technically and sound like an accompaniment in both versions. The test is whether the line, played alone, is a line.\n#Where the labels fail\nTwo habits cause most of the confusion around fugue. The first is assuming that every fugue has all the parts named above. Plenty of fugues begin with a subject and a countersubject that is never used again, or build their middle sections from new material rather than episodes, or dispense with stretto entirely. The vocabulary describes what a fugue is likely to contain, and the piece is the authority.\nThe second habit is treating the answer’s type as a fact about the piece rather than about the subject. A tonal answer is not a lesser version of a real answer, and a real answer is not a sign of a less sophisticated subject. The question that decides it is narrow: does the subject’s opening need to have its dominant emphasized, or would a literal transposition preserve the tonic-dominant relationship? When a subject begins on the dominant, or arpeggiates it, the answer usually employs the same adjustment in the other direction.\nThere is also a naming problem specific to the countersubject. Because it is identified retrospectively, two analysts can disagree about which line deserves the name, and both can be right about the counterpoint while differing about the label. This does not matter for analysis and matters a great deal for writing: if you are composing, you choose which line will return with the subject, and you write it accordingly.\n#Writing one\nStart with the head, not the whole subject. Compose three or four notes with a rhythm and an interval you would recognize if you heard them again after thirty seconds of other music, and check what chord they imply. Then extend to the length of a normal phrase, ending on a note that leaves the phrase open, and check that the answer at the fifth stays inside the voice’s range.\nThen answer it twice: literally, and with the opening interval adjusted if the head emphasizes the dominant. Sing or play both, and decide which one preserves the tonic-dominant relationship rather than merely transposing the notes. The ear training studio is useful at this stage, because hearing a fourth where you expected a fifth is the whole skill in miniature.\nOnly then write the countersubject, and write it with the inversion in mind from the first note. Play the subject and countersubject together on the virtual piano in both arrangements before you decide which one you prefer. If the second arrangement collapses, the countersubject is not finished, however good the first arrangement sounded.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "fugue", "subject", "answer" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/fugue-subject-and-answer.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/functional-harmony/", "url": "https://musics.name/articles/functional-harmony/", "title": "Functional Harmony: Tonic, Predominant, Dominant", "summary": "Tonic, predominant and dominant describe what a chord does to a phrase rather than what it is. This is how the three harmonic jobs divide the work.", "content_html": "A chord chart tells you which chords a piece uses. It does not tell you why the piece moves. The labels tonic, predominant and dominant answer the second question: they describe what a chord does to the phrase it sits in, not what notes it happens to contain.
\nThe clearest test uses three chords. In C major, take C, F and G. Played C–F–G–C they sound like a statement, a departure and a return. Played G–C–F–G the same three chords sound adrift, because the ear reads every chord against where it believes home to be and where the music seems to be heading. Function is that reading.
\nEvery diatonic chord in a key can be assigned one of three jobs.
\nTonic is the point of rest. It states the key, and it is the position a phrase measures its distance from. In C major the tonic job belongs to the C chord, but not to C alone: A minor and E minor share enough of its notes to stand in for it.
\nPredominant, also called subdominant, leaves home and aims at the dominant. F and D minor carry scale degree 4, the note that will eventually fall to scale degree 3 over the dominant.
\nDominant makes the return to the tonic sound required rather than merely possible. The leading tone and scale degree 4 are both present in the dominant seventh, G–B–D–F, and in the diminished triad on the leading tone, B–D–F; in both, the two notes form the tritone that has to resolve.
\n| Function | In major | In minor | What the chord does |
|---|---|---|---|
| Tonic | I, vi, iii | i, VI, III | states or restates the key, provides rest |
| Predominant | IV, ii, ii6 | iv, ii°6, VI | leaves the tonic and prepares the dominant |
| Dominant | V, V7, vii°, viiø7 | V, V7, vii°7 | makes the tonic’s arrival feel necessary |
The table names chords, but the job belongs to the context. A G major triad is the tonic of G major and the dominant of C major. In C major, A minor can be a soft rest, with vi standing in for I, or it can be a momentary new home after a D major chord has pointed at it. The same object, two jobs, decided by the phrase.
\nThe progressions that sound strongest move roots down a fifth, which is the same as up a fourth. A minor, D minor, G, C: three descending fifths in a row, arriving home. Each root becomes the fifth of the chord that follows it, so the bass makes large, unambiguous steps while the upper voices barely move. The harmony travels a long way and the voices do very little.
\nThe last stretch of that chain is the part that matters most: predominant to dominant. IV–V does the same work by a different route, with the bass stepping up instead of falling, and the two are interchangeable as preparations for V. D minor in first inversion is the more directional of the two, because its bass note is scale degree 4, climbing a step to 5 while the upper voices sink into the leading tone and fifth of the dominant.
\nTonic straight to dominant is also available, and common at the start of a phrase. It simply skips the departure.
\n\nThat is the whole cycle in one line: tonic, predominant, dominant, tonic.
\nBecause function depends on relations rather than identities, chords that share important notes can take over a job. A minor shares scale degree 1 and scale degree 3 with C major, so vi can substitute for I. E minor shares scale degree 3 and scale degree 5, but it also contains scale degree 7, the leading tone, which pulls the music toward the dominant. That is why iii is the weakest of the three tonic substitutes and why it so often appears as a passing chord on the way somewhere else.
\nOn the other side, ii and IV both contain scale degree 4, and ii6 and IV are close to interchangeable in front of V. In minor the same divisions hold, with iv and ii°6 as predominants and VI available in the tonic area. The dominant in minor needs the raised leading tone, which is why A minor’s dominant is E major rather than the E minor triad of the natural scale. Natural minor’s E minor and G major chords behave as modal colour, not as dominant function.
\nSubstituting changes the weight of a phrase without changing its syntax. A progression that reads tonic–predominant–dominant–tonic can be written vi–ii–V–I and still make the same argument.
\n\nInversions do not change what a chord is doing. They change how much it insists. I6 is tonic without being final; V6/5 is dominant with a softer profile than root-position V7.
\nThe cadential six-four is the standard example of function and identity separating. Its notes spell a tonic triad, but its bass note is scale degree 5 and it resolves into the dominant, so nobody hears it as a place to stop. Analysts disagree about whether to call it a tonic chord with two suspensions or part of an expanded dominant. For a player or a writer the practical answer is the same: it belongs to the cadence and it must lean forward.
\nAdding a seventh to a dominant triad tightens the same mechanism. The seventh is scale degree 4, which has to fall to scale degree 3, and the leading tone has to rise. Two obligations in one chord make the arrival harder to resist than a plain triad can manage. Jazz builds its basic unit, ii7–V7–I, from exactly this behaviour, and the added ninths and thirteenths of later practice change a chord’s colour without promoting it to a different job.
\nFunctional labels are an analysis tool built for tonal music, formulated around 1900 by Hugo Riemann and extended by later theorists. Applied everywhere, they become a category error.
\nModal polyphony before roughly 1600 does not fit. There, harmony emerges from independent lines, and a cadence is made by contrary motion and a raised leading tone rather than by a dominant chord doing its duty.
\nThe blues does not fit either. Its I–IV–V carries the same numerals as a tonal progression, but the tonic is a home rather than a goal and the dominant often sounds like a restatement of the tonic. When the third of the tonic chord is unstable by design, dominant function has nothing to attach to.
\nModal rock and folk progressions such as i–♭VII–♭VI work by repetition, register and rhythmic weight instead of tension and release. Post-tonal music abandons the premise altogether, and much of the world’s music never had tertian chords to give functions to.
\nAnalysis gets easier if you work in three passes. Find the cadence first, because that fixes the key (the article on cadences covers how). Then mark each chord as tonic, predominant or dominant instead of writing numerals. Only then look for the chord that refuses to fit its column; it is usually a chromatic preparation for a local tonic, which is the subject of secondary dominants.
\nTwo habits come out of this. A phrase that will not settle usually has a dominant with no predominant in front of it, so the arrival feels unmotivated. And a phrase that sounds harmonically vague often has its tonic substitutes doing work their weight cannot carry. The circle of fifths tool plays the diatonic chords of any key in sequence, which makes the difference between I, vi and iii audible in a few seconds rather than a few paragraphs.
", "content_text": "A chord chart tells you which chords a piece uses. It does not tell you why the piece moves. The labels tonic, predominant and dominant answer the second question: they describe what a chord does to the phrase it sits in, not what notes it happens to contain.\nThe clearest test uses three chords. In C major, take C, F and G. Played C–F–G–C they sound like a statement, a departure and a return. Played G–C–F–G the same three chords sound adrift, because the ear reads every chord against where it believes home to be and where the music seems to be heading. Function is that reading.\n#Three jobs, not three chords\nEvery diatonic chord in a key can be assigned one of three jobs.\nTonic is the point of rest. It states the key, and it is the position a phrase measures its distance from. In C major the tonic job belongs to the C chord, but not to C alone: A minor and E minor share enough of its notes to stand in for it.\nPredominant, also called subdominant, leaves home and aims at the dominant. F and D minor carry scale degree 4, the note that will eventually fall to scale degree 3 over the dominant.\nDominant makes the return to the tonic sound required rather than merely possible. The leading tone and scale degree 4 are both present in the dominant seventh, G–B–D–F, and in the diminished triad on the leading tone, B–D–F; in both, the two notes form the tritone that has to resolve.\nFunctionIn majorIn minorWhat the chord doesTonicI, vi, iiii, VI, IIIstates or restates the key, provides restPredominantIV, ii, ii6iv, ii°6, VIleaves the tonic and prepares the dominantDominantV, V7, vii°, viiø7V, V7, vii°7makes the tonic’s arrival feel necessary\nThe table names chords, but the job belongs to the context. A G major triad is the tonic of G major and the dominant of C major. In C major, A minor can be a soft rest, with vi standing in for I, or it can be a momentary new home after a D major chord has pointed at it. The same object, two jobs, decided by the phrase.\n#The engine underneath: roots falling by fifth\nThe progressions that sound strongest move roots down a fifth, which is the same as up a fourth. A minor, D minor, G, C: three descending fifths in a row, arriving home. Each root becomes the fifth of the chord that follows it, so the bass makes large, unambiguous steps while the upper voices barely move. The harmony travels a long way and the voices do very little.\nThe last stretch of that chain is the part that matters most: predominant to dominant. IV–V does the same work by a different route, with the bass stepping up instead of falling, and the two are interchangeable as preparations for V. D minor in first inversion is the more directional of the two, because its bass note is scale degree 4, climbing a step to 5 while the upper voices sink into the leading tone and fifth of the dominant.\nTonic straight to dominant is also available, and common at the start of a phrase. It simply skips the departure.\nC4+E4+G4 F4+A4+C5 G3+B3+D4+F4 C4+E4+G4\nThat is the whole cycle in one line: tonic, predominant, dominant, tonic.\n#Substitution inside a job\nBecause function depends on relations rather than identities, chords that share important notes can take over a job. A minor shares scale degree 1 and scale degree 3 with C major, so vi can substitute for I. E minor shares scale degree 3 and scale degree 5, but it also contains scale degree 7, the leading tone, which pulls the music toward the dominant. That is why iii is the weakest of the three tonic substitutes and why it so often appears as a passing chord on the way somewhere else.\nOn the other side, ii and IV both contain scale degree 4, and ii6 and IV are close to interchangeable in front of V. In minor the same divisions hold, with iv and ii°6 as predominants and VI available in the tonic area. The dominant in minor needs the raised leading tone, which is why A minor’s dominant is E major rather than the E minor triad of the natural scale. Natural minor’s E minor and G major chords behave as modal colour, not as dominant function.\nSubstituting changes the weight of a phrase without changing its syntax. A progression that reads tonic–predominant–dominant–tonic can be written vi–ii–V–I and still make the same argument.\nA3+C4+E4 D4+F4+A4 G3+B3+D4+F4 C4+E4+G4\n#Where function and chord identity part ways\nInversions do not change what a chord is doing. They change how much it insists. I6 is tonic without being final; V6/5 is dominant with a softer profile than root-position V7.\nThe cadential six-four is the standard example of function and identity separating. Its notes spell a tonic triad, but its bass note is scale degree 5 and it resolves into the dominant, so nobody hears it as a place to stop. Analysts disagree about whether to call it a tonic chord with two suspensions or part of an expanded dominant. For a player or a writer the practical answer is the same: it belongs to the cadence and it must lean forward.\nAdding a seventh to a dominant triad tightens the same mechanism. The seventh is scale degree 4, which has to fall to scale degree 3, and the leading tone has to rise. Two obligations in one chord make the arrival harder to resist than a plain triad can manage. Jazz builds its basic unit, ii7–V7–I, from exactly this behaviour, and the added ninths and thirteenths of later practice change a chord’s colour without promoting it to a different job.\n#What the model cannot describe\nFunctional labels are an analysis tool built for tonal music, formulated around 1900 by Hugo Riemann and extended by later theorists. Applied everywhere, they become a category error.\nModal polyphony before roughly 1600 does not fit. There, harmony emerges from independent lines, and a cadence is made by contrary motion and a raised leading tone rather than by a dominant chord doing its duty.\nThe blues does not fit either. Its I–IV–V carries the same numerals as a tonal progression, but the tonic is a home rather than a goal and the dominant often sounds like a restatement of the tonic. When the third of the tonic chord is unstable by design, dominant function has nothing to attach to.\nModal rock and folk progressions such as i–♭VII–♭VI work by repetition, register and rhythmic weight instead of tension and release. Post-tonal music abandons the premise altogether, and much of the world’s music never had tertian chords to give functions to.\n#Reading a phrase as functions\nAnalysis gets easier if you work in three passes. Find the cadence first, because that fixes the key (the article on cadences covers how). Then mark each chord as tonic, predominant or dominant instead of writing numerals. Only then look for the chord that refuses to fit its column; it is usually a chromatic preparation for a local tonic, which is the subject of secondary dominants.\nTwo habits come out of this. A phrase that will not settle usually has a dominant with no predominant in front of it, so the arrival feels unmotivated. And a phrase that sounds harmonically vague often has its tonic substitutes doing work their weight cannot carry. The circle of fifths tool plays the diatonic chords of any key in sequence, which makes the difference between I, vi and iii audible in a few seconds rather than a few paragraphs.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "chord function", "tonality", "harmonic analysis" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/functional-harmony.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/imitation-and-invertible-counterpoint/", "url": "https://musics.name/articles/imitation-and-invertible-counterpoint/", "title": "Imitation and Invertible Counterpoint", "summary": "Imitation makes one idea serve two lines. Invertible counterpoint makes those lines interchangeable, and the arithmetic behind it is small enough to check.", "content_html": "Two melodies that have nothing to do with each other can still be written together correctly, and the result will often sound like two people talking past each other. Imitation is the oldest solution to that problem: the second voice takes up the first voice’s idea, and the ear immediately hears a relationship that no amount of correct part-writing can manufacture.
\nInvertible counterpoint goes one step further and asks what happens if the two voices trade places. It sounds like a compositional parlour trick, and in the sixteenth century it often was one: a demonstration that a piece of counterpoint is so well made that it survives being turned upside down. But the technique is also the mechanism behind fugue subjects that combine with their countersubjects in any order, and behind development sections that shuffle the same two ideas through every register the orchestra owns.
\nIn its simplest form, one voice states a melodic idea and another voice enters with the same idea, usually at a different pitch level. The entry comes after the first voice has stated enough of the idea for the listener to recognize it: typically the length of the motive, a beat or a bar, sometimes longer.
\nEverything about the device depends on that entry interval. Entering at the fifth or the octave is the norm in tonal music because those intervals sit inside the tonic-dominant relationship that defines the key; an entry at the fifth announces a world with a tonic and a dominant in it. Entries at the second, third, sixth or seventh are common in later and modal repertoires, but they do not articulate a key in the same way, so they tend to read as colour rather than as an answer.
\nImitation can be strict or free. Strict imitation repeats the motive exactly, interval for interval, transposed. Free imitation keeps the rhythm and contour but alters the intervals. Between those two extremes lies the adjustment that tonal music needs constantly: tonal imitation, in which a motive is transposed only as far as the key allows. A motive that begins on the tonic and leaps to the dominant has to be treated carefully when it enters on the dominant, because a literal transposition would leap to the supertonic and change what the idea means harmonically. The traditional fix shortens the opening leap of a fifth to a fourth, which is exactly the adjustment that produces a tonal answer in fugue.
\nCanon is imitation with no exit: the following voice repeats the leader strictly, from the beginning to the end, so that every note of the piece is part of the imitative relationship. The simplest kind is a round, where the second voice enters after a fixed delay at the unison or octave and everyone can sing the whole melody.
\nThe technique gets richer when the entry is transposed or transformed. A canon at the fifth enters a fifth higher; a canon by inversion flips the leader’s intervals upside down, so that an ascending third in the leader becomes a descending third in the follower; a canon by retrograde plays the melody backwards. Mensuration canons change the speed of the following voice instead of its pitch, so the same notes are read in a different proportion.
\nThe Goldberg Variations show how far the device can be organized across a large work. Between the variations, Bach places canons at successively larger intervals, from the unison through the second, third, fourth, fifth, sixth, seventh and octave to the ninth, so that the whole set is a graded anthology of what imitation can do at each entry distance. The point is not display for its own sake. Writing a canon at the ninth is a genuine problem in register and interval choice, and the solutions are the same solutions a fugue needs.
\nInvertible counterpoint asks you to write two lines so that either one may be the upper voice. The lower voice may be moved above the upper (typically at the octave, the tenth or the twelfth), and both arrangements must still be consonant.
\nThe arithmetic that governs this is simple once you see it. When two voices are inverted at the octave, an interval and its inversion add up to nine: a second becomes a seventh, a third becomes a sixth, a fifth becomes a fourth, an octave becomes a unison. The list is worth memorizing because it tells you immediately which intervals survive the flip.
\n| Interval above | Inverts to | Usable in two-voice writing? |
|---|---|---|
| Unison | Octave | yes |
| Second | Seventh | no; dissonant both ways |
| Third | Sixth | yes; the safest interval available |
| Fourth | Fifth | no; the fourth is dissonant in two voices |
| Fifth | Fourth | no; see above |
| Sixth | Third | yes |
| Seventh | Second | no |
| Octave | Unison | yes |
The fourth is where this bites. In two-voice writing a perfect fifth above the lower voice inverts into a perfect fourth above the lower voice, and a fourth over the bass is a dissonance. So a strict two-voice exercise invertible at the octave cannot use fifths at all, which is a severe restriction: thirds and sixths become the entire vocabulary. Add a third voice and the restriction relaxes, since a fourth above the bass can be consonant if one of the upper voices supplies a fifth or a third above it. This is why invertible counterpoint is taught first in two voices and used mostly in three and four.
\nQuality inverts along with size. A major interval becomes minor when the voices swap; perfect intervals stay perfect; augmented and diminished trade places. So a major third above the lower voice comes back as a minor sixth below it, which is consonant in both directions and therefore perfectly usable, even though it does not sound identical.
\nHere is a pair of parallel thirds, then the same four pitch classes with the voices swapped:
\n\nThe interval above the lower voice is a major or minor third at every step. Now the same material inverted at the octave, with the formerly lower line on top:
\n\nEvery vertical interval is now a third’s inversion, a sixth, and the counterpoint still works. The pitch content is identical in the two versions; only the register, and therefore which line is on top, has changed.
\nInverting at the octave is not the only option, and composers often prefer a different interval because it permits intervals the octave forbids. At the tenth, an interval and its inversion add up to eleven: a third inverts to an octave, a fifth to a sixth, a sixth to a fifth, and an octave to a third. Thirds, fifths, sixths, octaves and tenths are all usable, which is a much larger palette than the octave allows.
\nAt the twelfth, the sum is thirteen: a third inverts to a tenth, a fifth to an octave, an octave to a fifth, and a tenth to a third. The sixteenth-century writers valued this arrangement because a fifth between the voices becomes an octave when the texture is flipped, a strong interval in both positions. But a sixth inverts to a seventh, so sixths are unavailable in strict two-voice writing invertible at the twelfth.
\nThe rule generalizes cleanly: an interval and its inversion must both be consonant. That is the whole discipline. Everything else, including which inversion interval to choose and whether the flip happens at a phrase boundary or in the middle of a line, is a compositional decision about where you want the strong intervals to land.
\nFugue is the form that lives on invertible counterpoint. A subject and its countersubject are written so that they can be combined in either order, at the octave or the twelfth, in whatever key the middle entries require; the exposition’s whole architecture depends on that property holding.
\nDevelopment sections use the same mechanism on a larger scale. Haydn and Beethoven frequently take two ideas from an exposition and combine them in invertible counterpoint, which is a way of proving that the two themes belong to the same piece. The coda of the finale of Mozart’s Symphony No. 41 does this with several distinct figures at once, stacked so that every voice is audible and none of them cancels another.
\nTwo misunderstandings are worth clearing up. First, imitation is not the same as sequence. A sequence repeats a motive in one voice at successively higher or lower pitch levels; imitation requires a second voice to take up the idea, and the two devices have different effects even when they use the same motive. Second, inversion in this sense has nothing to do with chord inversion or with melodic inversion by mirroring. Invertible counterpoint means swapping registers, not flipping intervals.
\nThe deeper caution is about writing invertible counterpoint where it will never be inverted. The technique imposes real costs, in restricted intervals and careful register planning, and paying those costs for a passage that will never trade places is wasted effort. It is worth doing when the material is going to combine in several orders, and it is worth doing as an exercise because it improves how you hear the independence of two lines, and for no other reason.
\nWrite the two lines as one piece, not as a melody and an accompaniment, then test the flip before you like either version. If the lower voice’s line sounds like an accompaniment when it moves on top, it was an accompaniment. Play both versions on the virtual piano, and check the interval at every arrival rather than checking once at the end.
\nThen set yourself a problem that forces the issue: take an eight-note idea, write a line against it, invert the pair at the octave, and repair whatever broke. Most first attempts break in the same two places: a fifth that inverts to a fourth, and a climax in the upper voice that becomes a high note buried in the middle of the texture when the parts swap. Both are register problems, and both are solved the same way, by deciding where the two lines are allowed to travel before writing a single note of either.
", "content_text": "Two melodies that have nothing to do with each other can still be written together correctly, and the result will often sound like two people talking past each other. Imitation is the oldest solution to that problem: the second voice takes up the first voice’s idea, and the ear immediately hears a relationship that no amount of correct part-writing can manufacture.\nInvertible counterpoint goes one step further and asks what happens if the two voices trade places. It sounds like a compositional parlour trick, and in the sixteenth century it often was one: a demonstration that a piece of counterpoint is so well made that it survives being turned upside down. But the technique is also the mechanism behind fugue subjects that combine with their countersubjects in any order, and behind development sections that shuffle the same two ideas through every register the orchestra owns.\n#Imitation: one idea, two entries\nIn its simplest form, one voice states a melodic idea and another voice enters with the same idea, usually at a different pitch level. The entry comes after the first voice has stated enough of the idea for the listener to recognize it: typically the length of the motive, a beat or a bar, sometimes longer.\nEverything about the device depends on that entry interval. Entering at the fifth or the octave is the norm in tonal music because those intervals sit inside the tonic-dominant relationship that defines the key; an entry at the fifth announces a world with a tonic and a dominant in it. Entries at the second, third, sixth or seventh are common in later and modal repertoires, but they do not articulate a key in the same way, so they tend to read as colour rather than as an answer.\nImitation can be strict or free. Strict imitation repeats the motive exactly, interval for interval, transposed. Free imitation keeps the rhythm and contour but alters the intervals. Between those two extremes lies the adjustment that tonal music needs constantly: tonal imitation, in which a motive is transposed only as far as the key allows. A motive that begins on the tonic and leaps to the dominant has to be treated carefully when it enters on the dominant, because a literal transposition would leap to the supertonic and change what the idea means harmonically. The traditional fix shortens the opening leap of a fifth to a fourth, which is exactly the adjustment that produces a tonal answer in fugue.\n#Canon as the limit case\nCanon is imitation with no exit: the following voice repeats the leader strictly, from the beginning to the end, so that every note of the piece is part of the imitative relationship. The simplest kind is a round, where the second voice enters after a fixed delay at the unison or octave and everyone can sing the whole melody.\nThe technique gets richer when the entry is transposed or transformed. A canon at the fifth enters a fifth higher; a canon by inversion flips the leader’s intervals upside down, so that an ascending third in the leader becomes a descending third in the follower; a canon by retrograde plays the melody backwards. Mensuration canons change the speed of the following voice instead of its pitch, so the same notes are read in a different proportion.\nThe Goldberg Variations show how far the device can be organized across a large work. Between the variations, Bach places canons at successively larger intervals, from the unison through the second, third, fourth, fifth, sixth, seventh and octave to the ninth, so that the whole set is a graded anthology of what imitation can do at each entry distance. The point is not display for its own sake. Writing a canon at the ninth is a genuine problem in register and interval choice, and the solutions are the same solutions a fugue needs.\n#Inverting the texture\nInvertible counterpoint asks you to write two lines so that either one may be the upper voice. The lower voice may be moved above the upper (typically at the octave, the tenth or the twelfth), and both arrangements must still be consonant.\nThe arithmetic that governs this is simple once you see it. When two voices are inverted at the octave, an interval and its inversion add up to nine: a second becomes a seventh, a third becomes a sixth, a fifth becomes a fourth, an octave becomes a unison. The list is worth memorizing because it tells you immediately which intervals survive the flip.\nInterval aboveInverts toUsable in two-voice writing?UnisonOctaveyesSecondSeventhno; dissonant both waysThirdSixthyes; the safest interval availableFourthFifthno; the fourth is dissonant in two voicesFifthFourthno; see aboveSixthThirdyesSeventhSecondnoOctaveUnisonyes\nThe fourth is where this bites. In two-voice writing a perfect fifth above the lower voice inverts into a perfect fourth above the lower voice, and a fourth over the bass is a dissonance. So a strict two-voice exercise invertible at the octave cannot use fifths at all, which is a severe restriction: thirds and sixths become the entire vocabulary. Add a third voice and the restriction relaxes, since a fourth above the bass can be consonant if one of the upper voices supplies a fifth or a third above it. This is why invertible counterpoint is taught first in two voices and used mostly in three and four.\nQuality inverts along with size. A major interval becomes minor when the voices swap; perfect intervals stay perfect; augmented and diminished trade places. So a major third above the lower voice comes back as a minor sixth below it, which is consonant in both directions and therefore perfectly usable, even though it does not sound identical.\nHere is a pair of parallel thirds, then the same four pitch classes with the voices swapped:\nC4+E4 D4+F4 E4+G4 F4+A4\nThe interval above the lower voice is a major or minor third at every step. Now the same material inverted at the octave, with the formerly lower line on top:\nE3+C4 F3+D4 G3+E4 A3+F4\nEvery vertical interval is now a third’s inversion, a sixth, and the counterpoint still works. The pitch content is identical in the two versions; only the register, and therefore which line is on top, has changed.\n#At the tenth and the twelfth\nInverting at the octave is not the only option, and composers often prefer a different interval because it permits intervals the octave forbids. At the tenth, an interval and its inversion add up to eleven: a third inverts to an octave, a fifth to a sixth, a sixth to a fifth, and an octave to a third. Thirds, fifths, sixths, octaves and tenths are all usable, which is a much larger palette than the octave allows.\nAt the twelfth, the sum is thirteen: a third inverts to a tenth, a fifth to an octave, an octave to a fifth, and a tenth to a third. The sixteenth-century writers valued this arrangement because a fifth between the voices becomes an octave when the texture is flipped, a strong interval in both positions. But a sixth inverts to a seventh, so sixths are unavailable in strict two-voice writing invertible at the twelfth.\nThe rule generalizes cleanly: an interval and its inversion must both be consonant. That is the whole discipline. Everything else, including which inversion interval to choose and whether the flip happens at a phrase boundary or in the middle of a line, is a compositional decision about where you want the strong intervals to land.\n#Where it shows up, and where the idea fails\nFugue is the form that lives on invertible counterpoint. A subject and its countersubject are written so that they can be combined in either order, at the octave or the twelfth, in whatever key the middle entries require; the exposition’s whole architecture depends on that property holding.\nDevelopment sections use the same mechanism on a larger scale. Haydn and Beethoven frequently take two ideas from an exposition and combine them in invertible counterpoint, which is a way of proving that the two themes belong to the same piece. The coda of the finale of Mozart’s Symphony No. 41 does this with several distinct figures at once, stacked so that every voice is audible and none of them cancels another.\nTwo misunderstandings are worth clearing up. First, imitation is not the same as sequence. A sequence repeats a motive in one voice at successively higher or lower pitch levels; imitation requires a second voice to take up the idea, and the two devices have different effects even when they use the same motive. Second, inversion in this sense has nothing to do with chord inversion or with melodic inversion by mirroring. Invertible counterpoint means swapping registers, not flipping intervals.\nThe deeper caution is about writing invertible counterpoint where it will never be inverted. The technique imposes real costs, in restricted intervals and careful register planning, and paying those costs for a passage that will never trade places is wasted effort. It is worth doing when the material is going to combine in several orders, and it is worth doing as an exercise because it improves how you hear the independence of two lines, and for no other reason.\n#Working with it\nWrite the two lines as one piece, not as a melody and an accompaniment, then test the flip before you like either version. If the lower voice’s line sounds like an accompaniment when it moves on top, it was an accompaniment. Play both versions on the virtual piano, and check the interval at every arrival rather than checking once at the end.\nThen set yourself a problem that forces the issue: take an eight-note idea, write a line against it, invert the pair at the octave, and repair whatever broke. Most first attempts break in the same two places: a fifth that inverts to a fourth, and a climax in the upper voice that becomes a high note buried in the middle of the texture when the parts swap. Both are register problems, and both are solved the same way, by deciding where the two lines are allowed to travel before writing a single note of either.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "imitation", "canon", "invertible counterpoint" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/imitation-and-invertible-counterpoint.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/instrument-ranges-for-arrangers/", "url": "https://musics.name/articles/instrument-ranges-for-arrangers/", "title": "Instrument Ranges for Arrangers: Playable, Comfortable, Usable", "summary": "A playable range and a usable register are different facts. Transposition arithmetic, endurance, doubling, and the notes that are another instrument.", "content_html": "Range charts are the first thing an arranger looks up and the last thing that answers the question. A note can be inside the printed range and still be a bad idea: unplayable in context, wrong in character, or a strain the player will pay for on the third rehearsal. Three separate facts hide behind the word “range”, and keeping them apart prevents most of the mistakes that make an arrangement sound written rather than played.
\nPlayable range is what a professional can produce on a good day, with the right reed, warm, and a phrase that gives them somewhere to breathe. It is the number in the reference book, and it is the least useful of the three.
\nComfortable tessitura is where the instrument sounds like itself and a player can stay for a whole movement without fatigue. This is where most of the material belongs. An arrangement that lives at the extremes of the comfortable range is already asking for trouble, even when every note is legal.
\nUsable register is the character question. A written C6 on a trumpet is within the instrument’s range and is not the same instrument as the same trumpet playing C4: it is louder, harder, brighter, and physically expensive. Writing for the top of a range changes the orchestration whether or not the arranger intended it, because the register decides how a line balances against everything else.
\n\nThe same four notes two octaves apart. On a real instrument the difference is larger than pitch: the upper statement will project through an ensemble and the lower one will not, so the same melodic idea placed in a different register becomes a different orchestral decision.
\nMost of the confusion in arranging for transposing instruments comes from treating the transposition as a fact to remember rather than a subtraction to perform.
\n| Notation | Instruments | Sounds relative to written | To hear concert C4, write |
|---|---|---|---|
| Concert pitch | flute, oboe, bassoon, trombone, tuba, violin, viola, cello | unison | C4 |
| B flat | clarinet in Bb, trumpet in Bb | major second lower | D4 |
| B flat, octave | tenor saxophone, bass clarinet | major ninth lower | D5 |
| E flat | alto saxophone, alto clarinet | major sixth lower | A4 |
| F | horn in F, cor anglais | perfect fifth lower | G4 |
| Octave up | piccolo | an octave higher | C3 |
| Octave down | double bass, guitar, contrabassoon | an octave lower | C5 |
The practical method is to write everything at concert pitch first, decide the music, and transpose at the end, because transposing while composing makes it easy to lose track of the sounding intervals. Two failure modes are worth naming. The first is the accidental unison: a part written for Bb trumpet and a part written for Eb alto sax, both notated in comfortable treble-clef territory, can end up sounding a sixth apart in a register where one of them is fighting the other. The second is the octave error in the low instruments, where a double bass or guitar part written at the pitch you want to hear will sound an octave below it.
\nBrass players lose range as they tire. A high passage that works at the start of a rehearsal may not be playable at the end of a two-hour session, so the top of the trumpet’s range should be treated as a budget rather than a location: a few notes, at a climax, in a piece that is not already demanding.
\nWoodwind limits are usually about air. The oboe has almost no reserve, which makes long sustained lines at a low dynamic genuinely hard, and the flute’s low register is quiet and consumes more air than the player’s phrase length assumes. Endurance for woodwinds is also phrase-shaped: writing continuous lines for four minutes without a rest is a request no breath mark can fix.
\nStrings pay in bow and in intonation. A long sustained note at a pianissimo dynamic requires slow bow management and will still have a direction the composer did not ask for; repeated high notes on a violin or a horn expose intonation and tire the hand. The double bass’s top register is a specialist area, and the cello above A4 changes into a tense, thin voice, which is a colour and not a substitute for a violin.
\nTwo notation traps also live here. Hand-stopping on the horn lowers the sounding pitch — by about a semitone when the bell is fully closed, which is why the player fingers a note a semitone higher to compensate — and changes the tone, so a composer who wants the stopped sound must know what it does to the note. Mutes are not cosmetic: they alter volume, timbre, and in the brass often the pitch centre, and a passage that switches mutes mid-phrase needs time in rehearsal that a page of text cannot grant.
\nDoubling is a decision about character, not about volume. Two unlike instruments playing the same note in the same octave do not simply get louder; they produce a composite timbre in which each instrument’s weaknesses are audible as colour. A flute and a violin in unison sound like neither, and the result usually carries a slight roughness because their attacks and vibrato differ.
\nWhat doubling is good at:
\nWhat it is bad at:
\n\nAn octave, then the same octave with a second octave above it. Adding the upper octave makes the pitch clearer, not just louder, which is why octave doubling is the safest way to reinforce a bass line. Adding the same pitch twice in the same octave would only make it thicker.
\nDoubling also creates register collisions. Two instruments written in the same sounding octave, each in the comfortable part of its own range, will balance in ways the score does not show, and the outcome depends on how the players hear each other in the room. When in doubt, separate the doublings by an octave and use register, not force, to solve the balance problem.
\nWritten ranges below are the practical ones, not the extreme claims of an instrument catalogue. Where an instrument is rarely played at the top of its range, that is stated.
\n| Instrument | Written range (practical) | Sounds | Transposition | Warning |
|---|---|---|---|---|
| Flute | C4 to C7 | as written | none | The top octave is loud and hard to control quietly; the low register is soft and air-hungry |
| Piccolo | D5 to about C8 | an octave higher | +8va | Cuts through an entire orchestra; use as a colour, not a melody line that must be soft |
| Oboe | Bb3 to A6 | as written | none | No air reserve; long phrases and rapid repeated notes are harder than they look |
| Clarinet in Bb | E3 to G6 | major second lower | Bb | The throat register around Bb3 to E4 is uneven in tone; the written top demands a strong player |
| Bass clarinet in Bb | Eb3 to C6 | major ninth lower | Bb | Its reason to exist is the low register; the altissimo is a specialist effect |
| Bassoon | Bb1 to E5, G5 in expert hands | as written | none | Reads tenor clef above about Bb3; the top is thin and reed-dependent |
| Contrabassoon | Bb1 to about Bb4 | an octave lower | -8vb | Slow response in the low register; a foundation, not a moving bass line |
| Horn in F | Bb2 to F5, C6 in expert hands | perfect fifth lower | F | The top of the range is a reputation held in public; hand-stopping alters pitch and colour |
| Trumpet in Bb | F#3 to C6 | major second lower | Bb | Above C6 the sound changes and the cost rises; high passages are an endurance budget |
| Tenor trombone | E2 to Bb4, F5 in expert hands | as written | none | Loud and direct; glissando is available and the slide makes wide leaps talkative |
| Tuba | D1 to F4 | as written | none | Very heavy; fast low passages turn into pitch fog |
| Alto saxophone in Eb | Bb3 to F#6 | major sixth lower | Eb | Written “low” territory sounds an octave above a bass line; altissimo above F#6 is an effect |
| Tenor saxophone in Bb | Bb3 to F#6 | major ninth lower | Bb | Sounds an octave below the alto’s written line; the same written note is a different job |
| Violin | G3 to E7, practical to A6 | as written | none | The top string at high dynamics dominates any ensemble; sustained high writing tires the hand |
| Viola | C3 to E6 | as written | none | Reads alto clef; the lower strings are covered and cannot be forced to project |
| Cello | C2 to G5, practical to E5 | as written | none | Bass and tenor clefs; above A4 it is a tense colour, not a violin |
| Double bass | E2 to G4, C5 in specialist hands | an octave lower | -8vb | Sounds an octave below written; the top register is rarely idiomatic |
| Guitar | E3 to E6 | an octave lower | -8vb | Written an octave above sounding; chord shapes do not survive transposition to other instruments |
The method that prevents most range problems is to write the line, then sing it at the pitch you intend, in the register you intend. A melody that is comfortable in your head is often sitting exactly where the instrument has no voice, and the sing-through finds it before the rehearsal does.
\nAfter that, check three things on every part: the extremes against the tessitura column rather than the range column, the transposition against the sounding column, and the low register of the whole arrangement against the bass line’s clarity. If the foundation is unclear, nothing above it will fix that, and if the top of a part is written at a player’s limit for more than a few bars, the arranger has borrowed a favour that a performance will have to repay.
", "content_text": "Range charts are the first thing an arranger looks up and the last thing that answers the question. A note can be inside the printed range and still be a bad idea: unplayable in context, wrong in character, or a strain the player will pay for on the third rehearsal. Three separate facts hide behind the word “range”, and keeping them apart prevents most of the mistakes that make an arrangement sound written rather than played.\n#Playable, comfortable, usable\nPlayable range is what a professional can produce on a good day, with the right reed, warm, and a phrase that gives them somewhere to breathe. It is the number in the reference book, and it is the least useful of the three.\nComfortable tessitura is where the instrument sounds like itself and a player can stay for a whole movement without fatigue. This is where most of the material belongs. An arrangement that lives at the extremes of the comfortable range is already asking for trouble, even when every note is legal.\nUsable register is the character question. A written C6 on a trumpet is within the instrument’s range and is not the same instrument as the same trumpet playing C4: it is louder, harder, brighter, and physically expensive. Writing for the top of a range changes the orchestration whether or not the arranger intended it, because the register decides how a line balances against everything else.\nC4 D4 E4 F4 C6 D6 E6 F6\nThe same four notes two octaves apart. On a real instrument the difference is larger than pitch: the upper statement will project through an ensemble and the lower one will not, so the same melodic idea placed in a different register becomes a different orchestral decision.\n#Transposition as arithmetic\nMost of the confusion in arranging for transposing instruments comes from treating the transposition as a fact to remember rather than a subtraction to perform.\nNotationInstrumentsSounds relative to writtenTo hear concert C4, writeConcert pitchflute, oboe, bassoon, trombone, tuba, violin, viola, cellounisonC4B flatclarinet in Bb, trumpet in Bbmajor second lowerD4B flat, octavetenor saxophone, bass clarinetmajor ninth lowerD5E flatalto saxophone, alto clarinetmajor sixth lowerA4Fhorn in F, cor anglaisperfect fifth lowerG4Octave uppiccoloan octave higherC3Octave downdouble bass, guitar, contrabassoonan octave lowerC5\nThe practical method is to write everything at concert pitch first, decide the music, and transpose at the end, because transposing while composing makes it easy to lose track of the sounding intervals. Two failure modes are worth naming. The first is the accidental unison: a part written for Bb trumpet and a part written for Eb alto sax, both notated in comfortable treble-clef territory, can end up sounding a sixth apart in a register where one of them is fighting the other. The second is the octave error in the low instruments, where a double bass or guitar part written at the pitch you want to hear will sound an octave below it.\n#Endurance, breath, and other limits that do not appear on the page\nBrass players lose range as they tire. A high passage that works at the start of a rehearsal may not be playable at the end of a two-hour session, so the top of the trumpet’s range should be treated as a budget rather than a location: a few notes, at a climax, in a piece that is not already demanding.\nWoodwind limits are usually about air. The oboe has almost no reserve, which makes long sustained lines at a low dynamic genuinely hard, and the flute’s low register is quiet and consumes more air than the player’s phrase length assumes. Endurance for woodwinds is also phrase-shaped: writing continuous lines for four minutes without a rest is a request no breath mark can fix.\nStrings pay in bow and in intonation. A long sustained note at a pianissimo dynamic requires slow bow management and will still have a direction the composer did not ask for; repeated high notes on a violin or a horn expose intonation and tire the hand. The double bass’s top register is a specialist area, and the cello above A4 changes into a tense, thin voice, which is a colour and not a substitute for a violin.\nTwo notation traps also live here. Hand-stopping on the horn lowers the sounding pitch — by about a semitone when the bell is fully closed, which is why the player fingers a note a semitone higher to compensate — and changes the tone, so a composer who wants the stopped sound must know what it does to the note. Mutes are not cosmetic: they alter volume, timbre, and in the brass often the pitch centre, and a passage that switches mutes mid-phrase needs time in rehearsal that a page of text cannot grant.\n#Doubling decisions\nDoubling is a decision about character, not about volume. Two unlike instruments playing the same note in the same octave do not simply get louder; they produce a composite timbre in which each instrument’s weaknesses are audible as colour. A flute and a violin in unison sound like neither, and the result usually carries a slight roughness because their attacks and vibrato differ.\nWhat doubling is good at:\nOctave doubling adds weight and makes a line easier to hear without thickening the same frequency band. It is the most reliable doubling technique there is.Unison doubling between close relatives, such as two clarinets, thickens a line without changing its identity, which is useful for the middle of a texture.Doubling a line at the same pitch with a much quieter instrument is colour, not power: it works when the quieter instrument adds an edge the primary instrument lacks, and fails when both are at a similar dynamic.\nWhat it is bad at:\nC3+C4 C3+C4+C5\nAn octave, then the same octave with a second octave above it. Adding the upper octave makes the pitch clearer, not just louder, which is why octave doubling is the safest way to reinforce a bass line. Adding the same pitch twice in the same octave would only make it thicker.\nDoubling also creates register collisions. Two instruments written in the same sounding octave, each in the comfortable part of its own range, will balance in ways the score does not show, and the outcome depends on how the players hear each other in the room. When in doubt, separate the doublings by an octave and use register, not force, to solve the balance problem.\n#The working chart\nWritten ranges below are the practical ones, not the extreme claims of an instrument catalogue. Where an instrument is rarely played at the top of its range, that is stated.\nInstrumentWritten range (practical)SoundsTranspositionWarningFluteC4 to C7as writtennoneThe top octave is loud and hard to control quietly; the low register is soft and air-hungryPiccoloD5 to about C8an octave higher+8vaCuts through an entire orchestra; use as a colour, not a melody line that must be softOboeBb3 to A6as writtennoneNo air reserve; long phrases and rapid repeated notes are harder than they lookClarinet in BbE3 to G6major second lowerBbThe throat register around Bb3 to E4 is uneven in tone; the written top demands a strong playerBass clarinet in BbEb3 to C6major ninth lowerBbIts reason to exist is the low register; the altissimo is a specialist effectBassoonBb1 to E5, G5 in expert handsas writtennoneReads tenor clef above about Bb3; the top is thin and reed-dependentContrabassoonBb1 to about Bb4an octave lower-8vbSlow response in the low register; a foundation, not a moving bass lineHorn in FBb2 to F5, C6 in expert handsperfect fifth lowerFThe top of the range is a reputation held in public; hand-stopping alters pitch and colourTrumpet in BbF#3 to C6major second lowerBbAbove C6 the sound changes and the cost rises; high passages are an endurance budgetTenor tromboneE2 to Bb4, F5 in expert handsas writtennoneLoud and direct; glissando is available and the slide makes wide leaps talkativeTubaD1 to F4as writtennoneVery heavy; fast low passages turn into pitch fogAlto saxophone in EbBb3 to F#6major sixth lowerEbWritten “low” territory sounds an octave above a bass line; altissimo above F#6 is an effectTenor saxophone in BbBb3 to F#6major ninth lowerBbSounds an octave below the alto’s written line; the same written note is a different jobViolinG3 to E7, practical to A6as writtennoneThe top string at high dynamics dominates any ensemble; sustained high writing tires the handViolaC3 to E6as writtennoneReads alto clef; the lower strings are covered and cannot be forced to projectCelloC2 to G5, practical to E5as writtennoneBass and tenor clefs; above A4 it is a tense colour, not a violinDouble bassE2 to G4, C5 in specialist handsan octave lower-8vbSounds an octave below written; the top register is rarely idiomaticGuitarE3 to E6an octave lower-8vbWritten an octave above sounding; chord shapes do not survive transposition to other instruments\n#How to use this\nThe method that prevents most range problems is to write the line, then sing it at the pitch you intend, in the register you intend. A melody that is comfortable in your head is often sitting exactly where the instrument has no voice, and the sing-through finds it before the rehearsal does.\nAfter that, check three things on every part: the extremes against the tessitura column rather than the range column, the transposition against the sounding column, and the low register of the whole arrangement against the bass line’s clarity. If the foundation is unclear, nothing above it will fix that, and if the top of a part is written at a player’s limit for more than a few bars, the arranger has borrowed a favour that a performance will have to repay.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "instrument ranges", "transposition", "arranging" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/instrument-ranges-for-arrangers.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/just-intonation/", "url": "https://musics.name/articles/just-intonation/", "title": "Just Intonation, Beating and Why Intervals Sound in Tune", "summary": "Pure intervals come from small whole-number ratios, and beating is what stops them sounding pure. What just intonation gives, and what it cannot fix.", "content_html": "“In tune” sounds like a matter of taste and is mostly a physical event. When two tones sound together and their frequencies stand in a simple whole-number ratio, their overtones land on top of one another and the combined sound holds still. When the ratio is not simple, the overtones nearly coincide, and the small difference between them is heard as a slow pulsation. That pulsation is beating, and it is what listeners mean when they say an interval is out of tune.
\nJust intonation is the system in which intervals are defined by small whole-number frequency ratios. The octave is 2:1, the fifth 3:2, the fourth 4:3, the major third 5:4, the minor third 6:5. Each of these ratios produces overtone coincidences: in a 3:2 fifth, every second overtone of the upper note lines up with every third overtone of the lower one, and in a 5:4 third the fifth overtone of the lower note lines up with the fourth overtone of the upper one. The fewer and lower the coincidences, and the more exactly they line up, the more stable the interval sounds.
\nIntervals are usually measured in cents, where one equal-tempered semitone is 100 cents and the octave is 1200. A ratio becomes cents through 1200 times the base-2 logarithm of the ratio, which is worth knowing because it lets the two systems be compared directly: the pure fifth of 3:2 is 702.0 cents, and equal temperament gives the same interval 700.
\n| Interval | Ratio | Just (cents) | Equal temperament | Difference |
|---|---|---|---|---|
| Octave | 2:1 | 1200.0 | 1200.0 | 0 |
| Perfect fifth | 3:2 | 702.0 | 700.0 | +2.0 |
| Perfect fourth | 4:3 | 498.0 | 500.0 | −2.0 |
| Major third | 5:4 | 386.3 | 400.0 | −13.7 |
| Minor third | 6:5 | 315.6 | 300.0 | +15.6 |
| Major sixth | 5:3 | 884.4 | 900.0 | −15.6 |
| Minor sixth | 8:5 | 813.7 | 800.0 | +13.7 |
| Major second | 9:8 | 203.9 | 200.0 | +3.9 |
| Minor second | 16:15 | 111.7 | 100.0 | +11.7 |
| Harmonic seventh | 7:4 | 968.8 | 1000.0 | −31.2 |
| Pythagorean major third | 81:64 | 407.8 | 400.0 | +7.8 |
The table is a summary of the whole argument. The perfect fifth and fourth are nearly the same in both systems, the thirds and sixths differ by roughly 14 to 16 cents, and the harmonic seventh of 7:4 sits more than a quarter of a semitone away from the equal-tempered minor seventh that occupies the same staff position.
\nBeating has a rate, and the rate is the arithmetic that makes tempering decisions for composers.
\nTwo tones whose frequencies differ by one cycle per second produce one beat per second. In a fifth, the coincidence that matters in the middle of the keyboard is between the third overtone of the lower note and the second overtone of the upper one. At A3 to E4, the equal-tempered fifth leaves those two partials about 0.7 cycles per second apart, which is a slow, almost imperceptible undulation. In a major third the relevant coincidence is between the fifth overtone of the lower note and the fourth overtone of the upper one, and those partials are higher and the cents error is larger, so the C4 to E4 third beats at roughly 10 cycles per second. The measurement is easy to reproduce: play the third, count the pulses in a second or two, then play the fifth below the same root.
\n\nThat third is equal-tempered, and it beats. A 5:4 third in the same register has no beat at all in theory. In practice a real piano string is slightly inharmonic, its overtones sit a little sharp of the exact multiples, so even a pure third beats faintly and tuners stretch the octaves to accommodate the effect. The theoretical case and the instrument’s behaviour differ by a small amount, and the direction of the difference is always the same.
\nThe consequence for any twelve-note tuning is a budget. Narrowing the fifth by 2 cents to reach equal temperament costs three quarters of a beat a second at that register. Widening the major third by 13.7 cents costs about ten. A keyboard that wants still, consonant thirds has to protect the 5:4 interval and let the fifths take the damage, which is what meantone temperaments do.
\nA single fixed set of twelve pitches cannot make every interval pure in every key. The reason is that the same interval can be reached by two different routes that do not agree. Four pure fifths up from C and two octaves down give an E at 407.8 cents, the Pythagorean third. A pure 5:4 third above the same C is 386.3 cents. The difference, 21.5 cents, is the syntonic comma, and no single E can be both. Similarly, twelve pure fifths exceed seven octaves by 23.5 cents, so a chain of fifths cannot be closed at all.
\nSequences that stay pure in performance do not solve the problem, they move it. Three pure major thirds stacked upward fall 41 cents short of an octave, the same 128:125 gap that separates G sharp from A flat. A choir or a string quartet that insists on pure thirds on every chord will drift downward as the phrase continues, and the drift is the comma finding its way back in. Unaccompanied ensembles therefore alternate: pure intervals where the chord is held and exposed, slight adjustments where the music has to arrive somewhere.
\nJust intonation is not a historical curiosity. It is what unaccompanied ensembles do when they are listening carefully. A barbershop quartet’s dominant seventh chord rings because the seventh is sung near 7:4, and the upper partials of the four voices coincide. Play the same chord the way a keyboard would, and the seventh sits high:
\n\nThat A flat is about 31 cents sharper than the harmonic seventh a quartet settles on. The gap is not a stylistic preference. It is what the ratio 7:4 costs when a fixed twelve-note tuning cannot supply it, and the ringing sound of the chord is the audience hearing the difference. The same interval is available on any brass instrument as the seventh partial of the harmonic series, which is why brass players tune chords from the series rather than from a keyboard, and why that partial sounds noticeably flat when played against a piano.
\nString quartets work the same way, with the cello as the reference. The third of a sustained chord is the note most likely to be adjusted, and the leading tone is usually played a little high, because a raised leading tone that resolves upward sounds better when it is close to its destination. In ensemble playing this is a set of habits rather than a theory, and the habit is always the same: find the beat, and slow it down.
\nDrone-based traditions make the reference pitch explicit. Indian classical music fixes the tonic and fifth on the drone and treats the melodic intonation of other degrees as a variable that can be inflected expressively, which is a different use of the same physics. Turkish and Arabic makam practice includes intervals that sit between the equal-tempered minor and major third, often around 350 cents, close to an 11:9 ratio. Indonesian gamelan ensembles are tuned as a set rather than to a standard: each ensemble has its own tuning, and paired instruments are deliberately tuned a few cycles apart so that they produce a shimmering beat. A piano tuner spends an hour removing exactly that effect.
\nComposers have also treated just intonation as a system rather than a resource. Harry Partch built instruments for a forty-three-tone scale and wrote for the intonation he had built, and Ben Johnston wrote string quartets that specify just ratios in the score. Both cases depend on performers who can hear the difference and adjust, which makes the intonation part of the performance rather than a setting of the instrument.
\nFor a performer, the useful skill is measuring beats by ear. Play the interval, listen to the top of the sound, and count the pulses. Then change one note by the smallest amount you can and hear the rate fall. When the pulses slow to nothing you are at the pure interval, and the position of that interval in the chord tells you which note to adjust: the third and the seventh carry the intonation of a chord, and the fifth is forgiving enough that it rarely needs attention.
\nFor a composer or arranger, the practical consequences are about register and voicing. The cents error is fixed, but its audibility is not: the same wide third beats slowly in the bass and quickly in the treble, which is why a major third low in a chord sounds thick rather than sharp, and why arrangers keep thirds out of the bottom of a voicing and let the fifth or the octave carry the foundation. Held final chords sound more settled when the third is high enough to beat quickly and low enough not to scream.
\nA tuner measures the same quantity this article has been describing. The chromatic tuner reports the deviation of a played note from equal temperament in cents, so a fifth reading plus 2 and a third reading minus 14 is the table above, measured live. Hearing the beat and reading the cents are two ways of observing one number.
", "content_text": "“In tune” sounds like a matter of taste and is mostly a physical event. When two tones sound together and their frequencies stand in a simple whole-number ratio, their overtones land on top of one another and the combined sound holds still. When the ratio is not simple, the overtones nearly coincide, and the small difference between them is heard as a slow pulsation. That pulsation is beating, and it is what listeners mean when they say an interval is out of tune.\n#Ratios the ear can check\nJust intonation is the system in which intervals are defined by small whole-number frequency ratios. The octave is 2:1, the fifth 3:2, the fourth 4:3, the major third 5:4, the minor third 6:5. Each of these ratios produces overtone coincidences: in a 3:2 fifth, every second overtone of the upper note lines up with every third overtone of the lower one, and in a 5:4 third the fifth overtone of the lower note lines up with the fourth overtone of the upper one. The fewer and lower the coincidences, and the more exactly they line up, the more stable the interval sounds.\nIntervals are usually measured in cents, where one equal-tempered semitone is 100 cents and the octave is 1200. A ratio becomes cents through 1200 times the base-2 logarithm of the ratio, which is worth knowing because it lets the two systems be compared directly: the pure fifth of 3:2 is 702.0 cents, and equal temperament gives the same interval 700.\nIntervalRatioJust (cents)Equal temperamentDifferenceOctave2:11200.01200.00Perfect fifth3:2702.0700.0+2.0Perfect fourth4:3498.0500.0−2.0Major third5:4386.3400.0−13.7Minor third6:5315.6300.0+15.6Major sixth5:3884.4900.0−15.6Minor sixth8:5813.7800.0+13.7Major second9:8203.9200.0+3.9Minor second16:15111.7100.0+11.7Harmonic seventh7:4968.81000.0−31.2Pythagorean major third81:64407.8400.0+7.8\nThe table is a summary of the whole argument. The perfect fifth and fourth are nearly the same in both systems, the thirds and sixths differ by roughly 14 to 16 cents, and the harmonic seventh of 7:4 sits more than a quarter of a semitone away from the equal-tempered minor seventh that occupies the same staff position.\n#Beating is the mechanism\nBeating has a rate, and the rate is the arithmetic that makes tempering decisions for composers.\nTwo tones whose frequencies differ by one cycle per second produce one beat per second. In a fifth, the coincidence that matters in the middle of the keyboard is between the third overtone of the lower note and the second overtone of the upper one. At A3 to E4, the equal-tempered fifth leaves those two partials about 0.7 cycles per second apart, which is a slow, almost imperceptible undulation. In a major third the relevant coincidence is between the fifth overtone of the lower note and the fourth overtone of the upper one, and those partials are higher and the cents error is larger, so the C4 to E4 third beats at roughly 10 cycles per second. The measurement is easy to reproduce: play the third, count the pulses in a second or two, then play the fifth below the same root.\nC4+E4\nThat third is equal-tempered, and it beats. A 5:4 third in the same register has no beat at all in theory. In practice a real piano string is slightly inharmonic, its overtones sit a little sharp of the exact multiples, so even a pure third beats faintly and tuners stretch the octaves to accommodate the effect. The theoretical case and the instrument’s behaviour differ by a small amount, and the direction of the difference is always the same.\nThe consequence for any twelve-note tuning is a budget. Narrowing the fifth by 2 cents to reach equal temperament costs three quarters of a beat a second at that register. Widening the major third by 13.7 cents costs about ten. A keyboard that wants still, consonant thirds has to protect the 5:4 interval and let the fifths take the damage, which is what meantone temperaments do.\n#What just intonation cannot fix\nA single fixed set of twelve pitches cannot make every interval pure in every key. The reason is that the same interval can be reached by two different routes that do not agree. Four pure fifths up from C and two octaves down give an E at 407.8 cents, the Pythagorean third. A pure 5:4 third above the same C is 386.3 cents. The difference, 21.5 cents, is the syntonic comma, and no single E can be both. Similarly, twelve pure fifths exceed seven octaves by 23.5 cents, so a chain of fifths cannot be closed at all.\nSequences that stay pure in performance do not solve the problem, they move it. Three pure major thirds stacked upward fall 41 cents short of an octave, the same 128:125 gap that separates G sharp from A flat. A choir or a string quartet that insists on pure thirds on every chord will drift downward as the phrase continues, and the drift is the comma finding its way back in. Unaccompanied ensembles therefore alternate: pure intervals where the chord is held and exposed, slight adjustments where the music has to arrive somewhere.\n#Where it survives in practice\nJust intonation is not a historical curiosity. It is what unaccompanied ensembles do when they are listening carefully. A barbershop quartet’s dominant seventh chord rings because the seventh is sung near 7:4, and the upper partials of the four voices coincide. Play the same chord the way a keyboard would, and the seventh sits high:\nBb3+D4+F4+Ab4\nThat A flat is about 31 cents sharper than the harmonic seventh a quartet settles on. The gap is not a stylistic preference. It is what the ratio 7:4 costs when a fixed twelve-note tuning cannot supply it, and the ringing sound of the chord is the audience hearing the difference. The same interval is available on any brass instrument as the seventh partial of the harmonic series, which is why brass players tune chords from the series rather than from a keyboard, and why that partial sounds noticeably flat when played against a piano.\nString quartets work the same way, with the cello as the reference. The third of a sustained chord is the note most likely to be adjusted, and the leading tone is usually played a little high, because a raised leading tone that resolves upward sounds better when it is close to its destination. In ensemble playing this is a set of habits rather than a theory, and the habit is always the same: find the beat, and slow it down.\nDrone-based traditions make the reference pitch explicit. Indian classical music fixes the tonic and fifth on the drone and treats the melodic intonation of other degrees as a variable that can be inflected expressively, which is a different use of the same physics. Turkish and Arabic makam practice includes intervals that sit between the equal-tempered minor and major third, often around 350 cents, close to an 11:9 ratio. Indonesian gamelan ensembles are tuned as a set rather than to a standard: each ensemble has its own tuning, and paired instruments are deliberately tuned a few cycles apart so that they produce a shimmering beat. A piano tuner spends an hour removing exactly that effect.\nComposers have also treated just intonation as a system rather than a resource. Harry Partch built instruments for a forty-three-tone scale and wrote for the intonation he had built, and Ben Johnston wrote string quartets that specify just ratios in the score. Both cases depend on performers who can hear the difference and adjust, which makes the intonation part of the performance rather than a setting of the instrument.\n#Working with it\nFor a performer, the useful skill is measuring beats by ear. Play the interval, listen to the top of the sound, and count the pulses. Then change one note by the smallest amount you can and hear the rate fall. When the pulses slow to nothing you are at the pure interval, and the position of that interval in the chord tells you which note to adjust: the third and the seventh carry the intonation of a chord, and the fifth is forgiving enough that it rarely needs attention.\nFor a composer or arranger, the practical consequences are about register and voicing. The cents error is fixed, but its audibility is not: the same wide third beats slowly in the bass and quickly in the treble, which is why a major third low in a chord sounds thick rather than sharp, and why arrangers keep thirds out of the bottom of a voicing and let the fifth or the octave carry the foundation. Held final chords sound more settled when the third is high enough to beat quickly and low enough not to scream.\nA tuner measures the same quantity this article has been describing. The chromatic tuner reports the deviation of a played note from equal temperament in cents, so a fifth reading plus 2 and a third reading minus 14 is the table above, measured live. Hearing the beat and reading the cents are two ways of observing one number.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "just intonation", "intervals", "acoustics" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/just-intonation.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/key-signatures-order-of-sharps-and-flats/", "url": "https://musics.name/articles/key-signatures-order-of-sharps-and-flats/", "title": "Key Signatures and the Order of Sharps and Flats", "summary": "Sharps arrive in the order F C G D A E B and flats in reverse. The order is the circle of fifths written out, one signature at a time.", "content_html": "The accidentals in a key signature never appear in an arbitrary order. Sharps are added one at a time as F, C, G, D, A, E, B. Flats are added as B, E, A, D, G, C, F. Both lists are the circle of fifths spelled out, and the reason they are constant is that each key inherits the previous key’s alterations and adds exactly one more.
\nStart from C major, which needs no accidentals. Move the tonic up a perfect fifth to G, and the scale requires one alteration: the seventh degree must be raised to F♯ so that it sits a semitone below the tonic. That is the one sharp in the signature.
\nMove up another fifth to D major and the same requirement appears again. F♯ stays, because it is still the third of the key, and C♯ joins it as the new leading tone. Every further step up a fifth keeps the existing sharps and adds the next one in the cycle. A major has F♯, C♯ and G♯; E major adds D♯; B major adds A♯.
\nFlats run the same ladder downhill. Moving the tonic up a fourth, which is the same as moving it down a fifth, adds a flat, and the new flat is always the fourth degree of the key. F major takes B♭, B♭ major keeps B♭ and adds E♭, E♭ major adds A♭.
\nBecause the two ladders meet, most keys have two names. All fifteen signatures, from seven flats to seven sharps, are listed below with their relative minors.
\n| Signature | Major key | Accidentals, in the order they appear | Relative minor |
|---|---|---|---|
| none | C major | none | A minor |
| 1 sharp | G major | F♯ | E minor |
| 2 sharps | D major | F♯ C♯ | B minor |
| 3 sharps | A major | F♯ C♯ G♯ | F♯ minor |
| 4 sharps | E major | F♯ C♯ G♯ D♯ | C♯ minor |
| 5 sharps | B major | F♯ C♯ G♯ D♯ A♯ | G♯ minor |
| 6 sharps | F♯ major | F♯ C♯ G♯ D♯ A♯ E♯ | D♯ minor |
| 7 sharps | C♯ major | F♯ C♯ G♯ D♯ A♯ E♯ B♯ | A♯ minor |
| 1 flat | F major | B♭ | D minor |
| 2 flats | B♭ major | B♭ E♭ | G minor |
| 3 flats | E♭ major | B♭ E♭ A♭ | C minor |
| 4 flats | A♭ major | B♭ E♭ A♭ D♭ | F minor |
| 5 flats | D♭ major | B♭ E♭ A♭ D♭ G♭ | B♭ minor |
| 6 flats | G♭ major | B♭ E♭ A♭ D♭ G♭ C♭ | E♭ minor |
| 7 flats | C♭ major | B♭ E♭ A♭ D♭ G♭ C♭ F♭ | A♭ minor |
Two rules replace counting accidentals.
\nWith sharps, the last sharp in the signature is the leading tone, so the tonic is a semitone above it. Three sharps are F♯, C♯ and G♯, the last of which is G♯, and a semitone above G♯ is A: A major. Six sharps end on E♯, and a semitone above E♯ is F♯: F♯ major.
\nWith flats, the last flat is the fourth degree of the key, so the tonic is a fourth below it, which is the same as a fifth above it. Two flats are B♭ and E♭, and a fourth below E♭ is B♭: B♭ major. One flat does not break the pattern, though it does defeat the shortcut of counting backwards from the end; the single flat is the fourth degree, and a fourth below B♭ is F, so one flat means F major.
\nTo move from a major key to its relative minor, count down a minor third from the tonic. The signature is unchanged, so A major with three sharps is also F♯ minor, and B♭ major with two flats is also G minor.
\n\n\nPlaying a scale from the signature is the check that a reader actually knows the key rather than recognising a shape. The two examples above begin a D major and an F major scale; the first prints two sharps, the second a single flat.
\nF♯ major with six sharps and G♭ major with six flats contain the same twelve pitches in the same order. B major with five sharps and C♭ major with seven flats are also the same sounds, as are C♯ major with seven sharps and D♭ major with five flats.
\nBoth spellings exist because a major scale uses each letter name exactly once. D♭ major needs a seventh degree that is a C of some kind, so it is written C♭ rather than B♮; spelling that degree as B would leave the scale with two B’s and no C, and the notation would stop showing the interval structure it exists to show. Spelling is also how the notation records function: in a passage built around the dominant of F♯ major, the altered note is derived from C♯, so it belongs to a C letter name, not a D♭.
\nPublishers normally prefer the spelling with fewer accidentals in the signature, which is why B major is far more common than C♭ major and D♭ major more common than C♯ major. Between six sharps and six flats there is no arithmetic advantage, and the choice follows the harmony and the instruments: a passage that modulates to the subdominant of F♯ major moves to B major and drops a sharp, while the same music written in G♭ major moves to C♭ major and adds a flat.
\nThe signature of a minor key is the signature of its relative major. A minor and C major have none; E minor and G major have one sharp; D minor and F major have one flat. Nothing in the signature tells you that the piece is minor. What tells you is the harmony, the first and last bass notes, and the fact that the seventh degree is raised often enough to establish a leading tone.
\nRaised degrees in minor are written as accidentals in the music, never in the signature. A natural minor contains G♮; A harmonic minor raises it to G♯ wherever the dominant harmony appears, and those G♯s are written measure by measure because the signature says nothing about them.
\n\n\nThe difference between those two examples is one semitone and an entire harmonic function. The raised seventh is what turns a mode into a key with a dominant.
\nAn accidental lasts until the barline, in that octave only. A sharp printed on F4 applies to that F for the rest of the measure. It does not apply to F5, and it does not survive the barline. Accidentals are per octave and per measure, which is why a chord spanning two octaves sometimes needs the same accidental printed twice.
\nCourtesy accidentals are editorial. Publishers print cautionary naturals and sharps in parentheses, or without parentheses, to help the reader after a chromatic passage. Their presence and their formatting vary between editions, so their absence never proves that a note is unaltered. Read the signature and the previous accidentals in the measure, not the parentheses.
\nA note tied across a barline continues to sound at its altered pitch without a new accidental, because it is the same sounding note continued, not a new one.
\nIn music written before functional tonality settled, and in a great deal of music written after it dissolved, a signature reports which notes are inflected rather than which key the piece is in. A Renaissance piece with one flat is not necessarily in F major or D minor; if it is Dorian, the flat is there because Dorian has a low sixth degree, and the final, the cadences and the range of the voices are what identify the mode. Reading a modal piece as if its signature named a key produces analyses that fight the music.
\nThe same caution applies to modern scores that carry no signature at all while remaining tonal: no signature means no systematic alteration, not the absence of a key.
\nA modern keyboard hides its compromise. Every key is equally in tune, which also means every key is equally out of tune, and a major third in C major is exactly the same size as a major third in F sharp major. Before the nineteenth century the compromise was visible, adjustable and chosen, and the tuning an instrument carried was part of why a piece in E flat major sounded different from a piece in C major.
\nStart with the two intervals the ear trusts without help. The octave is a 2:1 frequency ratio and the fifth is 3:2, and both can be tuned by listening rather than counting. Now stack twelve fifths and compare the result with seven octaves. The fifths overshoot by roughly 23.5 cents, an interval with its own name, the Pythagorean comma. The circle of fifths does not close, and every fixed-pitch tuning is a decision about where to put that error.
\nThe two obvious places to put it are the fifth and the third. Pythagorean tuning keeps all twelve fifths pure and lets the thirds pay: four pure fifths reduced by two octaves give a major third of about 407.8 cents, which is 21.5 cents wider than the pure 5:4 third of 386.3 cents. That gap is the syntonic comma, and a third widened by it sounds restless and rough. Equal temperament does the opposite. It narrows every fifth by 2 cents to 700 cents, and the resulting major third of 400 cents is 13.7 cents wider than pure.
\nThe reason the two errors are not equivalent is beating. When two tones sound together, their overtones collide wherever the frequencies nearly coincide, and the surviving difference is heard as a pulsation. In the octave A3 to E4 an equal-tempered fifth has its coincident partials about 0.7 cycles per second apart. A major third in the same register, such as C4 to E4, has them about 10 cycles per second apart. The fifth is out by 2 cents and barely pulses; the third is out by 13.7 cents and pulses ten times a second, on higher partials where the ear is more sensitive. Any temperament that wants stable triads has to spend its error on the fifth, not the third.
\nThe tuning that spends it in exactly that way is meantone, and the name describes the whole tone. Two whole tones were in circulation: the large one of 9:8, about 203.9 cents, and the small one of 10:9, about 182.4. Meantone takes the mean of the two, 193.2 cents, and the simplest way to reach it is to narrow every fifth by exactly a quarter of the syntonic comma. The fifth becomes 696.6 cents, and four of those fifths reduced by two octaves give 386.3 cents: a major third that is pure.
\n\nIn equal temperament, as your browser plays it, that third beats about ten times a second at this register. In quarter-comma meantone it does not beat at all. What the tuning gives up in exchange is visible in the fifth: 696.6 cents is 5.4 cents narrow, so a meantone fifth beats about two and a half times as fast as an equal-tempered one. In a triad that is the right trade, because the third is the interval that decides whether a chord sounds settled, and the fifth can absorb a few cents without anyone noticing.
\nThe price of the trade is the wolf. A keyboard has twelve keys in the octave, and eleven narrowed fifths are enough to connect all twelve of them. The interval that would close the circle is left to absorb everything remaining, and it comes out at about 737.6 cents, some 36 cents wider than a pure fifth. That is the wolf fifth, G sharp up to E flat, and it does not function as a fifth at all: it wobbles, it beats furiously, and no player writes a cadence that needs it.
\nThe wolf is really a spelling problem, and the spellings are the interesting part. On the keyboard the note between G and A is one key, but the tuning has to decide whether it is G sharp or A flat, and the two meanings are not the same pitch. Three pure major thirds, C to E to G sharp to B sharp, fall short of the octave by 41 cents, and that is exactly the distance between G sharp and A flat. The ratio behind it is 128:125.
\n\nYour browser plays those two names as the same pitch, because equal temperament has decided they are. On a meantone keyboard they were 41 cents apart, about two fifths of a semitone, and only one of them could exist on any given instrument.
\nThat single decision sets the working range of the instrument. With the black keys tuned as F sharp, C sharp, G sharp, B flat and E flat, the keys from E flat major round to A major are available with pure or near-pure thirds, and the neighbours on the circle of fifths are not. A flat major needs a pitch the tuning does not have; B major needs D sharp as a separate note from E flat, and does not get it. A so-called wolf also waits for anyone who insists on A flat: the keyboard’s G sharp is 41 cents flatter than the A flat that key needs, and every triad built on it sounds unacceptable.
\nCollecting the sizes side by side, with the wolf in the same frame, shows what each system decided to spend:
\n| Interval | Just intonation | Quarter-comma meantone | Equal temperament |
|---|---|---|---|
| Major third | 386.3 | 386.3 | 400.0 |
| Minor third | 315.6 | 310.3 | 300.0 |
| Perfect fifth | 702.0 | 696.6 | 700.0 |
| Major sixth | 884.4 | 889.7 | 900.0 |
| Whole tone | 203.9 and 182.4 | 193.2 | 200.0 |
| The wolf interval | not applicable | 737.6 | not applicable |
The obvious mechanical answer was to add keys. Split sharps turn the single G sharp or A flat into two levers, giving fourteen or more pitches to the octave, and harpsichords and organs were built that way for the music that needed them. The extreme case is Nicola Vicentino’s archicembalo, described in his treatise of 1555, with thirty-one steps to the octave: the tuning problem converted into an instrument-building problem.
\nMost players took the cheaper route and stayed inside the range their tuning allowed. That is why sixteenth- and seventeenth-century keyboard music clusters around two or three accidentals, and why pieces were transposed to suit the instrument rather than the singer’s preference alone. The limits of a temperament are written into the repertoire.
\nBy the late seventeenth century the requirement had changed. Composers wanted all twenty-four keys available on one keyboard without retuning, and the way to get them was to stop making all the fifths equal. A well temperament distributes the comma unevenly: a few fifths are left almost pure, one or two are narrowed more than the rest, and every key ends up with a usable third. Andreas Werckmeister published schemes of this kind in the 1690s, and others followed, Vallotti’s among them in the next century.
\nThe difference from equal temperament is not the amount of compromise but its distribution. In a well temperament the third in C major can sit within about four cents of pure while the third in a remote key is wider than 20 cents, so two keys sound measurably different in stability and brightness. In equal temperament every major third is 13.7 cents wide, which is better than the worst of a well temperament and worse than the best of it. Equal temperament is not the most accurate tuning on offer; it is the one with no variation, and the variation is what players meant by key colour.
\nThe two books of preludes and fugues in every major and minor key, finished in 1722 and 1742, are evidence that all twenty-four keys were playable on Bach’s keyboard. They are not evidence that all twenty-four sounded alike. Whether the instrument was tuned in a well temperament or in something closer to equal is one of the more durable arguments in the field, and a much-discussed reading from 2005 proposed a specific well temperament decoded from the decoration on the title page. That reading has been contested since, and it does not need to be settled here. The music supports the weaker claim comfortably: in Bach’s keyboard the remote keys work, and they do not sound like the home keys.
\nPeriod-instrument players tune by ear, and the method is the same one this article has been describing. Set the fifths a little narrow so the thirds land where the temperament wants them, count the beats as you go, and listen for the interval that refuses to settle. For late Renaissance and early seventeenth-century keyboard music, quarter-comma meantone is the usual choice; later Baroque repertoire is often played in a sixth- or fifth-comma variant or a well temperament; nineteenth-century music normally wants equal temperament, because that is what it was written on.
\nAny listener can test the arithmetic without an instrument. Play a major third and listen for the pulsation at the top of it, then play the fifth below the same root and notice how much slower the pulsation is. The chromatic tuner on this site displays the same quantity as a cents deviation, and the two readings together are the whole argument: the third is out by more than six times as much as the fifth, and that ratio is why meantone sacrificed one interval to protect the other.
", "content_text": "A modern keyboard hides its compromise. Every key is equally in tune, which also means every key is equally out of tune, and a major third in C major is exactly the same size as a major third in F sharp major. Before the nineteenth century the compromise was visible, adjustable and chosen, and the tuning an instrument carried was part of why a piece in E flat major sounded different from a piece in C major.\n#Twelve pure fifths do not close the circle\nStart with the two intervals the ear trusts without help. The octave is a 2:1 frequency ratio and the fifth is 3:2, and both can be tuned by listening rather than counting. Now stack twelve fifths and compare the result with seven octaves. The fifths overshoot by roughly 23.5 cents, an interval with its own name, the Pythagorean comma. The circle of fifths does not close, and every fixed-pitch tuning is a decision about where to put that error.\nThe two obvious places to put it are the fifth and the third. Pythagorean tuning keeps all twelve fifths pure and lets the thirds pay: four pure fifths reduced by two octaves give a major third of about 407.8 cents, which is 21.5 cents wider than the pure 5:4 third of 386.3 cents. That gap is the syntonic comma, and a third widened by it sounds restless and rough. Equal temperament does the opposite. It narrows every fifth by 2 cents to 700 cents, and the resulting major third of 400 cents is 13.7 cents wider than pure.\nThe reason the two errors are not equivalent is beating. When two tones sound together, their overtones collide wherever the frequencies nearly coincide, and the surviving difference is heard as a pulsation. In the octave A3 to E4 an equal-tempered fifth has its coincident partials about 0.7 cycles per second apart. A major third in the same register, such as C4 to E4, has them about 10 cycles per second apart. The fifth is out by 2 cents and barely pulses; the third is out by 13.7 cents and pulses ten times a second, on higher partials where the ear is more sensitive. Any temperament that wants stable triads has to spend its error on the fifth, not the third.\n#Quarter-comma meantone, and the interval it gives up\nThe tuning that spends it in exactly that way is meantone, and the name describes the whole tone. Two whole tones were in circulation: the large one of 9:8, about 203.9 cents, and the small one of 10:9, about 182.4. Meantone takes the mean of the two, 193.2 cents, and the simplest way to reach it is to narrow every fifth by exactly a quarter of the syntonic comma. The fifth becomes 696.6 cents, and four of those fifths reduced by two octaves give 386.3 cents: a major third that is pure.\nC4+E4\nIn equal temperament, as your browser plays it, that third beats about ten times a second at this register. In quarter-comma meantone it does not beat at all. What the tuning gives up in exchange is visible in the fifth: 696.6 cents is 5.4 cents narrow, so a meantone fifth beats about two and a half times as fast as an equal-tempered one. In a triad that is the right trade, because the third is the interval that decides whether a chord sounds settled, and the fifth can absorb a few cents without anyone noticing.\nThe price of the trade is the wolf. A keyboard has twelve keys in the octave, and eleven narrowed fifths are enough to connect all twelve of them. The interval that would close the circle is left to absorb everything remaining, and it comes out at about 737.6 cents, some 36 cents wider than a pure fifth. That is the wolf fifth, G sharp up to E flat, and it does not function as a fifth at all: it wobbles, it beats furiously, and no player writes a cadence that needs it.\nThe wolf is really a spelling problem, and the spellings are the interesting part. On the keyboard the note between G and A is one key, but the tuning has to decide whether it is G sharp or A flat, and the two meanings are not the same pitch. Three pure major thirds, C to E to G sharp to B sharp, fall short of the octave by 41 cents, and that is exactly the distance between G sharp and A flat. The ratio behind it is 128:125.\nG#3 Ab3\nYour browser plays those two names as the same pitch, because equal temperament has decided they are. On a meantone keyboard they were 41 cents apart, about two fifths of a semitone, and only one of them could exist on any given instrument.\nThat single decision sets the working range of the instrument. With the black keys tuned as F sharp, C sharp, G sharp, B flat and E flat, the keys from E flat major round to A major are available with pure or near-pure thirds, and the neighbours on the circle of fifths are not. A flat major needs a pitch the tuning does not have; B major needs D sharp as a separate note from E flat, and does not get it. A so-called wolf also waits for anyone who insists on A flat: the keyboard’s G sharp is 41 cents flatter than the A flat that key needs, and every triad built on it sounds unacceptable.\nCollecting the sizes side by side, with the wolf in the same frame, shows what each system decided to spend:\nIntervalJust intonationQuarter-comma meantoneEqual temperamentMajor third386.3386.3400.0Minor third315.6310.3300.0Perfect fifth702.0696.6700.0Major sixth884.4889.7900.0Whole tone203.9 and 182.4193.2200.0The wolf intervalnot applicable737.6not applicable\n#Split keys, and how far players went\nThe obvious mechanical answer was to add keys. Split sharps turn the single G sharp or A flat into two levers, giving fourteen or more pitches to the octave, and harpsichords and organs were built that way for the music that needed them. The extreme case is Nicola Vicentino’s archicembalo, described in his treatise of 1555, with thirty-one steps to the octave: the tuning problem converted into an instrument-building problem.\nMost players took the cheaper route and stayed inside the range their tuning allowed. That is why sixteenth- and seventeenth-century keyboard music clusters around two or three accidentals, and why pieces were transposed to suit the instrument rather than the singer’s preference alone. The limits of a temperament are written into the repertoire.\n#Well temperaments: every key usable, no two keys alike\nBy the late seventeenth century the requirement had changed. Composers wanted all twenty-four keys available on one keyboard without retuning, and the way to get them was to stop making all the fifths equal. A well temperament distributes the comma unevenly: a few fifths are left almost pure, one or two are narrowed more than the rest, and every key ends up with a usable third. Andreas Werckmeister published schemes of this kind in the 1690s, and others followed, Vallotti’s among them in the next century.\nThe difference from equal temperament is not the amount of compromise but its distribution. In a well temperament the third in C major can sit within about four cents of pure while the third in a remote key is wider than 20 cents, so two keys sound measurably different in stability and brightness. In equal temperament every major third is 13.7 cents wide, which is better than the worst of a well temperament and worse than the best of it. Equal temperament is not the most accurate tuning on offer; it is the one with no variation, and the variation is what players meant by key colour.\n#What the Well-Tempered Clavier does not settle\nThe two books of preludes and fugues in every major and minor key, finished in 1722 and 1742, are evidence that all twenty-four keys were playable on Bach’s keyboard. They are not evidence that all twenty-four sounded alike. Whether the instrument was tuned in a well temperament or in something closer to equal is one of the more durable arguments in the field, and a much-discussed reading from 2005 proposed a specific well temperament decoded from the decoration on the title page. That reading has been contested since, and it does not need to be settled here. The music supports the weaker claim comfortably: in Bach’s keyboard the remote keys work, and they do not sound like the home keys.\n#Tuning a keyboard today\nPeriod-instrument players tune by ear, and the method is the same one this article has been describing. Set the fifths a little narrow so the thirds land where the temperament wants them, count the beats as you go, and listen for the interval that refuses to settle. For late Renaissance and early seventeenth-century keyboard music, quarter-comma meantone is the usual choice; later Baroque repertoire is often played in a sixth- or fifth-comma variant or a well temperament; nineteenth-century music normally wants equal temperament, because that is what it was written on.\nAny listener can test the arithmetic without an instrument. Play a major third and listen for the pulsation at the top of it, then play the fifth below the same root and notice how much slower the pulsation is. The chromatic tuner on this site displays the same quantity as a cents deviation, and the two readings together are the whole argument: the third is out by more than six times as much as the fifth, and that ratio is why meantone sacrificed one interval to protect the other.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "temperament", "tuning", "keyboard" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/meantone-and-well-temperament.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/meter-and-hypermeter/", "url": "https://musics.name/articles/meter-and-hypermeter/", "title": "Meter, Beat Hierarchy and Hypermeter in Performance", "summary": "Meter is a pattern of strong and weak positions, not a time signature: how it is built, how it stretches past the bar, and where performance contradicts it.", "content_html": "Three words get used as if they meant the same thing in rehearsal: meter, time signature, beat. They are related and they are not the same, and the difference shows up exactly where the music becomes interesting, in syncopation, hemiola, and phrase endings that fall the wrong way round the barline.
\nMeter is a pattern of strong and weak positions that a listener infers from a rhythm. A time signature is a written instruction recording how many beats a bar contains and which note value carries the beat. The beat is the pulse a listener taps.
\nThe relationship runs one way. Notation records a meter; it does not create one. Given almost any sequence of durations, a listener will impose a metrical pattern, and that pattern follows a few strong preferences: duple grouping over triple, the beginnings of repeated figures over their middles, and the low and long notes over the high and short ones. A composer writing against those preferences is writing against a real force, not a convention.
\nTwo consequences are worth carrying into the practice room. First, the pattern is heard even when nobody accents anything, which is how an ensemble agrees on the meter without discussion. Second, a printed barline and a heard downbeat are different pieces of information. Most of the time they coincide, which is why the exceptions are worth studying.
\nTwo numbers classify any meter: how many beats a bar holds, and how each beat divides. Counting the beats gives duple, triple or quadruple; counting the division gives simple, where each beat splits in two, or compound, where each beat splits in three.
\n| Meter type | Common signatures | Beats per bar | Division of each beat |
|---|---|---|---|
| Simple duple | 2/4, 2/2 | 2 | in two |
| Simple triple | 3/4, 3/8 | 3 | in two |
| Simple quadruple | 4/4, 4/8 | 4 | in two |
| Compound duple | 6/8, 6/4 | 2 | in three |
| Compound triple | 9/8, 9/16 | 3 | in three |
| Compound quadruple | 12/8 | 4 | in three |
| Irregular | 5/8, 7/8, 11/8 | varies | grouped additively |
The table’s point is that the top number alone tells you nothing. 6/8 and 3/4 both contain six eighth notes. In 3/4 those eighths belong to three beats and divide each beat in two; in 6/8 they belong to two beats and divide each beat in three. A waltz is not a slow jig, and a player who counts three in a bar of 6/8 will finish the bar in the right place while placing the weight in the wrong one.
\nIrregular meters are grouped additively, and the grouping has to be decided before anyone plays. 7/8 as 2+2+3 is a different piece from 7/8 as 3+2+2 even when the notes are identical, because the grouping decides where the beats are, which notes are strong, and how a conductor’s hand moves. Composers who want a particular grouping usually print it above the staff, and ensembles that ignore it spend rehearsal time discovering the problem by ear.
\nInside a bar, positions are not equal. In 4/4 the downbeat is the strongest position, beat 3 is next, and beats 2 and 4 are weak. In 3/4 the downbeat leads and the other two answer it. The hierarchy is metrical rather than dynamic: a performer may accent a weak beat and play the downbeat softly, and the meter survives, because the hierarchy is the frame against which accents are measured.
\nSyncopation is what happens when the accents contradict that frame. An onset on a weak position, an accent placed early, or a note sustained across a strong position all displace the accent the meter promises. None of it works without an established meter. A rhythm that is syncopated in one metre is unremarkable in another, and the same figure can be a driving groove or a limp depending on which pattern the listener latched onto first.
\nHemiola is a regrouping of the same total duration. Two bars of 3/4 contain six quarter notes; heard as three groups of two, they become three units stretched across two bars, and the notated downbeat of the second bar loses its weight. The device is common in triple-meter dances and at phrase endings, where the shift makes the final downbeat land harder than the meter alone would make it.
\n\nSix quarter notes under two bars of 3/4. Grouped three and three they are two arpeggios, one per bar. Grouped two and two and two they are three pairs cutting across the barline, which is the hemiola, and both readings use exactly the same notes.
\n\nOne note per beat in 4/4, with the highest note arriving on the weakest position in the bar. That clash between registral weight and metrical weight is one of the plainest ways to keep a phrase open at its fourth beat.
\nMeasures group into larger units that behave like beats. The term comes from mid-twentieth-century writing on phrase rhythm, and the observation behind it is older than the name: a four-bar phrase is heard as a hypermeasure with four hyperbeats, the first of which carries the weight a downbeat carries in a bar.
\nWhat reveals the grouping is the same evidence that reveals a bar’s meter. Harmonic rhythm that changes once per bar makes each bar behave like a beat. Melodic contour and grouping point at the unit. Cadences mark the ends of units. And the last bar of a four-bar group tends to act as a pickup into the next group, with a rising line, an upbeat harmony, or a rhythmic figure that leans forward, so the ear hears the phrase continuing rather than stopping.
\nHypermeter is a strong expectation, which is why irregular phrases register as irregular rather than as neutral. A five-bar phrase is heard as a four-bar unit with an extra bar, and a seven-bar phrase as a four plus a three. Composers use that expectation deliberately, inserting a bar to delay an arrival or extending the penultimate unit so the final cadence arrives a beat later than the ear predicted.
\nTwo distinctions keep the idea honest. Hypermeter is not the same thing as phrase structure: a phrase can begin on a hyperbeat that is not a hyperdownbeat, most obviously when it starts with a bar of pickup. And the heard hyperdownbeat is not the same thing as the notated first bar: a phrase that begins mid-bar shifts the grouping relative to the barlines, and performers who follow the notation mechanically will place the weight in the wrong bars.
\nFour situations account for most of the disagreements, and only one of them is a problem.
\nA hemiola puts a second grouping on top of the notated one for a short span, usually at a cadence. A displaced phrase beginning shifts the hypermetre by a bar. A syncopated groove keeps the notated bar exactly where it is while the accents repeat with a period that does not match it, which is the territory of cross-rhythm. And an additive meter resolves the disagreement in advance: 7/8 printed as 3+2+2 has the grouping built into the notation, so the barline already follows the beats.
\nWhat is left is a reading habit. A barline is a container, not an accent. Counting beats mechanically will place the player in the right bar at the right time while telling them nothing about where the weight goes, and weight is what listeners hear.
\nTempo is not meter. A pulse at 100 beats per minute in 4/4 and a pulse at 50 in 2/2 can describe the same passage, and only one of those descriptions matches the notation. Groove is not meter either: the unequal pairs of a swung eighth line live inside the meter and do not move the beats.
\nDynamics are not meter: a pianissimo downbeat is still a downbeat. Note values are not meter: whole notes fit in 3/4 and sixteenths fit in 4/4, and neither tells you where the strong positions are. Rubato stretches the meter without abolishing it, since a listener follows a slowing beat rather than losing the pattern. Even the metronome is only partly relevant: a click on every beat marks the pulse, but the hierarchy appears only when the click accents certain beats, which is why practising with a click on all four beats and practising with a click on beats 1 and 3 are different exercises.
\nThe test of all of it fits in one practice session. Count the passage out loud while tapping the beats; then tap the phrase grouping instead. When the two taps disagree and you can still hear where the music is going, you are hearing meter and hypermeter as the separate layers they are.
", "content_text": "Three words get used as if they meant the same thing in rehearsal: meter, time signature, beat. They are related and they are not the same, and the difference shows up exactly where the music becomes interesting, in syncopation, hemiola, and phrase endings that fall the wrong way round the barline.\n#Meter, time signature, beat\nMeter is a pattern of strong and weak positions that a listener infers from a rhythm. A time signature is a written instruction recording how many beats a bar contains and which note value carries the beat. The beat is the pulse a listener taps.\nThe relationship runs one way. Notation records a meter; it does not create one. Given almost any sequence of durations, a listener will impose a metrical pattern, and that pattern follows a few strong preferences: duple grouping over triple, the beginnings of repeated figures over their middles, and the low and long notes over the high and short ones. A composer writing against those preferences is writing against a real force, not a convention.\nTwo consequences are worth carrying into the practice room. First, the pattern is heard even when nobody accents anything, which is how an ensemble agrees on the meter without discussion. Second, a printed barline and a heard downbeat are different pieces of information. Most of the time they coincide, which is why the exceptions are worth studying.\n#Simple, compound, irregular\nTwo numbers classify any meter: how many beats a bar holds, and how each beat divides. Counting the beats gives duple, triple or quadruple; counting the division gives simple, where each beat splits in two, or compound, where each beat splits in three.\nMeter typeCommon signaturesBeats per barDivision of each beatSimple duple2/4, 2/22in twoSimple triple3/4, 3/83in twoSimple quadruple4/4, 4/84in twoCompound duple6/8, 6/42in threeCompound triple9/8, 9/163in threeCompound quadruple12/84in threeIrregular5/8, 7/8, 11/8variesgrouped additively\nThe table’s point is that the top number alone tells you nothing. 6/8 and 3/4 both contain six eighth notes. In 3/4 those eighths belong to three beats and divide each beat in two; in 6/8 they belong to two beats and divide each beat in three. A waltz is not a slow jig, and a player who counts three in a bar of 6/8 will finish the bar in the right place while placing the weight in the wrong one.\nIrregular meters are grouped additively, and the grouping has to be decided before anyone plays. 7/8 as 2+2+3 is a different piece from 7/8 as 3+2+2 even when the notes are identical, because the grouping decides where the beats are, which notes are strong, and how a conductor’s hand moves. Composers who want a particular grouping usually print it above the staff, and ensembles that ignore it spend rehearsal time discovering the problem by ear.\n#Beat hierarchy\nInside a bar, positions are not equal. In 4/4 the downbeat is the strongest position, beat 3 is next, and beats 2 and 4 are weak. In 3/4 the downbeat leads and the other two answer it. The hierarchy is metrical rather than dynamic: a performer may accent a weak beat and play the downbeat softly, and the meter survives, because the hierarchy is the frame against which accents are measured.\nSyncopation is what happens when the accents contradict that frame. An onset on a weak position, an accent placed early, or a note sustained across a strong position all displace the accent the meter promises. None of it works without an established meter. A rhythm that is syncopated in one metre is unremarkable in another, and the same figure can be a driving groove or a limp depending on which pattern the listener latched onto first.\nHemiola is a regrouping of the same total duration. Two bars of 3/4 contain six quarter notes; heard as three groups of two, they become three units stretched across two bars, and the notated downbeat of the second bar loses its weight. The device is common in triple-meter dances and at phrase endings, where the shift makes the final downbeat land harder than the meter alone would make it.\nG3 B3 D4 G3 B3 D4\nSix quarter notes under two bars of 3/4. Grouped three and three they are two arpeggios, one per bar. Grouped two and two and two they are three pairs cutting across the barline, which is the hemiola, and both readings use exactly the same notes.\nC4 E4 G4 C5\nOne note per beat in 4/4, with the highest note arriving on the weakest position in the bar. That clash between registral weight and metrical weight is one of the plainest ways to keep a phrase open at its fourth beat.\n#Hypermeter\nMeasures group into larger units that behave like beats. The term comes from mid-twentieth-century writing on phrase rhythm, and the observation behind it is older than the name: a four-bar phrase is heard as a hypermeasure with four hyperbeats, the first of which carries the weight a downbeat carries in a bar.\nWhat reveals the grouping is the same evidence that reveals a bar’s meter. Harmonic rhythm that changes once per bar makes each bar behave like a beat. Melodic contour and grouping point at the unit. Cadences mark the ends of units. And the last bar of a four-bar group tends to act as a pickup into the next group, with a rising line, an upbeat harmony, or a rhythmic figure that leans forward, so the ear hears the phrase continuing rather than stopping.\nHypermeter is a strong expectation, which is why irregular phrases register as irregular rather than as neutral. A five-bar phrase is heard as a four-bar unit with an extra bar, and a seven-bar phrase as a four plus a three. Composers use that expectation deliberately, inserting a bar to delay an arrival or extending the penultimate unit so the final cadence arrives a beat later than the ear predicted.\nTwo distinctions keep the idea honest. Hypermeter is not the same thing as phrase structure: a phrase can begin on a hyperbeat that is not a hyperdownbeat, most obviously when it starts with a bar of pickup. And the heard hyperdownbeat is not the same thing as the notated first bar: a phrase that begins mid-bar shifts the grouping relative to the barlines, and performers who follow the notation mechanically will place the weight in the wrong bars.\n#Where the meter and the barline disagree\nFour situations account for most of the disagreements, and only one of them is a problem.\nA hemiola puts a second grouping on top of the notated one for a short span, usually at a cadence. A displaced phrase beginning shifts the hypermetre by a bar. A syncopated groove keeps the notated bar exactly where it is while the accents repeat with a period that does not match it, which is the territory of cross-rhythm. And an additive meter resolves the disagreement in advance: 7/8 printed as 3+2+2 has the grouping built into the notation, so the barline already follows the beats.\nWhat is left is a reading habit. A barline is a container, not an accent. Counting beats mechanically will place the player in the right bar at the right time while telling them nothing about where the weight goes, and weight is what listeners hear.\n#What meter is not\nTempo is not meter. A pulse at 100 beats per minute in 4/4 and a pulse at 50 in 2/2 can describe the same passage, and only one of those descriptions matches the notation. Groove is not meter either: the unequal pairs of a swung eighth line live inside the meter and do not move the beats.\nDynamics are not meter: a pianissimo downbeat is still a downbeat. Note values are not meter: whole notes fit in 3/4 and sixteenths fit in 4/4, and neither tells you where the strong positions are. Rubato stretches the meter without abolishing it, since a listener follows a slowing beat rather than losing the pattern. Even the metronome is only partly relevant: a click on every beat marks the pulse, but the hierarchy appears only when the click accents certain beats, which is why practising with a click on all four beats and practising with a click on beats 1 and 3 are different exercises.\nThe test of all of it fits in one practice session. Count the passage out loud while tapping the beats; then tap the phrase grouping instead. When the two taps disagree and you can still hear where the music is going, you are hearing meter and hypermeter as the separate layers they are.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "meter", "hypermeter", "phrase structure" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/meter-and-hypermeter.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/modulation-strategies/", "url": "https://musics.name/articles/modulation-strategies/", "title": "Modulation: How a Key Change Is Confirmed", "summary": "A modulation is not a chromatic chord; it is a new key confirmed by cadence. The routes into one, and how to read them off the page.", "content_html": "A piece that ends in G major and began in C major has modulated. The change of key is not a single event on one beat but a process spread over several, and the process has a rule: a key exists when its tonic has been confirmed by a cadence. Everything else, including the chromatic chords that make the change possible, is preparation.
\nThree things have to happen for a modulation to count.
\nThe music has to leave the old key, usually by way of a chord or a chromatic motion that belongs to the new one. The dominant of the new key has to appear, because a key is defined by its dominant as much as by its tonic. And that dominant has to resolve to the new tonic in an arrival the ear accepts as a cadence. An authentic cadence in the new key is the standard confirmation; a half cadence on the new dominant is usually not enough on its own, since a passage can sit on a new dominant for bars without ever committing to the key it implies.
\nThere is a fourth, informal condition: the new key has to outlive the arrival. If the music reaches G major, cadences, and leaves within a measure, the analyst is better off describing a tonicization. The label is a claim about duration and confirmation, and both can be read off the page.
\nThe commonest route is a chord that is diatonic in both keys, called a pivot chord. Two keys a fifth apart share four triads, because the diatonic sets overlap heavily.
\n| Chord | In C major | In G major |
|---|---|---|
| C major | I | IV |
| E minor | iii | vi |
| G major | V | I |
| A minor | vi | ii |
A progression from C major into G major can therefore move through A minor and interpret the same chord twice. In C major the A minor triad is vi, a weak tonic substitute. In G major it is ii, the pre-dominant, and it leads directly to D, which is V in G, and from there to a cadence on G. The pivot chord has done no work by itself; the work is done by the reinterpretation and by the accidental that follows.
\n\nThe F sharp in the fourth chord is the new leading tone, and it is the audible marker of the change, since the key signature of C major does not contain it. In a classical score this moment is where the modulation becomes visible: a persistent accidental, a new dominant, then a cadence.
\nTwo practical points follow. The first is that a chord which functions as the dominant in the old key makes a poor pivot, whatever its diatonic status in the new one. Using G major as the pivot out of C major keeps both feet in C, because G is the old dominant and the ear has already heard it as such. The second is that the pivot does not decide the key. Aim the progression at A minor instead of G and the same pivot leads to a completely different destination, so the decision is made by the cadence at the end of the phrase, not by the chord in the middle.
\nA modulation with no pivot at all is called direct, and it works because the ear accepts a change of key at a boundary. The phrase ends in one key and the next phrase begins in another, with no shared chord between them. The join usually falls at a cadence or a rest, which is why the device is common in song and in popular music, where a verse can end in one key and a chorus begin in another without any harmonic bridge.
\nThe strength of direct modulation is that it resets the ear without argument. Its cost is that the new key sounds like a new section rather than a continuation, and a piece that uses the device too often stops sounding like it has a key at all. A tonality needs time to be established, and every repeated jump spends some of the listener’s attention on the mechanics.
\nChromatic third relations belong to the same family. Moving directly from C major to A flat major, or from C major to E major, changes the tonic by a third and leaves the ear without a functional connection to hold on to. Hungarian and Russian repertoire of the nineteenth century, and a great deal of film music afterwards, uses these shifts deliberately for exactly that effect.
\nThree chromatic devices deserve separate mention because they allow a modulation into a key that the pivot table cannot reach.
\nThe first is the secondary dominant used as an entrance. A chord that tonicizes a scale degree in the old key can also be the dominant of the new one, and the same chord then serves two functions in sequence. A D major triad in C major is V/V; if the music instead follows it with a cadence in G, it was V of G all along. Nothing about the chord changes; only the resolution tells you which reading was correct.
\nThe second is the diminished seventh. The chord is symmetric, built from stacked minor thirds, and it repeats at the octave every three semitones. The four pitches B, D, F and A flat spell the leading-tone seventh of C minor, and the same four pitches, respelled, spell the leading-tone seventh of E flat minor, G flat minor and A minor. A composer who wants to land in a key a minor third away therefore needs no pivot in the ordinary sense: the same chord serves, and the destination is chosen by what follows it. This is one reason the chord appears so often at moments of harmonic surprise.
\nThe third is enharmonic reinterpretation at the augmented sixth. The German sixth in C, spelled A flat, C, E flat, F sharp, contains the same four pitches as an A flat dominant seventh when F sharp is written as G flat, and an A flat dominant seventh resolves to D flat, not to C.
\n\nThe notation records the intention, and the two resolutions are a semitone apart in opposite directions. A modulation may pass through such a chord without any change of spelling in performance, because the player reads pitch and the analyst reads function.
\nKeys are related by distance on the circle of fifths, and the distance predicts how easily a modulation will be heard. From C major, G major and F major are one step away and differ by one accidental each; A minor shares the signature entirely; D minor and E minor are one step away plus the relative shift. These are the modulations that need the least preparation. Keys two or three steps away, such as D major or A major from C, need a pivot sequence or a chromatic chord to be convincing.
\nBeyond that, the relationship is usually described by interval rather than by circle: to the third, to the sixth, to the flattened mediant. These are the relationships that colour a large structure rather than a phrase, and they are worth planning before the notes are written. A movement that begins in C and ends in E flat has told the listener something before the first cadence arrives.
\nFor the transitions themselves, the practical order of work is the destination first. Choose the key, then choose the cadence that will confirm it, then decide which of the chords before that cadence can be shared with the old key. Working backwards from the confirmation keeps the pivot from becoming the point of the process.
\nFour signs, taken together, settle the question.
\nTrack accidentals and see whether they persist. A single chromatic note that resolves in the next chord is a tonicization; a chromatic note that becomes part of the prevailing scale is a new key. Watch for the leading tone of the new key and see where it resolves. Follow the bass: a modulation almost always involves a move to the new dominant, and a cadential approach to it in the bass is the clearest single indicator. Finally, ask whether the old key returns. If it does, the middle section was a departure; if it does not, the piece has changed its tonal centre.
\nThe common analytical error is to read a single chromatic chord as a key change. A D major chord in C major is a strong signal, and it is also entirely consistent with a phrase that stays in C. The secondary dominants discussion covers the case in which the chord resolves and is forgotten. Modulation is what happens when the chord is followed by a cadence that insists on the new key, and the cadence is the only evidence that counts.
\nThe circle of fifths explorer on this site shows the shared diatonic chords of any two keys side by side, which makes the pivot table above a matter of clicking two keys rather than rebuilding the arithmetic.
", "content_text": "A piece that ends in G major and began in C major has modulated. The change of key is not a single event on one beat but a process spread over several, and the process has a rule: a key exists when its tonic has been confirmed by a cadence. Everything else, including the chromatic chords that make the change possible, is preparation.\n#The confirmation requirement\nThree things have to happen for a modulation to count.\nThe music has to leave the old key, usually by way of a chord or a chromatic motion that belongs to the new one. The dominant of the new key has to appear, because a key is defined by its dominant as much as by its tonic. And that dominant has to resolve to the new tonic in an arrival the ear accepts as a cadence. An authentic cadence in the new key is the standard confirmation; a half cadence on the new dominant is usually not enough on its own, since a passage can sit on a new dominant for bars without ever committing to the key it implies.\nThere is a fourth, informal condition: the new key has to outlive the arrival. If the music reaches G major, cadences, and leaves within a measure, the analyst is better off describing a tonicization. The label is a claim about duration and confirmation, and both can be read off the page.\n#Pivot-chord modulation\nThe commonest route is a chord that is diatonic in both keys, called a pivot chord. Two keys a fifth apart share four triads, because the diatonic sets overlap heavily.\nChordIn C majorIn G majorC majorIIVE minoriiiviG majorVIA minorviii\nA progression from C major into G major can therefore move through A minor and interpret the same chord twice. In C major the A minor triad is vi, a weak tonic substitute. In G major it is ii, the pre-dominant, and it leads directly to D, which is V in G, and from there to a cadence on G. The pivot chord has done no work by itself; the work is done by the reinterpretation and by the accidental that follows.\nC4+E4+G4 A3+C4+E4 D4+F#4+A4+C5 G3+B3+D4\nThe F sharp in the fourth chord is the new leading tone, and it is the audible marker of the change, since the key signature of C major does not contain it. In a classical score this moment is where the modulation becomes visible: a persistent accidental, a new dominant, then a cadence.\nTwo practical points follow. The first is that a chord which functions as the dominant in the old key makes a poor pivot, whatever its diatonic status in the new one. Using G major as the pivot out of C major keeps both feet in C, because G is the old dominant and the ear has already heard it as such. The second is that the pivot does not decide the key. Aim the progression at A minor instead of G and the same pivot leads to a completely different destination, so the decision is made by the cadence at the end of the phrase, not by the chord in the middle.\n#Direct and phrase modulation\nA modulation with no pivot at all is called direct, and it works because the ear accepts a change of key at a boundary. The phrase ends in one key and the next phrase begins in another, with no shared chord between them. The join usually falls at a cadence or a rest, which is why the device is common in song and in popular music, where a verse can end in one key and a chorus begin in another without any harmonic bridge.\nThe strength of direct modulation is that it resets the ear without argument. Its cost is that the new key sounds like a new section rather than a continuation, and a piece that uses the device too often stops sounding like it has a key at all. A tonality needs time to be established, and every repeated jump spends some of the listener’s attention on the mechanics.\nChromatic third relations belong to the same family. Moving directly from C major to A flat major, or from C major to E major, changes the tonic by a third and leaves the ear without a functional connection to hold on to. Hungarian and Russian repertoire of the nineteenth century, and a great deal of film music afterwards, uses these shifts deliberately for exactly that effect.\n#Chromatic routes\nThree chromatic devices deserve separate mention because they allow a modulation into a key that the pivot table cannot reach.\nThe first is the secondary dominant used as an entrance. A chord that tonicizes a scale degree in the old key can also be the dominant of the new one, and the same chord then serves two functions in sequence. A D major triad in C major is V/V; if the music instead follows it with a cadence in G, it was V of G all along. Nothing about the chord changes; only the resolution tells you which reading was correct.\nThe second is the diminished seventh. The chord is symmetric, built from stacked minor thirds, and it repeats at the octave every three semitones. The four pitches B, D, F and A flat spell the leading-tone seventh of C minor, and the same four pitches, respelled, spell the leading-tone seventh of E flat minor, G flat minor and A minor. A composer who wants to land in a key a minor third away therefore needs no pivot in the ordinary sense: the same chord serves, and the destination is chosen by what follows it. This is one reason the chord appears so often at moments of harmonic surprise.\nThe third is enharmonic reinterpretation at the augmented sixth. The German sixth in C, spelled A flat, C, E flat, F sharp, contains the same four pitches as an A flat dominant seventh when F sharp is written as G flat, and an A flat dominant seventh resolves to D flat, not to C.\nAb3+C4+Eb4+F#4 Db4+F4+Ab4\nThe notation records the intention, and the two resolutions are a semitone apart in opposite directions. A modulation may pass through such a chord without any change of spelling in performance, because the player reads pitch and the analyst reads function.\n#Planning key relationships\nKeys are related by distance on the circle of fifths, and the distance predicts how easily a modulation will be heard. From C major, G major and F major are one step away and differ by one accidental each; A minor shares the signature entirely; D minor and E minor are one step away plus the relative shift. These are the modulations that need the least preparation. Keys two or three steps away, such as D major or A major from C, need a pivot sequence or a chromatic chord to be convincing.\nBeyond that, the relationship is usually described by interval rather than by circle: to the third, to the sixth, to the flattened mediant. These are the relationships that colour a large structure rather than a phrase, and they are worth planning before the notes are written. A movement that begins in C and ends in E flat has told the listener something before the first cadence arrives.\nFor the transitions themselves, the practical order of work is the destination first. Choose the key, then choose the cadence that will confirm it, then decide which of the chords before that cadence can be shared with the old key. Working backwards from the confirmation keeps the pivot from becoming the point of the process.\n#Reading a modulation off the score\nFour signs, taken together, settle the question.\nTrack accidentals and see whether they persist. A single chromatic note that resolves in the next chord is a tonicization; a chromatic note that becomes part of the prevailing scale is a new key. Watch for the leading tone of the new key and see where it resolves. Follow the bass: a modulation almost always involves a move to the new dominant, and a cadential approach to it in the bass is the clearest single indicator. Finally, ask whether the old key returns. If it does, the middle section was a departure; if it does not, the piece has changed its tonal centre.\nThe common analytical error is to read a single chromatic chord as a key change. A D major chord in C major is a strong signal, and it is also entirely consistent with a phrase that stays in C. The secondary dominants discussion covers the case in which the chord resolves and is forgotten. Modulation is what happens when the chord is followed by a cadence that insists on the new key, and the cadence is the only evidence that counts.\nThe circle of fifths explorer on this site shows the shared diatonic chords of any two keys side by side, which makes the pivot table above a matter of clicking two keys rather than rebuilding the arithmetic.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "modulation", "pivot chord", "key relationships" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/modulation-strategies.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/music-after-tonality/", "url": "https://musics.name/articles/music-after-tonality/", "title": "After Tonality: What Composers Replaced It With", "summary": "When chromatic harmony outgrew the tonal hierarchy, composers tried serial rules, rhythm, folk modes, and sound mass. What each method actually solved.", "content_html": "The tonal system was not abandoned because composers got bored of it. It was abandoned, in a handful of works written between roughly 1908 and 1923, because the chromatic language that had been built on top of it could no longer be explained by it. What followed was not one replacement but a series of answers to a specific question: if there is no hierarchy of pitches, what tells a listener that a passage is going somewhere?
\nBy the end of the nineteenth century, Wagner’s harmony was using chords for their colour and their tendency rather than their function, and the pieces still resolved. That is the point. Tristan und Isolde (1859) delays resolution past what any earlier theory could justify, but the delay is only meaningful because a resolution is expected. The tonal hierarchy was not destroyed; it was made to carry more weight than it could hold.
\nSchoenberg’s move around 1908 was to stop pretending. In the Three Piano Pieces op. 11 (1909) and Erwartung op. 17 (1909) the language is called free atonality: no key signature, no tonal centre asserted for long enough to hear, chords chosen for interval content rather than function. It solved the immediate problem of expressing what the late Romantic idiom could no longer reach, and it created a new one. Without a scale, every interval is the same distance from every other interval. Nothing is more dissonant than anything else, so nothing can be more urgent than anything else, so a piece has no way to build an expectation it then denies.
\nThe twelve-tone method, worked out by Schoenberg in the early 1920s and used completely in the Suite for Piano op. 25 (composed 1921 to 1923), answers that question with an ordering rather than a hierarchy. A fixed sequence of the twelve pitch classes governs the work; each of the twelve sounds once before any is repeated; the sequence appears in four forms (prime, inversion, retrograde, and retrograde inversion) and at all twelve transpositions, giving 48 usable forms. The row supplies both the material and the relationship between materials, because the intervals of the row are the same intervals in every form.
\nTake a row built for demonstration:
\n\nAnd the same row in retrograde:
\n\nThe two are the same information reversed, and they do not sound like the same music. That gap is the practical heart of the method. A row is a lexicon, not a theme: the composer still decides which forms appear in which register, in which rhythm, and against which other forms, and those decisions are what a listener actually hears. It is why Webern’s Variations op. 27 (1936) and Berg’s Violin Concerto (1935), both twelve-tone works, sound nothing alike. Berg’s concerto quotes a Bach chorale and lets tonal triads sound inside a serial frame; Webern’s variations reduce the surface to a few intervals and a great deal of silence.
\nThe method spread from pitch into everything else in the early 1950s, when Boulez’s Structures Ia (1952) and Stockhausen’s Kreuzspiel (1951) applied serial ordering to durations, dynamics, and articulation. Total serialism has a reputation for sounding inhuman, which is a fair description of some of it, but the aim was ordinary: to keep all twelve values of each parameter in circulation, the same way the pitch row does.
\n| The problem chromaticism left | The method proposed | Representative works |
|---|---|---|
| No hierarchy to create direction | serial ordering of the twelve pitch classes | Schoenberg, Suite for Piano op. 25 (1921–23); Webern, Variations op. 27 (1936) |
| Pitch hierarchy had become an automatic habit | rhythm and register as primary structure | Stravinsky, The Rite of Spring (1913) |
| A single scale could not carry the material | folk-derived modes and irregular metre | Bartók, Music for Strings, Percussion and Celesta (1936) |
| Chord identity had become irrelevant | mass, density, and register as the material | Varèse, Ionisation (1931) |
| The score could not survive as a fixed object | stochastic process, chance, and tape | Xenakis, Metastaseis (1954); Cage, 4'33\" (1952) |
| Harmony had become a set of conventions disconnected from sound | spectra as the source of harmony | Grisey, Partiels (1975) |
Stravinsky’s answer was to stop asking pitch to do the structural work. In The Rite of Spring (1913) the primary events are accents, changing metres, and register blocks; harmony is often static or repeated while the rhythm moves, which inverts the practice of the previous two centuries. His neoclassicism from Pulcinella (1920) onward took tonal material seriously again, but used it the way a collage uses photographs: quotation, reframing, and irony rather than development. He began using twelve-tone procedures himself in the 1950s, and the move is less surprising than it looks once you notice that both are systems for organising fixed material.
\nBartók derived his scales from the folk music he collected and analysed, which gave him modes, pentatonic fragments, and asymmetrical rhythms outside the common-practice vocabulary. He also built harmony around symmetry: in much of his music, a pitch and its tritone or a chord and its inversion can substitute for one another, which turns a tonal axis into a geometric one.
\n\nAn augmented triad, the simplest symmetrical division of the octave at four-semitone intervals. Around it, transposition and inversion are the same operation, which is exactly the property Bartók’s harmonic language exploits and the reason his chords can stand in for one another without a tonal function to explain it.
\nIves took the opposite route: instead of a new system, layered old ones. In works such as Three Places in New England (written 1903 to 1914) two tonal groups can play in different keys and different tempos at once, and the interest lies in the collision rather than in any resolution of it.
\nVarèse’s premise was that composition concerns masses and densities rather than themes and development. Hyperprism (1923), Intégrales (1925), and Ionisation (1931), the last for percussion ensemble and usually described as the first concert work for percussion alone, are built from blocks of sound that expand, collide, and recede. There is no harmony in the tonal sense because there is nothing to be consonant with.
\nCage’s contribution was to hand the decisions to procedures that did not represent his taste: chance operations, the prepared piano (Sonatas and Interludes, written 1946 to 1948), and the silent 4'33\" (1952), in which the material is whatever the room produces. Xenakis built his masses with mathematics, treating large numbers of individual lines as a single statistical object; Metastaseis (1954) sets sixty-one musicians in glissandi that read as shifting densities, and his later work uses sieves and stochastic distributions to choose pitches.
\nThe spectralists of the 1970s went back to acoustics. Rather than inventing a scale, they analysed the sound of an instrument and derived harmony from its partials, so that consonant relationships are those the ear already finds in a resonant body. Grisey’s Partiels (1975) takes the spectrum of a low trombone note and orchestrates its components; Murail works the same territory with different material. Inharmonic spectra and difference tones give the music a natural roughness-to-smoothness scale that serialism had deliberately refused.
\nThe honest summary is that the twentieth century did not replace tonality. It produced a plural repertoire, and the tonal hierarchy kept its position in film music, jazz, popular music, and the work of composers such as Shostakovich, Britten, and Ravel, who were writing at the same time as the serialists and were no less serious about it.
\nWhat the new methods gained was control over parameters tonality had always taken for granted, and whole vocabularies of timbre that the earlier system had no use for. What they lost was a shared frame of expectation that listeners had already learned, which is why the difficulty of twentieth-century repertoire is not a matter of dissonance but of reference: dissonance is easy to hear, and an interval with no home is harder to hold in memory.
\n\nA cadence in C. Everything about it is predictable, including the fact that you heard where it was going before it arrived. The century after 1908 is largely a record of composers deciding what they were willing to give up in order to stop being predictable, and each of the methods above names a different thing that was given up.
", "content_text": "The tonal system was not abandoned because composers got bored of it. It was abandoned, in a handful of works written between roughly 1908 and 1923, because the chromatic language that had been built on top of it could no longer be explained by it. What followed was not one replacement but a series of answers to a specific question: if there is no hierarchy of pitches, what tells a listener that a passage is going somewhere?\n#The problem chromaticism left behind\nBy the end of the nineteenth century, Wagner’s harmony was using chords for their colour and their tendency rather than their function, and the pieces still resolved. That is the point. Tristan und Isolde (1859) delays resolution past what any earlier theory could justify, but the delay is only meaningful because a resolution is expected. The tonal hierarchy was not destroyed; it was made to carry more weight than it could hold.\nSchoenberg’s move around 1908 was to stop pretending. In the Three Piano Pieces op. 11 (1909) and Erwartung op. 17 (1909) the language is called free atonality: no key signature, no tonal centre asserted for long enough to hear, chords chosen for interval content rather than function. It solved the immediate problem of expressing what the late Romantic idiom could no longer reach, and it created a new one. Without a scale, every interval is the same distance from every other interval. Nothing is more dissonant than anything else, so nothing can be more urgent than anything else, so a piece has no way to build an expectation it then denies.\n#Twelve tones instead of seven\nThe twelve-tone method, worked out by Schoenberg in the early 1920s and used completely in the Suite for Piano op. 25 (composed 1921 to 1923), answers that question with an ordering rather than a hierarchy. A fixed sequence of the twelve pitch classes governs the work; each of the twelve sounds once before any is repeated; the sequence appears in four forms (prime, inversion, retrograde, and retrograde inversion) and at all twelve transpositions, giving 48 usable forms. The row supplies both the material and the relationship between materials, because the intervals of the row are the same intervals in every form.\nTake a row built for demonstration:\nC4 E4 F4 Bb3 D4 G4 Ab4 B3 Eb4 A3 Db4 F#4\nAnd the same row in retrograde:\nF#4 Db4 A3 Eb4 B3 Ab4 G4 D4 Bb3 F4 E4 C4\nThe two are the same information reversed, and they do not sound like the same music. That gap is the practical heart of the method. A row is a lexicon, not a theme: the composer still decides which forms appear in which register, in which rhythm, and against which other forms, and those decisions are what a listener actually hears. It is why Webern’s Variations op. 27 (1936) and Berg’s Violin Concerto (1935), both twelve-tone works, sound nothing alike. Berg’s concerto quotes a Bach chorale and lets tonal triads sound inside a serial frame; Webern’s variations reduce the surface to a few intervals and a great deal of silence.\nThe method spread from pitch into everything else in the early 1950s, when Boulez’s Structures Ia (1952) and Stockhausen’s Kreuzspiel (1951) applied serial ordering to durations, dynamics, and articulation. Total serialism has a reputation for sounding inhuman, which is a fair description of some of it, but the aim was ordinary: to keep all twelve values of each parameter in circulation, the same way the pitch row does.\nThe problem chromaticism leftThe method proposedRepresentative worksNo hierarchy to create directionserial ordering of the twelve pitch classesSchoenberg, Suite for Piano op. 25 (1921–23); Webern, Variations op. 27 (1936)Pitch hierarchy had become an automatic habitrhythm and register as primary structureStravinsky, The Rite of Spring (1913)A single scale could not carry the materialfolk-derived modes and irregular metreBartók, Music for Strings, Percussion and Celesta (1936)Chord identity had become irrelevantmass, density, and register as the materialVarèse, Ionisation (1931)The score could not survive as a fixed objectstochastic process, chance, and tapeXenakis, Metastaseis (1954); Cage, 4'33\" (1952)Harmony had become a set of conventions disconnected from soundspectra as the source of harmonyGrisey, Partiels (1975)\n#Rhythm, folk material, and juxtaposition\nStravinsky’s answer was to stop asking pitch to do the structural work. In The Rite of Spring (1913) the primary events are accents, changing metres, and register blocks; harmony is often static or repeated while the rhythm moves, which inverts the practice of the previous two centuries. His neoclassicism from Pulcinella (1920) onward took tonal material seriously again, but used it the way a collage uses photographs: quotation, reframing, and irony rather than development. He began using twelve-tone procedures himself in the 1950s, and the move is less surprising than it looks once you notice that both are systems for organising fixed material.\nBartók derived his scales from the folk music he collected and analysed, which gave him modes, pentatonic fragments, and asymmetrical rhythms outside the common-practice vocabulary. He also built harmony around symmetry: in much of his music, a pitch and its tritone or a chord and its inversion can substitute for one another, which turns a tonal axis into a geometric one.\nC4+E4+G#4\nAn augmented triad, the simplest symmetrical division of the octave at four-semitone intervals. Around it, transposition and inversion are the same operation, which is exactly the property Bartók’s harmonic language exploits and the reason his chords can stand in for one another without a tonal function to explain it.\nIves took the opposite route: instead of a new system, layered old ones. In works such as Three Places in New England (written 1903 to 1914) two tonal groups can play in different keys and different tempos at once, and the interest lies in the collision rather than in any resolution of it.\n#Sound itself as material\nVarèse’s premise was that composition concerns masses and densities rather than themes and development. Hyperprism (1923), Intégrales (1925), and Ionisation (1931), the last for percussion ensemble and usually described as the first concert work for percussion alone, are built from blocks of sound that expand, collide, and recede. There is no harmony in the tonal sense because there is nothing to be consonant with.\nCage’s contribution was to hand the decisions to procedures that did not represent his taste: chance operations, the prepared piano (Sonatas and Interludes, written 1946 to 1948), and the silent 4'33\" (1952), in which the material is whatever the room produces. Xenakis built his masses with mathematics, treating large numbers of individual lines as a single statistical object; Metastaseis (1954) sets sixty-one musicians in glissandi that read as shifting densities, and his later work uses sieves and stochastic distributions to choose pitches.\nThe spectralists of the 1970s went back to acoustics. Rather than inventing a scale, they analysed the sound of an instrument and derived harmony from its partials, so that consonant relationships are those the ear already finds in a resonant body. Grisey’s Partiels (1975) takes the spectrum of a low trombone note and orchestrates its components; Murail works the same territory with different material. Inharmonic spectra and difference tones give the music a natural roughness-to-smoothness scale that serialism had deliberately refused.\n#What was actually lost, and what never left\nThe honest summary is that the twentieth century did not replace tonality. It produced a plural repertoire, and the tonal hierarchy kept its position in film music, jazz, popular music, and the work of composers such as Shostakovich, Britten, and Ravel, who were writing at the same time as the serialists and were no less serious about it.\nWhat the new methods gained was control over parameters tonality had always taken for granted, and whole vocabularies of timbre that the earlier system had no use for. What they lost was a shared frame of expectation that listeners had already learned, which is why the difficulty of twentieth-century repertoire is not a matter of dissonance but of reference: dissonance is easy to hear, and an interval with no home is harder to hold in memory.\nC4+E4+G4 G3+B3+D4 C4+E4+G4\nA cadence in C. Everything about it is predictable, including the fact that you heard where it was going before it arrived. The century after 1908 is largely a record of composers deciding what they were willing to give up in order to stop being predictable, and each of the methods above names a different thing that was given up.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "atonality", "serialism", "modernism" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/music-after-tonality.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/polyrhythm-and-cross-rhythm/", "url": "https://musics.name/articles/polyrhythm-and-cross-rhythm/", "title": "Polyrhythm and Cross-Rhythm: Counting the Distinction", "summary": "Polyrhythm, cross-rhythm and polymeter are not interchangeable. Counting the ratios shows what separates a divided span from a pattern set against a metre.", "content_html": "A percussionist and a conductor can look at the same bar and disagree about what to call it, because two terms get used as if they were synonyms. They are not. One describes a texture in which two parts divide the same span differently; the other describes a pattern that contradicts the metre it sits in. A passage can be both, and often is, but a passage can also be one without the other.
\nA polyrhythm is a simultaneity. Two or more parts divide the same span of time into different numbers of equal parts and play them together, so the span has two internal grids at once: three against two, four against three, five against four. The parts agree about where the span begins and ends and disagree about everything inside it.
\nA cross-rhythm is a relationship between a pattern and a metre. Onsets or accents fall where the prevailing metre does not expect them, and the conflict is audible whether one part is playing or five. Hemiola in triple metre is the commonest case, and it is described in the article on meter and beat hierarchy: two bars of 3/4 regrouped as three groups of two.
\nThe distinction is that a polyrhythm needs two parts to exist while a cross-rhythm needs only one part and a metre. Writers on rhythm do not all draw the line in the same place, and some prefer to treat cross-rhythm as the general principle with polyrhythm as its special case. The stable part of the distinction is the one worth keeping: a texture against a pattern.
\nThe arithmetic of a polyrhythm is the arithmetic of the lowest common multiple. If one part plays three equal notes across a span and another plays two, both can be plotted on a grid of six, because six is the first number both three and two divide into.
\n| Ratio | Parts | Smallest grid | Onsets on the grid |
|---|---|---|---|
| 3:2 | three against two | 6 | three part at 1, 3, 5; two part at 1, 4 |
| 4:3 | four against three | 12 | four part at 1, 4, 7, 10; three part at 1, 5, 9 |
| 5:4 | five against four | 20 | five part at 1, 5, 9, 13, 17; four part at 1, 6, 11, 16 |
Read the 3:2 row carefully, because it corrects the most common beginner’s error. The two parts meet at the start of the span and nowhere else. There is no mid-point where the hands come back together: on the six-unit grid the three-part has a note on unit 5 while the two-part has already played its last note on unit 4. The composite order a listener hears is therefore together, triplet, duplet, triplet. Anyone waiting for a second coincidence halfway will lose the pattern and blame their hands.
\n\nThe one onset both parts share: the beginning of the cycle. Everything that follows is placed between two of these.
\n\n\nThree evenly spread pitches, then two over the same span. Spread the first group across a bar and the second across the same bar, and the two grids no longer line up anywhere except at the edges.
\n\n\nFive against four uses a grid of twenty and behaves like the others: the parts coincide at the cycle boundary, and every remaining onset in one part falls between two onsets of the other.
\nA single line can contradict its metre. The tresillo figure, three notes grouped 3+3+2 across a duple bar, is a cross-rhythm in one voice, and it appears in Cuban music, in New Orleans second-line patterns, and in a great deal of popular music built on those foundations. The clave does the same work over two bars: five strokes placed so that they avoid the plain beats and define the cycle by their own shape instead.
\nBecause a cross-rhythm is a relationship to a metre, it can also be heard as an ambiguity rather than a conflict. A repeating three-note figure inside 4/4 will line up with the bar every three bars, and listeners may hear the pattern as primary and the bar as secondary. Nothing in the notation changes when that happens; the listener’s anchor does.
\nThis is why the two terms are worth separating in analysis. Calling a tresillo a polyrhythm tells you nothing about what to listen for, and calling a four-against-three texture a cross-rhythm hides the fact that two grid systems are running simultaneously.
\nTwo parts can also run in different metres at once, sharing a fast pulse but not a bar. A part in 3/4 against a part in 4/4 realigns after twelve beats, which is three bars of the first and four of the second. The relationship between the parts changes continuously and returns to its starting point when the two bar lengths find their common multiple.
\nPolyrhythm and polymeter differ in what is shared. Polyrhythm shares the span and divides it differently; polymeter shares the pulse and divides it into different bars. Phasing, the process behind some of Steve Reich’s early pieces, is a third mechanism again: two identical patterns run at slightly different speeds and drift in and out of alignment, so the relationship changes without any grid being divided at all.
\nThe counting method is the lowest common multiple, spoken out loud. For three against two, count six even units, tap the three-part on units 1, 3 and 5 and the two-part on 1 and 4. For four against three, count twelve and tap 1, 4, 7, 10 against 1, 5, 9. The counting is scaffolding, not the skill.
\nThe skill is the composite. Ensembles that use polyrhythm constantly, from Ewe drumming to Cuban percussion sections, teach interlocking parts as one composite pattern rather than as two ratios to be reconciled: a player learns their own figure against the bell pattern that lays out the cycle, and the polyrhythm is a by-product of everyone knowing their own line. The bell in an Ewe ensemble marks the cycle; the sounding result is layered.
\nThe motor problem is separate from the conceptual one. Knowing that three against two produces the sequence together, triplet, duplet, triplet does not produce the feel, and the feel rarely produces a correct verbal account of the arithmetic. The practical route between them is slow practice at a tempo where the grid is countable, then gradual speed increase until the figure stops being two acts and becomes one gesture, then singing one part while tapping the other, which is where the pattern either lives in the body or does not.
\nStandard notation handles polyrhythm by making one part take tuplets: the part playing three against the beat is written with a bracket, while the part playing two keeps the plain note values. The choice of which part receives the tuplet is arbitrary, and it is usually wrong for one of the two players, who then has to read a rhythm they would never count that way. Chords have the same problem in reverse: a texture of several layers can be written as one line of tuplets after another, at which point the score records the arithmetic accurately and communicates the pattern badly.
\nA sequencer avoids the puzzle by giving both layers one fine grid. Once the grid is set to twelve subdivisions to the bar, four against three can be placed exactly, and five against four needs twenty. What the grid cannot supply is what makes the figure audible: accents on the onsets, a slight length difference between the longer and shorter notes, and the small timing inequalities players introduce. That is the same gap that appears in swing timing, where a written ratio and a played one are two different claims. A perfectly quantised polyrhythm sounds like what it is, a demonstration that the arithmetic works, and the demonstration is worth hearing once before the figure is played by a person.
", "content_text": "A percussionist and a conductor can look at the same bar and disagree about what to call it, because two terms get used as if they were synonyms. They are not. One describes a texture in which two parts divide the same span differently; the other describes a pattern that contradicts the metre it sits in. A passage can be both, and often is, but a passage can also be one without the other.\n#Two definitions that survive scrutiny\nA polyrhythm is a simultaneity. Two or more parts divide the same span of time into different numbers of equal parts and play them together, so the span has two internal grids at once: three against two, four against three, five against four. The parts agree about where the span begins and ends and disagree about everything inside it.\nA cross-rhythm is a relationship between a pattern and a metre. Onsets or accents fall where the prevailing metre does not expect them, and the conflict is audible whether one part is playing or five. Hemiola in triple metre is the commonest case, and it is described in the article on meter and beat hierarchy: two bars of 3/4 regrouped as three groups of two.\nThe distinction is that a polyrhythm needs two parts to exist while a cross-rhythm needs only one part and a metre. Writers on rhythm do not all draw the line in the same place, and some prefer to treat cross-rhythm as the general principle with polyrhythm as its special case. The stable part of the distinction is the one worth keeping: a texture against a pattern.\n#One span, two divisions\nThe arithmetic of a polyrhythm is the arithmetic of the lowest common multiple. If one part plays three equal notes across a span and another plays two, both can be plotted on a grid of six, because six is the first number both three and two divide into.\nRatioPartsSmallest gridOnsets on the grid3:2three against two6three part at 1, 3, 5; two part at 1, 44:3four against three12four part at 1, 4, 7, 10; three part at 1, 5, 95:4five against four20five part at 1, 5, 9, 13, 17; four part at 1, 6, 11, 16\nRead the 3:2 row carefully, because it corrects the most common beginner’s error. The two parts meet at the start of the span and nowhere else. There is no mid-point where the hands come back together: on the six-unit grid the three-part has a note on unit 5 while the two-part has already played its last note on unit 4. The composite order a listener hears is therefore together, triplet, duplet, triplet. Anyone waiting for a second coincidence halfway will lose the pattern and blame their hands.\nC4+G4\nThe one onset both parts share: the beginning of the cycle. Everything that follows is placed between two of these.\nC4 G4 C5\nC4 C5\nThree evenly spread pitches, then two over the same span. Spread the first group across a bar and the second across the same bar, and the two grids no longer line up anywhere except at the edges.\nC4 D4 E4 F4 G4\nC4 E4 G4 C5\nFive against four uses a grid of twenty and behaves like the others: the parts coincide at the cycle boundary, and every remaining onset in one part falls between two onsets of the other.\n#Cross-rhythm with one part\nA single line can contradict its metre. The tresillo figure, three notes grouped 3+3+2 across a duple bar, is a cross-rhythm in one voice, and it appears in Cuban music, in New Orleans second-line patterns, and in a great deal of popular music built on those foundations. The clave does the same work over two bars: five strokes placed so that they avoid the plain beats and define the cycle by their own shape instead.\nBecause a cross-rhythm is a relationship to a metre, it can also be heard as an ambiguity rather than a conflict. A repeating three-note figure inside 4/4 will line up with the bar every three bars, and listeners may hear the pattern as primary and the bar as secondary. Nothing in the notation changes when that happens; the listener’s anchor does.\nThis is why the two terms are worth separating in analysis. Calling a tresillo a polyrhythm tells you nothing about what to listen for, and calling a four-against-three texture a cross-rhythm hides the fact that two grid systems are running simultaneously.\n#Polymeter is a third thing\nTwo parts can also run in different metres at once, sharing a fast pulse but not a bar. A part in 3/4 against a part in 4/4 realigns after twelve beats, which is three bars of the first and four of the second. The relationship between the parts changes continuously and returns to its starting point when the two bar lengths find their common multiple.\nPolyrhythm and polymeter differ in what is shared. Polyrhythm shares the span and divides it differently; polymeter shares the pulse and divides it into different bars. Phasing, the process behind some of Steve Reich’s early pieces, is a third mechanism again: two identical patterns run at slightly different speeds and drift in and out of alignment, so the relationship changes without any grid being divided at all.\n#Counting it, then hearing it\nThe counting method is the lowest common multiple, spoken out loud. For three against two, count six even units, tap the three-part on units 1, 3 and 5 and the two-part on 1 and 4. For four against three, count twelve and tap 1, 4, 7, 10 against 1, 5, 9. The counting is scaffolding, not the skill.\nThe skill is the composite. Ensembles that use polyrhythm constantly, from Ewe drumming to Cuban percussion sections, teach interlocking parts as one composite pattern rather than as two ratios to be reconciled: a player learns their own figure against the bell pattern that lays out the cycle, and the polyrhythm is a by-product of everyone knowing their own line. The bell in an Ewe ensemble marks the cycle; the sounding result is layered.\nThe motor problem is separate from the conceptual one. Knowing that three against two produces the sequence together, triplet, duplet, triplet does not produce the feel, and the feel rarely produces a correct verbal account of the arithmetic. The practical route between them is slow practice at a tempo where the grid is countable, then gradual speed increase until the figure stops being two acts and becomes one gesture, then singing one part while tapping the other, which is where the pattern either lives in the body or does not.\n#Notation and the sequencer\nStandard notation handles polyrhythm by making one part take tuplets: the part playing three against the beat is written with a bracket, while the part playing two keeps the plain note values. The choice of which part receives the tuplet is arbitrary, and it is usually wrong for one of the two players, who then has to read a rhythm they would never count that way. Chords have the same problem in reverse: a texture of several layers can be written as one line of tuplets after another, at which point the score records the arithmetic accurately and communicates the pattern badly.\nA sequencer avoids the puzzle by giving both layers one fine grid. Once the grid is set to twelve subdivisions to the bar, four against three can be placed exactly, and five against four needs twenty. What the grid cannot supply is what makes the figure audible: accents on the onsets, a slight length difference between the longer and shorter notes, and the small timing inequalities players introduce. That is the same gap that appears in swing timing, where a written ratio and a played one are two different claims. A perfectly quantised polyrhythm sounds like what it is, a demonstration that the arithmetic works, and the demonstration is worth hearing once before the figure is played by a person.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "polyrhythm", "cross-rhythm", "meter" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/polyrhythm-and-cross-rhythm.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/reading-intervals-at-sight/", "url": "https://musics.name/articles/reading-intervals-at-sight/", "title": "Reading Intervals at Sight: Generic and Specific", "summary": "Naming an interval at sight takes two passes: count the staff positions first, then judge what the accidental does to the size.", "content_html": "An interval is described by two properties at once, and readers who collapse them into one get stuck. The first is the generic interval: how many staff positions the notes span, counted inclusively from the lower note. C to E is a third because three letter names sit inside it. The second property is quality: major, minor, perfect, augmented or diminished. Quality states how many semitones that span actually contains.
\nThe separation matters because spelling and sound are independent. F to B spans four positions and six semitones. B to F spans six semitones as well, but only five positions. The first is an augmented fourth, the second a diminished fifth. On a keyboard they use the same two keys in reverse order, but they are not the same interval, they do not resolve the same way, and a reader who counts only semitones can neither tell them apart nor spell them correctly on paper.
\nThe generic interval comes from the staff, not from the pitch. Count the lower note as one, then add one for every line or space upward until you reach the upper note. Adjacent lines, or adjacent spaces, span three positions, which is a third. Lines with exactly one line between them, or spaces with exactly one space between them, span five positions, which is a fifth.
\nThe geometry is regular enough to memorise in a minute, and it is the fastest filter in sight reading: odd-numbered intervals keep the same kind of position, even-numbered intervals change it.
\n| Generic interval | Positions counted | Position of the upper note |
|---|---|---|
| Unison | 1 | same line or same space |
| Second | 2 | line to space, or space to line |
| Third | 3 | line to line, or space to space |
| Fourth | 4 | line to space, or space to line |
| Fifth | 5 | line to line, or space to space |
| Sixth | 6 | line to space, or space to line |
| Seventh | 7 | line to line, or space to space |
| Octave | 8 | same letter, same kind of position, one staff higher |
Quality is measured against the major scale built on the lower note. If the upper note belongs to that scale, the interval is major, which applies to seconds, thirds, sixths and sevenths, or perfect, which applies to unisons, fourths, fifths and octaves. Lower the upper note by a semitone and a major interval becomes minor; lower it again and it becomes diminished; raise it and the interval becomes augmented. Perfect intervals skip the major and minor stages entirely, so a perfect fifth lowered by a semitone is diminished and raised by a semitone is augmented.
\n| Interval | Semitones | Relation to the major scale on the lower note |
|---|---|---|
| Perfect unison | 0 | 1 |
| Minor second | 1 | 2 lowered |
| Major second | 2 | 2 |
| Minor third | 3 | 3 lowered |
| Major third | 4 | 3 |
| Perfect fourth | 5 | 4 |
| Augmented fourth | 6 | 4 raised |
| Diminished fifth | 6 | 5 lowered |
| Perfect fifth | 7 | 5 |
| Minor sixth | 8 | 6 lowered |
| Major sixth | 9 | 6 |
| Minor seventh | 10 | 7 lowered |
| Major seventh | 11 | 7 |
| Perfect octave | 12 | 8 |
The two readings have to agree. A fifth stays a fifth whether it is perfect, diminished or augmented; only the width changes.
\n\n\nCounting from the lower note every time is correct and slow. Fluent readers recognise a small set of shapes and measure everything else against them.
\nThe perfect fifth is the most useful anchor on the staff: both notes sit on the same kind of position with one line or one space between them, and the shape is unmistakable even in peripheral vision. It is also the interval that outlines the triad, so it appears constantly in chordal music. The octave is the second anchor: same letter, same kind of position, one staff higher.
\nThirds are the interval you read most often in melodic writing, and they are the ones beginners misjudge, because a third looks wider than it sounds. Two positions are skipped: a third from a line lands on the next line, not on the next space.
\nSkilled readers do not identify every interval from scratch. They read the outer interval of a leap or a chord, then place the inner notes by their distance from the outer ones. A reader who owns the fifth can find any note on the staff within two positions, because fifths cycle through the staff in a closed pattern: count up by fifths from a landmark line and you reach the same letters in a different register, then adjust by step.
\nFive short passes, in this order, for a few weeks:
\nA motet by Palestrina and a trio sonata by Corelli are separated by about a century, and the difference between them is not a matter of ornament. In the motet, harmony is what happens while four or five independent lines are doing something else. In the sonata, two upper lines move above a bass that decides what the harmony is at every moment, and a player at a keyboard fills in the rest. That change in the division of labour is the birth of tonal harmony, and it is visible in the notation long before anyone wrote a theory of it.
\nSixteenth-century polyphony is written in partbooks, one voice per book, and each line is conceived as a melody with its own shape and its own cadences. The vertical sonorities that pass by are consequences of the lines, not the subject of the composition. In that world a chord has no root in any practical sense, because nothing in the music asks the question. The lowest sounding voice is simply the voice that happens to be lowest.
\nOnce a composer writes a single melody singing above a single bass line, the bass becomes the harmonic ground. The top voice carries the tune, the bass carries the foundation, and everything between is accompaniment. This is also where inversion becomes a working idea rather than an abstraction: the same chord can sit above different notes in the bass, and the player recognises it as the same chord because the intervals above the bass change while the identity does not.
\nThe intellectual pressure for the change came from Florence. Vincenzo Galilei’s Dialogo della musica antica et della moderna of 1581 argued that modern counterpoint had buried the text under its own ingenuity, and the circle around Bardi’s camerata set out to recover what they believed was a Greek union of words and music. The result, within twenty years, was opera: Peri’s Euridice in 1600, Monteverdi’s Orfeo in 1607. In both, the essential texture is a voice and a bass, and the harmony is supplied by a continuo player reading figures.
\nMonteverdi’s fifth book of madrigals, published in 1605, adds a continuo part to a genre that had done without one, and the quarrel that followed is as instructive as the music. Giovanni Maria Artusi attacked the harmonic liberties of the moderns in 1600, and Monteverdi’s defence, the appeal to a seconda pratica in which the text governs what the rules permit, is the first clear statement that the older contrapuntal rules were one style among several rather than a law of nature.
\nThe practical driver behind all of this is easy to miss: someone had to play the chords, and to do that the chords had to be written down. Lodovico Viadana’s Cento concerti ecclesiastici of 1602 is the standard early example of church music printed with a figured bass part for organ, and within a generation the continuo part is standard equipment from the church to the chamber to the opera house. Once harmony is notated as figures under a bass, the harmonic content of a passage is an explicit object that a composer can plan and a theorist can describe.
\nThe older system organised pitch by mode. Eight church modes, four authentic and four plagal, each with a final and a characteristic range, and the theory of the period treats them as different species of melody with different expressive characters. Two developments narrowed that to the major and minor scales.
\nThe first was slow and took most of the fifteenth century: the third and the sixth were gradually accepted as consonances rather than tolerated dissonances, which is what makes a chord of root, third and fifth possible in the first place. Fifteenth-century treatises such as Tinctoris’s Liber de arte contrapuncti of 1477 catalogue consonances in a way that already includes them.
\nThe second was theoretical and happened in a single generation. Heinrich Glarean’s Dodecachordon of 1547 argued for twelve modes rather than eight, adding Ionian and Aeolian, the modes that correspond to what later centuries called major and minor. Gioseffo Zarlino’s Le istitutioni harmoniche of 1558 worked the major third into a systematic account of harmony. By the later seventeenth century, practical theory had two keys rather than twelve modes, and the modes survived in the church tones, in transposed performance practice, and in vernacular music that never needed the theory.
\nThe reason the change stuck is structural. When the chord is the unit of harmony, the third is the note that decides the chord’s quality, and there are two qualities. Transposition then becomes trivial: a key is a set of relationships, not a fixed set of pitches, so the same piece can be moved to another key without becoming another piece.
\nJean-Philippe Rameau’s Traité de l’harmonie of 1722 gave the practice a theory. Its central idea is the fundamental bass, the root of each chord, which is not necessarily the sounding bass note: an inverted chord has a sounding bass that is not its fundamental. Once the fundamental is distinguished from the bass, chord identity is stable across inversions, and a piece can be described as a sequence of roots with bass notes underneath them.
\nRameau also argued that harmony derives from the harmonic series of a sounding string, which was a strong claim that has been argued about ever since. What survived is the descriptive apparatus rather than the physics: roots, inversions, and the tendency of roots to move by fifth. The teaching tradition built on it turned the description into a drill, above all in the Neapolitan conservatories of the eighteenth century, where a student harmonised a scale either in the standard formula of the rule of the octave or against a partimento bass that left the chords unwritten. A modern harmony exercise descending from a figured or unfigured bass is a descendant of that pedagogy.
\nWhat makes a key a key is not the notes available but the cadence that keeps confirming a centre. In modal polyphony, cadences come in several types and land on different finals, and the leading tone that gives a tonal cadence its direction was often not written at all: performers supplied it as musica ficta, an accidental understood rather than notated.
\nBy the Baroque the raised leading tone is in the notation, and the cadential motion of a root falling a fifth into the tonic, with the leading tone rising to the tonic in an upper voice, is the standard punctuation of the style. That is the shift from mode to key in one sentence: the leading tone makes a final into a centre.
\n\nThe G sharp is what defines the key. Without it the chord is a minor seventh chord on E, and the arrival on A would be an ending rather than a confirmation. With it, the chord is a dominant seventh and the A is a tonic.
\n\nThe same motion in C major. The cadence transposes, which is exactly what modal finals do not do, and it is why a piece can move to another key, confirm it by cadence, and be understood to have modulated. The cadence types themselves are a vocabulary built on this single motion.
\nThe vocabulary most people learn first, tonic, subdominant and dominant with their substitutions, is younger than the music it describes. Hugo Riemann’s function theory, set out in the 1890s, grouped chords by the role they play rather than by their root, and it is the source of the T, S and D labels. Heinrich Schenker’s later account, culminating in Der freie Satz of 1935, reduced the same repertoire to voice leading and a background structure, and treated the surface chords as elaborations of it. Both are nineteenth- and twentieth-century attempts to describe a practice that had been running for two hundred years.
\nThe disagreement matters for anyone analysing old music, because the labels will almost always fit something. A Palestrina cadence can be given a Roman numeral, and a Josquin passage can be assigned functions, and the analysis will look plausible while describing the wrong thing: in that repertoire the vertical sonorities are consequences of the lines, the cadences have different weights and different kinds, and the piece is organised by mode rather than by a tonic that everything pulls towards.
\n| Aspect | Before about 1600 | After about 1650 |
|---|---|---|
| Texture | imitative polyphony, parts conceived separately | melody above a bass, chords supplied between them |
| What determines the chord | the meeting of independent lines | the bass note and the figures above it |
| Notation | partbooks, one voice per book | score and continuo part with figures |
| Chord identity | a sonority with no practical root | a root that survives inversion |
| Cadence | several types, with the leading tone often implied | one cadential motion, applied in every key |
| Pitch organisation | eight or twelve modes with different finals | two keys, transposable |
| Theory of the period | mode as a melodic species | the fundamental bass, then function |
The practical consequence is a reading habit. For music written before 1600, follow the lines, note where each voice cadences, and identify the final the piece keeps returning to; for music written after, follow the bass, mark the cadences that confirm the key, and label a function only where you can hear the role it plays. The chord-function model describes the second repertoire extremely well and the first one poorly, and the date of the change is the reason.
", "content_text": "A motet by Palestrina and a trio sonata by Corelli are separated by about a century, and the difference between them is not a matter of ornament. In the motet, harmony is what happens while four or five independent lines are doing something else. In the sonata, two upper lines move above a bass that decides what the harmony is at every moment, and a player at a keyboard fills in the rest. That change in the division of labour is the birth of tonal harmony, and it is visible in the notation long before anyone wrote a theory of it.\n#A texture in which the bass decides\nSixteenth-century polyphony is written in partbooks, one voice per book, and each line is conceived as a melody with its own shape and its own cadences. The vertical sonorities that pass by are consequences of the lines, not the subject of the composition. In that world a chord has no root in any practical sense, because nothing in the music asks the question. The lowest sounding voice is simply the voice that happens to be lowest.\nOnce a composer writes a single melody singing above a single bass line, the bass becomes the harmonic ground. The top voice carries the tune, the bass carries the foundation, and everything between is accompaniment. This is also where inversion becomes a working idea rather than an abstraction: the same chord can sit above different notes in the bass, and the player recognises it as the same chord because the intervals above the bass change while the identity does not.\n#Monody, the seconda pratica and the keyboard under the bass\nThe intellectual pressure for the change came from Florence. Vincenzo Galilei’s Dialogo della musica antica et della moderna of 1581 argued that modern counterpoint had buried the text under its own ingenuity, and the circle around Bardi’s camerata set out to recover what they believed was a Greek union of words and music. The result, within twenty years, was opera: Peri’s Euridice in 1600, Monteverdi’s Orfeo in 1607. In both, the essential texture is a voice and a bass, and the harmony is supplied by a continuo player reading figures.\nMonteverdi’s fifth book of madrigals, published in 1605, adds a continuo part to a genre that had done without one, and the quarrel that followed is as instructive as the music. Giovanni Maria Artusi attacked the harmonic liberties of the moderns in 1600, and Monteverdi’s defence, the appeal to a seconda pratica in which the text governs what the rules permit, is the first clear statement that the older contrapuntal rules were one style among several rather than a law of nature.\nThe practical driver behind all of this is easy to miss: someone had to play the chords, and to do that the chords had to be written down. Lodovico Viadana’s Cento concerti ecclesiastici of 1602 is the standard early example of church music printed with a figured bass part for organ, and within a generation the continuo part is standard equipment from the church to the chamber to the opera house. Once harmony is notated as figures under a bass, the harmonic content of a passage is an explicit object that a composer can plan and a theorist can describe.\n#From twelve modes to two keys\nThe older system organised pitch by mode. Eight church modes, four authentic and four plagal, each with a final and a characteristic range, and the theory of the period treats them as different species of melody with different expressive characters. Two developments narrowed that to the major and minor scales.\nThe first was slow and took most of the fifteenth century: the third and the sixth were gradually accepted as consonances rather than tolerated dissonances, which is what makes a chord of root, third and fifth possible in the first place. Fifteenth-century treatises such as Tinctoris’s Liber de arte contrapuncti of 1477 catalogue consonances in a way that already includes them.\nThe second was theoretical and happened in a single generation. Heinrich Glarean’s Dodecachordon of 1547 argued for twelve modes rather than eight, adding Ionian and Aeolian, the modes that correspond to what later centuries called major and minor. Gioseffo Zarlino’s Le istitutioni harmoniche of 1558 worked the major third into a systematic account of harmony. By the later seventeenth century, practical theory had two keys rather than twelve modes, and the modes survived in the church tones, in transposed performance practice, and in vernacular music that never needed the theory.\nThe reason the change stuck is structural. When the chord is the unit of harmony, the third is the note that decides the chord’s quality, and there are two qualities. Transposition then becomes trivial: a key is a set of relationships, not a fixed set of pitches, so the same piece can be moved to another key without becoming another piece.\n#Rameau names the fundamental bass\nJean-Philippe Rameau’s Traité de l’harmonie of 1722 gave the practice a theory. Its central idea is the fundamental bass, the root of each chord, which is not necessarily the sounding bass note: an inverted chord has a sounding bass that is not its fundamental. Once the fundamental is distinguished from the bass, chord identity is stable across inversions, and a piece can be described as a sequence of roots with bass notes underneath them.\nRameau also argued that harmony derives from the harmonic series of a sounding string, which was a strong claim that has been argued about ever since. What survived is the descriptive apparatus rather than the physics: roots, inversions, and the tendency of roots to move by fifth. The teaching tradition built on it turned the description into a drill, above all in the Neapolitan conservatories of the eighteenth century, where a student harmonised a scale either in the standard formula of the rule of the octave or against a partimento bass that left the chords unwritten. A modern harmony exercise descending from a figured or unfigured bass is a descendant of that pedagogy.\n#The cadence becomes the engine of a key\nWhat makes a key a key is not the notes available but the cadence that keeps confirming a centre. In modal polyphony, cadences come in several types and land on different finals, and the leading tone that gives a tonal cadence its direction was often not written at all: performers supplied it as musica ficta, an accidental understood rather than notated.\nBy the Baroque the raised leading tone is in the notation, and the cadential motion of a root falling a fifth into the tonic, with the leading tone rising to the tonic in an upper voice, is the standard punctuation of the style. That is the shift from mode to key in one sentence: the leading tone makes a final into a centre.\nE3+G#3+B3+D4 A3+C4+E4\nThe G sharp is what defines the key. Without it the chord is a minor seventh chord on E, and the arrival on A would be an ending rather than a confirmation. With it, the chord is a dominant seventh and the A is a tonic.\nG3+B3+D4+F4 C4+E4+G4\nThe same motion in C major. The cadence transposes, which is exactly what modal finals do not do, and it is why a piece can move to another key, confirm it by cadence, and be understood to have modulated. The cadence types themselves are a vocabulary built on this single motion.\n#Function is a nineteenth-century description\nThe vocabulary most people learn first, tonic, subdominant and dominant with their substitutions, is younger than the music it describes. Hugo Riemann’s function theory, set out in the 1890s, grouped chords by the role they play rather than by their root, and it is the source of the T, S and D labels. Heinrich Schenker’s later account, culminating in Der freie Satz of 1935, reduced the same repertoire to voice leading and a background structure, and treated the surface chords as elaborations of it. Both are nineteenth- and twentieth-century attempts to describe a practice that had been running for two hundred years.\nThe disagreement matters for anyone analysing old music, because the labels will almost always fit something. A Palestrina cadence can be given a Roman numeral, and a Josquin passage can be assigned functions, and the analysis will look plausible while describing the wrong thing: in that repertoire the vertical sonorities are consequences of the lines, the cadences have different weights and different kinds, and the piece is organised by mode rather than by a tonic that everything pulls towards.\nAspectBefore about 1600After about 1650Textureimitative polyphony, parts conceived separatelymelody above a bass, chords supplied between themWhat determines the chordthe meeting of independent linesthe bass note and the figures above itNotationpartbooks, one voice per bookscore and continuo part with figuresChord identitya sonority with no practical roota root that survives inversionCadenceseveral types, with the leading tone often impliedone cadential motion, applied in every keyPitch organisationeight or twelve modes with different finalstwo keys, transposableTheory of the periodmode as a melodic speciesthe fundamental bass, then function\nThe practical consequence is a reading habit. For music written before 1600, follow the lines, note where each voice cadences, and identify the final the piece keeps returning to; for music written after, follow the bass, mark the cadences that confirm the key, and label a function only where you can hear the role it plays. The chord-function model describes the second repertoire extremely well and the first one poorly, and the date of the change is the reason.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "tonality", "basso continuo", "modes" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/rise-of-functional-tonality.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/romantic-chromaticism/", "url": "https://musics.name/articles/romantic-chromaticism/", "title": "Romantic Chromaticism and the Expanding Tonic", "summary": "Chromatic harmony widened a key without leaving it: the resources that do the work, the Tristan chord, and how to analyse it without over-labelling.", "content_html": "Nineteenth-century harmony is usually described as chromatic, and the word hides the important part. A chromatic passage is not a passage that has left its key. It is a passage that has widened the key until it includes notes and chords the older vocabulary could not explain, and it stays tonal precisely because the widening is measured against a centre that the music keeps returning to. The history of the century is a history of how far that expansion could go before the centre stopped being audible.
\nChromaticism itself is not a nineteenth-century invention. Gesualdo’s madrigals at the end of the sixteenth century go further in some bars than anything before Wagner, and Bach’s Chromatic Fantasy and Fugue is built out of exactly the resource the twentieth century would later claim as its own. Carl Philipp Emanuel Bach’s Versuch über die wahre Art das Clavier zu spielen of 1753 codified an expressive, harmonically restless keyboard style a generation before the Romantics.
\nWhat changes in the nineteenth century is the status of the chromatic chord. In earlier music it is an event, a moment of heightened expression that the surrounding diatonic harmony frames and explains. In Schumann, Chopin, Liszt and Wagner it becomes the ordinary medium: a phrase may pass through four chromatic chords on its way to a cadence, and the cadence still arrives where the key promised. The tonic does not lose its authority, but the area it governs grows much larger.
\nFour devices account for most of the sound, and each one extends the key in a different direction.
\nThe chromatic mediant moves between chords whose roots lie a third apart. C major and A flat major share a single note, and their roots are a major third apart on the flat side: the relationship is outside the diatonic circle of fifths and produces an immediate change of colour, often at a phrase boundary where a functional connection would be expected.
\n\nThe diminished seventh is a chord of four notes, each a minor third from the next, and that symmetry is a licence. The same four pitches can be spelled four ways, and each spelling points at a different destination, so the chord can be used for a modulation that does not declare where it is going until it arrives.
\n| Spelling | Resolves into |
|---|---|
| B, D, F, A flat | C, major or minor |
| D, F, A flat, C flat | E flat, major or minor |
| F, A flat, C flat, E double flat | G flat, major or minor |
| A flat, C flat, E double flat, G double flat | A, major or minor |
That chord is one sonority with four available meanings, and the analysis depends on which spelling the music chooses next.
\nAugmented sixth chords and the Neapolitan do a similar job with more direction, and they belong to the section of the vocabulary that arrives from mixture rather than from free chromaticism. Their function is usually clear: they prepare a dominant. The Romantic extension of them is that they are used in sequence, which weakens the preparation and turns a functional chord into a colour.
\nThe last resource is not a chord at all. Delaying resolution is what makes chromatic harmony sound chromatic, because the ear is given the notes but not the arrival. A suspension held for a whole bar, a dominant that resolves to another dominant, a cadence that is prepared and then withdrawn: all of these widen the phrase without adding new chords.
\nSchubert’s harmonic language is the clearest early evidence that third relations had become a normal resource rather than a special effect. His late music regularly places a movement or a section in a key a third away from the home key, as the String Quintet in C major of 1828 does by putting its slow movement in E major, and the Winterreise songs move through harmonic territory that the opening key does not obviously own. Where a Classical composer would modulate to the dominant to create tension, Schubert’s music often shifts sideways, and the effect is a change of light rather than an increase in pressure.
\nChopin’s contribution is the conversion of chromatic harmony into texture. In the Nocturnes and the Mazurkas, and in the G minor Ballade, chromatic inner voices and mixture are not events the music stops for; they are the fabric, and the melody floats above them. This is also where chromaticism becomes an interpretive problem, because a texture made of chromatic lines has to be voiced, and voicing is a decision the notation does not make. A keyboard soloist has to choose which of several moving lines the ear should follow.
\nBeethoven’s late quartets are the immediate precedent for both, and Liszt’s B minor Sonata of 1853 shows where the technique leads when it is applied to form: a single continuous movement in which themes are transformed by chromatic alteration rather than set against each other.
\nThe chord that begins Wagner’s Tristan und Isolde, completed in 1859 and premiered in 1865, is spelled F, B, D sharp and G sharp. It resolves immediately to a chord of E, with the G sharp held over as the third of that chord, in a progression that circles A minor without settling into it.
\n\nRead enharmonically, F, B, D sharp and G sharp is F, A flat, C flat and E flat: a half-diminished seventh chord on F, with its root in the bass and the enharmonic equivalents of its third, fifth and seventh stacked above it. Every note has an obligation. The B is the diminished fifth of the chord and wants to resolve; the G sharp is a leading tone; the F and the D sharp are the suspended weight of the sound. The reason the chord has been discussed for a century and a half is not that it is unprecedented but that it is overdetermined: it can be spelled to lead several places, it delays rather than establishes, and the delay is the point.
\nWhat the chord does not do is end tonality. It resolves, it belongs to a key, and the music around it spends the rest of the opera returning to that key with a new intensity. The claim that tonality collapsed at this moment is a twentieth-century summary that flatters the twentieth century. What collapsed, and only gradually, was the assumption that a key has to be confirmed at every level of a phrase.
\nThe temptation in analysing this repertoire is to give every chromatic chord a name that implies a function. Sharp-four chords, borrowed chords, applied dominants and slashed numerals are all available, and a passage can be described almost entirely in them. The descriptions are often true and sometimes useless, because they can make an undirected chromatic colour look like a functional progression.
\nThe procedure that keeps analysis honest runs in a fixed order. Find the key area first and say how securely it is established. Then identify which notes are prolonged and which are ornamental: a chromatic note that behaves as an appoggiatura or a suspension is a voice-leading event, and no chord label changes that. Only then ask whether the resulting chord has a function, meaning whether its resolution confirms it. If the resolution is vague or withheld, the honest label is a description of the spelling and the voice leading rather than a function, and the functional model has reached the edge of what it can describe.
\nThe same discipline applies to the whole repertory. Two analysts can disagree about a passage and both be right, if one is describing the voice leading and the other is describing a chromatic route to the dominant. The disagreement is only a problem when a label is presented as an explanation.
\nChromatic textures put most of their difficulty in places the notation does not mark. Intonation is the first: in a chain of chromatic thirds or an extended augmented sixth, each player has to decide which line the ear is following and tune towards it, and a chord that is theoretically pure can sound sour if the players have chosen different leaders. The leading tones in a dense chromatic passage are the notes most likely to be played too low, and the resolution of a suspension is the note most likely to arrive early.
\nPacing is the second. Chromatic harmony invites a slowing that the notation does not ask for, because every dissonance sounds like an arrival. The nineteenth-century repertoire is full of passages that should move through their chromaticism at a steady tempo, and the habit of treating each chord as an event is what turns a passage of tension into a series of stops. The test is whether the music still has somewhere to go after the passage ends.
\nVoicing is the third, and for keyboard players it is the hardest. A chromatic inner line has to be audible without becoming the melody, which is a matter of touch rather than volume, and a sustained pedal turns an intended progression of separate sonorities into one blurred chord. The practical method is to mark, in the score, where each tendency tone resolves, and to play the passage slowly enough that the resolutions can be heard as resolutions. If you cannot say which note goes where, the voicing decision has not been made, and the chord remains an effect rather than a progression.
", "content_text": "Nineteenth-century harmony is usually described as chromatic, and the word hides the important part. A chromatic passage is not a passage that has left its key. It is a passage that has widened the key until it includes notes and chords the older vocabulary could not explain, and it stays tonal precisely because the widening is measured against a centre that the music keeps returning to. The history of the century is a history of how far that expansion could go before the centre stopped being audible.\n#Expansion, not abandonment\nChromaticism itself is not a nineteenth-century invention. Gesualdo’s madrigals at the end of the sixteenth century go further in some bars than anything before Wagner, and Bach’s Chromatic Fantasy and Fugue is built out of exactly the resource the twentieth century would later claim as its own. Carl Philipp Emanuel Bach’s Versuch über die wahre Art das Clavier zu spielen of 1753 codified an expressive, harmonically restless keyboard style a generation before the Romantics.\nWhat changes in the nineteenth century is the status of the chromatic chord. In earlier music it is an event, a moment of heightened expression that the surrounding diatonic harmony frames and explains. In Schumann, Chopin, Liszt and Wagner it becomes the ordinary medium: a phrase may pass through four chromatic chords on its way to a cadence, and the cadence still arrives where the key promised. The tonic does not lose its authority, but the area it governs grows much larger.\n#The resources that do the widening\nFour devices account for most of the sound, and each one extends the key in a different direction.\nThe chromatic mediant moves between chords whose roots lie a third apart. C major and A flat major share a single note, and their roots are a major third apart on the flat side: the relationship is outside the diatonic circle of fifths and produces an immediate change of colour, often at a phrase boundary where a functional connection would be expected.\nC4+E4+G4 Ab3+C4+Eb4\nThe diminished seventh is a chord of four notes, each a minor third from the next, and that symmetry is a licence. The same four pitches can be spelled four ways, and each spelling points at a different destination, so the chord can be used for a modulation that does not declare where it is going until it arrives.\nSpellingResolves intoB, D, F, A flatC, major or minorD, F, A flat, C flatE flat, major or minorF, A flat, C flat, E double flatG flat, major or minorA flat, C flat, E double flat, G double flatA, major or minor\nB3+D4+F4+Ab4\nThat chord is one sonority with four available meanings, and the analysis depends on which spelling the music chooses next.\nAugmented sixth chords and the Neapolitan do a similar job with more direction, and they belong to the section of the vocabulary that arrives from mixture rather than from free chromaticism. Their function is usually clear: they prepare a dominant. The Romantic extension of them is that they are used in sequence, which weakens the preparation and turns a functional chord into a colour.\nThe last resource is not a chord at all. Delaying resolution is what makes chromatic harmony sound chromatic, because the ear is given the notes but not the arrival. A suspension held for a whole bar, a dominant that resolves to another dominant, a cadence that is prepared and then withdrawn: all of these widen the phrase without adding new chords.\n#Schubert and Chopin: the practice becomes systematic\nSchubert’s harmonic language is the clearest early evidence that third relations had become a normal resource rather than a special effect. His late music regularly places a movement or a section in a key a third away from the home key, as the String Quintet in C major of 1828 does by putting its slow movement in E major, and the Winterreise songs move through harmonic territory that the opening key does not obviously own. Where a Classical composer would modulate to the dominant to create tension, Schubert’s music often shifts sideways, and the effect is a change of light rather than an increase in pressure.\nChopin’s contribution is the conversion of chromatic harmony into texture. In the Nocturnes and the Mazurkas, and in the G minor Ballade, chromatic inner voices and mixture are not events the music stops for; they are the fabric, and the melody floats above them. This is also where chromaticism becomes an interpretive problem, because a texture made of chromatic lines has to be voiced, and voicing is a decision the notation does not make. A keyboard soloist has to choose which of several moving lines the ear should follow.\nBeethoven’s late quartets are the immediate precedent for both, and Liszt’s B minor Sonata of 1853 shows where the technique leads when it is applied to form: a single continuous movement in which themes are transformed by chromatic alteration rather than set against each other.\n#The Tristan chord, stated plainly\nThe chord that begins Wagner’s Tristan und Isolde, completed in 1859 and premiered in 1865, is spelled F, B, D sharp and G sharp. It resolves immediately to a chord of E, with the G sharp held over as the third of that chord, in a progression that circles A minor without settling into it.\nF3+B3+D#4+G#4 E3+G#3+B3\nRead enharmonically, F, B, D sharp and G sharp is F, A flat, C flat and E flat: a half-diminished seventh chord on F, with its root in the bass and the enharmonic equivalents of its third, fifth and seventh stacked above it. Every note has an obligation. The B is the diminished fifth of the chord and wants to resolve; the G sharp is a leading tone; the F and the D sharp are the suspended weight of the sound. The reason the chord has been discussed for a century and a half is not that it is unprecedented but that it is overdetermined: it can be spelled to lead several places, it delays rather than establishes, and the delay is the point.\nWhat the chord does not do is end tonality. It resolves, it belongs to a key, and the music around it spends the rest of the opera returning to that key with a new intensity. The claim that tonality collapsed at this moment is a twentieth-century summary that flatters the twentieth century. What collapsed, and only gradually, was the assumption that a key has to be confirmed at every level of a phrase.\n#Where the analysis over-labels\nThe temptation in analysing this repertoire is to give every chromatic chord a name that implies a function. Sharp-four chords, borrowed chords, applied dominants and slashed numerals are all available, and a passage can be described almost entirely in them. The descriptions are often true and sometimes useless, because they can make an undirected chromatic colour look like a functional progression.\nThe procedure that keeps analysis honest runs in a fixed order. Find the key area first and say how securely it is established. Then identify which notes are prolonged and which are ornamental: a chromatic note that behaves as an appoggiatura or a suspension is a voice-leading event, and no chord label changes that. Only then ask whether the resulting chord has a function, meaning whether its resolution confirms it. If the resolution is vague or withheld, the honest label is a description of the spelling and the voice leading rather than a function, and the functional model has reached the edge of what it can describe.\nThe same discipline applies to the whole repertory. Two analysts can disagree about a passage and both be right, if one is describing the voice leading and the other is describing a chromatic route to the dominant. The disagreement is only a problem when a label is presented as an explanation.\n#What it costs the performer\nChromatic textures put most of their difficulty in places the notation does not mark. Intonation is the first: in a chain of chromatic thirds or an extended augmented sixth, each player has to decide which line the ear is following and tune towards it, and a chord that is theoretically pure can sound sour if the players have chosen different leaders. The leading tones in a dense chromatic passage are the notes most likely to be played too low, and the resolution of a suspension is the note most likely to arrive early.\nPacing is the second. Chromatic harmony invites a slowing that the notation does not ask for, because every dissonance sounds like an arrival. The nineteenth-century repertoire is full of passages that should move through their chromaticism at a steady tempo, and the habit of treating each chord as an event is what turns a passage of tension into a series of stops. The test is whether the music still has somewhere to go after the passage ends.\nVoicing is the third, and for keyboard players it is the hardest. A chromatic inner line has to be audible without becoming the melody, which is a matter of touch rather than volume, and a sustained pedal turns an intended progression of separate sonorities into one blurred chord. The practical method is to mark, in the score, where each tendency tone resolves, and to play the passage slowly enough that the resolutions can be heard as resolutions. If you cannot say which note goes where, the voicing decision has not been made, and the chord remains an effect rather than a progression.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "chromaticism", "harmony", "nineteenth century" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/romantic-chromaticism.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/scientific-pitch-notation/", "url": "https://musics.name/articles/scientific-pitch-notation/", "title": "Scientific Pitch Notation: What C4 Actually Means", "summary": "Middle C is C4 in scientific pitch notation, C3 on some manufacturers' labelling, and MIDI note 60 in every sequencer. The numbering is a convention.", "content_html": "Scientific pitch notation writes a letter, an optional accidental and an octave number: C4, B♭3, F♯5. The letter and the accidental are uncontroversial. The number is where documentation, instrument charts and software quietly disagree, and a single octave of disagreement is enough to send a part into the wrong register.
\nIn scientific pitch notation the octave number increments at C, not at A. C4, D4, E4, F4, G4, A4 and B4 all belong to the same octave, and the next number up begins at C5. A4 and B4 come before C5, which is why a scale counted from A crosses an octave boundary in the middle.
\nThe choice follows the instrument. On a piano keyboard the pattern of seven letter names restarts at each C, and the register boundary sits where the pattern restarts. Numbering the octaves from A would place a boundary between G and A, where the layout offers no such break. Every convention that numbers octaves from the note the pattern restarts on agrees with the keyboard; the disagreements are about which number sits at the bottom, not about where the boundary is.
\nThe reference note of the whole system is middle C, the C on the first ledger line below the treble staff and the first ledger line above the bass staff. On a standard piano it is the C nearest the middle of the keyboard, sounding at about 261.6 Hz.
\nThe practical consequence is that a label cannot be trusted on its own. A sample library whose documentation says its range starts at C3 may mean C3 or C4 in scientific pitch notation, and a patch labelled C4 may sound an octave away from the C4 you wrote in the score. The sounding pitch is the fact; the printed number is a naming choice. When they conflict, check the instrument’s key mapping or measure the output rather than the label.
\n| Note | Scientific pitch notation | Helmholtz notation | MIDI number | Yamaha-style label |
|---|---|---|---|---|
| Two octaves below middle C | C2 | C | 36 | C1 |
| One octave below middle C | C3 | c | 48 | C2 |
| Middle C | C4 | c′ | 60 | C3 |
| One octave above middle C | C5 | c″ | 72 | C4 |
| Concert A | A4 | a′ | 69 | A3 |
Helmholtz notation is the older continental system, and it uses capital and lowercase letters with primes rather than numbers. It still appears in nineteenth-century editions and in some German-language reference works, and the primes are easy to miscount in small print.
\nModern pitch reference is anchored at A4 = 440 Hz, the tuning frequency specified by ISO 16. Orchestras do not always follow it exactly: many European ensembles tune a little higher, and period-instrument groups working in Baroque repertoire often use 415 Hz, about a semitone below modern pitch, while some Classical-era projects sit near 430 Hz. For documentation purposes, 440 Hz for A4 is the default and the exceptions are stated explicitly.
\nThe relationship between a MIDI note number and its frequency is logarithmic in one direction and exponential in the other. With n as the MIDI note number:
\nf = 440 × 2^((n − 69)/12)
\nWork one case carefully. C4 is MIDI note 60, so the exponent is (60 − 69) / 12, which is −0.75. Two to the power of −0.75 is approximately 0.5946, and 440 multiplied by 0.5946 gives 261.6 Hz, the familiar frequency of middle C.
\nThe formula also handles octaves without special cases, because an octave in MIDI numbers is 12 steps, and 2 raised to the power of 12/12 is exactly 2. A5 is MIDI 81, one octave above A4, so it doubles to 880 Hz. C5 is MIDI 72, which is 2^0.25 × 440, or 523.25 Hz.
\nThe inverse is worth having as well:
\nn = 69 + 12 × log₂(f / 440)
\nThat version turns a measured frequency into a note number, which is what a tuner does when it reports both a pitch name and a deviation in cents. A semitone is 100 cents, and the conversion between a frequency ratio and cents is 1200 × log₂(ratio), so a measured 441 Hz against a target of 440 Hz is about 3.9 cents sharp. Most musicians cannot tune a sustained interval reliably to better than a few cents by ear, which is a useful thing to know before chasing a reading on a meter.
\n| Note | MIDI number | Frequency |
|---|---|---|
| C4 | 60 | 261.63 Hz |
| A4 | 69 | 440.00 Hz |
| C5 | 72 | 523.25 Hz |
| A5 | 81 | 880.00 Hz |
Notation software. Many programs have an octave setting that decides whether middle C is called C3 or C4, and it affects note entry by name, MIDI import and the octave numbers printed in range warnings. It is worth setting once per project and leaving it alone, because mixed conventions inside a single file are far harder to debug than a consistent wrong one.
\nSamplers and virtual instruments. A patch that claims to play C3 may sound an octave under or over the C3 in your score, since the developer chose a labelling convention before choosing the sample root. The reliable check is a single held note against a tuner, not the library’s documentation.
\nInstrument ranges. A range chart that says an instrument descends to C3 conveys nothing until it also says whether that is sounding or written pitch and which octave convention is in use. Reference material is only usable when the convention is printed with it. Assume the chart is wrong until the convention is stated.
\nTransposing instruments. A written C on a B♭ clarinet sounds B♭3. Written and sounding octaves differ by more than the transposition interval in the case of instruments such as the tenor saxophone, which sounds a major ninth below its written pitch, so a written C4 emerges as B♭2. Documenting a range means stating the frame of reference as well as the pitches.
\nGuitar and double bass. The lowest open string of a guitar sounds E2, about 82.4 Hz, and is written E3, because guitar music is notated an octave above sounding pitch. Both numbers are correct: one for the player’s page, one for the concert-pitch measurement. Exported MIDI depends on the software’s convention, which is why the same tablature can arrive in a DAW an octave away from what the player hears.
\nIn C major, a D major triad contains an F sharp. The key signature does not supply it and the chord is not an error. For as long as it sounds, the music treats G as a temporary tonic, and the D major triad is the dominant of that temporary tonic. Analysis writes it V/V and reads it “five of five”. The device is common enough that a page of Bach or Mozart will usually contain one.
\nA secondary dominant is the audible half of tonicization: a chord other than the tonic is raised, for a moment, to the status of a local tonic. The raising is not a decision taken by the analyst. It is what the ear does when a chord arrives by its own dominant, and the signal is the leading tone. In C major the leading tone is B. When another note starts behaving like a leading tone to a note that is not C, a tonicization is under way.
\nTwo qualifications keep the term honest. First, a tonicization operates at the level of a chord or a short progression, not a key. Nothing has been confirmed by cadence, and the phrase keeps its original direction once the chromatic note resolves. Second, the chord being tonicized is normally diatonic in the home key. V/V, V/ii, V/vi and V/iii in major, V/iv and V/III in minor, are the working set. A chromatic chord can be tonicized, but the labelling quickly becomes unreadable.
\nThe difference between a tonicization and a modulation is a difference in what happens afterwards. A tonicization resolves and is forgotten. A modulation is confirmed by a cadence in the new key, and the new key persists.
\nThe procedure is mechanical, and it is worth doing on paper before it becomes automatic.
\nThe chromatic note is the entire point. In a major key the note that creates each secondary dominant is predictable, because it is the leading tone of the target raised out of the home scale.
\n\n| Figure | Chord in C major | Notes | Resolves to | Chromatic note |
|---|---|---|---|---|
| V/V | D major | D–F sharp–A | V (G–B–D) | F sharp, raised 4 |
| V7/V | D dominant seventh | D–F sharp–A–C | V | F sharp |
| V/ii | A major | A–C sharp–E | ii (D–F–A) | C sharp, raised 1 |
| V/vi | E major | E–G sharp–B | vi (A–C–E) | G sharp, raised 5 |
| V/iii | B major | B–D sharp–F sharp | iii (E–G–B) | D sharp, raised 2 |
| V/IV | C major | C–E–G | IV (F–A–C) | none |
The last row is the interesting one. V/IV is spelled note for note like the tonic. A C major triad can only be heard as the dominant of F if the music then reaches F, and by that point most analysts simply write I and let the progression carry the meaning. The label is a statement about resolution, so it is earned by the resolution and not by the spelling.
\nIn a minor key the arithmetic shifts. In C minor, the leading tone is already raised by convention, so the chromatic note comes from elsewhere: raising the third degree, E flat to E natural, produces the C major triad that acts as V of F minor, and raising the seventh degree, B flat to B natural, produces the G major triad that acts as V of C minor itself.
\nThree situations produce secondary dominants more often than the rest.
\nBefore a dominant. A phrase that would simply move IV–V can move IV–V/V–V instead, which makes the dominant feel earned rather than stated. This is the most common use in classical repertoire, and it very often precedes a half cadence, since V/V points at V and V points at nothing.
\nIn sequences. Chained secondary dominants, each resolving down a fifth to the next, produce the circle-of-fifths motion that sits behind a great deal of tonal music. In C major, A7–D7–G7–C is V/ii, V7/V, V7 and I, and every chord is heard as the dominant of the one after it.
\nIn the approach to a chromatic chord. A borrowed chord arriving from a secondary dominant sounds purposeful rather than abrupt. The same machinery also supplies the dominant for chords that the key signature cannot support, which is why V/iv turns up in minor-key passages with an F minor chord in them.
\n\nThe chain is also where the notation strains. When each chord in a sequence is the dominant of the next, labelling every step V of something describes the mechanics correctly and says nothing about the passage’s key, which has not changed at all.
\nThe chromatic note resolves, the raised degree returns to its diatonic form, and the phrase continues. That return is the cancellation. If the chromatic note resolves upward by semitone into the target chord and the target chord then behaves as a tonic for a full phrase, with its own dominant and its own cadence, the process has become a modulation instead, and the label should change accordingly.
\nThe distinction matters for analysis because the two devices look identical at the moment of arrival. A V/V and the dominant of a new key are the same chord. What separates them is the cadence that does or does not follow.
\nThe chromatic note behaves as a leading tone and resolves up by semitone: the F sharp of V/V moves to G. It should not be doubled, and it should not leap away from its resolution. When the secondary dominant carries a seventh, as in V7/V, that seventh resolves down by step, so the C of an open D7 moves to the B of G.
\nSpell by function rather than by convenience. V/ii in C major is written with C sharp because the note is the leading tone to D. Written as D flat it would be describing a descent, and the resolution would no longer make sense.
\nA secondary dominant is a chord like any other, and the voice-leading rules that govern V govern V/x as well. Because the chords are usually in root position with the bass moving by fifth, parallel octaves between the bass and a doubled root are a standing temptation, and the resolution into the target chord is exactly where they tend to appear.
\nDominant function does not require a root. In place of V/ii, the diminished seventh on the raised fourth degree, C sharp, E, G, B flat in C major, supplies the same pull towards D minor without a root position dominant. These substituted chords are common in minor keys and in sequences, where an even chord-to-chord motion is wanted.
\nThe raised note is the announcement. In a C major context, an F sharp means G is coming, a C sharp means D is coming, a G sharp means A is coming, and the ear registers this faster than the labels can be written down. Play I–V/V–V–I in C major and stop. Then play I–V/V–vi. The second version builds the same dominant and does not deliver the tonic it promised, and the surprise is audible as a harmonic event rather than as an analysis. That comparison, run in several keys, does more for recognition than any amount of labelling.
\nThe ear training studio on this site includes a dominant-function drill for exactly that comparison.
", "content_text": "In C major, a D major triad contains an F sharp. The key signature does not supply it and the chord is not an error. For as long as it sounds, the music treats G as a temporary tonic, and the D major triad is the dominant of that temporary tonic. Analysis writes it V/V and reads it “five of five”. The device is common enough that a page of Bach or Mozart will usually contain one.\n#Tonicization, not modulation\nA secondary dominant is the audible half of tonicization: a chord other than the tonic is raised, for a moment, to the status of a local tonic. The raising is not a decision taken by the analyst. It is what the ear does when a chord arrives by its own dominant, and the signal is the leading tone. In C major the leading tone is B. When another note starts behaving like a leading tone to a note that is not C, a tonicization is under way.\nTwo qualifications keep the term honest. First, a tonicization operates at the level of a chord or a short progression, not a key. Nothing has been confirmed by cadence, and the phrase keeps its original direction once the chromatic note resolves. Second, the chord being tonicized is normally diatonic in the home key. V/V, V/ii, V/vi and V/iii in major, V/iv and V/III in minor, are the working set. A chromatic chord can be tonicized, but the labelling quickly becomes unreadable.\nThe difference between a tonicization and a modulation is a difference in what happens afterwards. A tonicization resolves and is forgotten. A modulation is confirmed by a cadence in the new key, and the new key persists.\n#How to spell one\nThe procedure is mechanical, and it is worth doing on paper before it becomes automatic.\nName the chord you intend to tonicize. Take ii in C major, which is D minor.Treat the root of that chord as a temporary tonic: D.Build the dominant of D, a major triad rooted a fifth above: A, with the notes A, C sharp, E.Spell the chord with the leading tone of D, whether or not the home key contains that note.\nThe chromatic note is the entire point. In a major key the note that creates each secondary dominant is predictable, because it is the leading tone of the target raised out of the home scale.\nC4+E4+G4 D4+F#4+A4 G3+B3+D4\nFigureChord in C majorNotesResolves toChromatic noteV/VD majorD–F sharp–AV (G–B–D)F sharp, raised 4V7/VD dominant seventhD–F sharp–A–CVF sharpV/iiA majorA–C sharp–Eii (D–F–A)C sharp, raised 1V/viE majorE–G sharp–Bvi (A–C–E)G sharp, raised 5V/iiiB majorB–D sharp–F sharpiii (E–G–B)D sharp, raised 2V/IVC majorC–E–GIV (F–A–C)none\nThe last row is the interesting one. V/IV is spelled note for note like the tonic. A C major triad can only be heard as the dominant of F if the music then reaches F, and by that point most analysts simply write I and let the progression carry the meaning. The label is a statement about resolution, so it is earned by the resolution and not by the spelling.\nIn a minor key the arithmetic shifts. In C minor, the leading tone is already raised by convention, so the chromatic note comes from elsewhere: raising the third degree, E flat to E natural, produces the C major triad that acts as V of F minor, and raising the seventh degree, B flat to B natural, produces the G major triad that acts as V of C minor itself.\n#Where they cluster\nThree situations produce secondary dominants more often than the rest.\nBefore a dominant. A phrase that would simply move IV–V can move IV–V/V–V instead, which makes the dominant feel earned rather than stated. This is the most common use in classical repertoire, and it very often precedes a half cadence, since V/V points at V and V points at nothing.\nIn sequences. Chained secondary dominants, each resolving down a fifth to the next, produce the circle-of-fifths motion that sits behind a great deal of tonal music. In C major, A7–D7–G7–C is V/ii, V7/V, V7 and I, and every chord is heard as the dominant of the one after it.\nIn the approach to a chromatic chord. A borrowed chord arriving from a secondary dominant sounds purposeful rather than abrupt. The same machinery also supplies the dominant for chords that the key signature cannot support, which is why V/iv turns up in minor-key passages with an F minor chord in them.\nA3+C#4+E4 D4+F#4+A4 G3+B3+D4 C4+E4+G4\nThe chain is also where the notation strains. When each chord in a sequence is the dominant of the next, labelling every step V of something describes the mechanics correctly and says nothing about the passage’s key, which has not changed at all.\n#What cancels a tonicization\nThe chromatic note resolves, the raised degree returns to its diatonic form, and the phrase continues. That return is the cancellation. If the chromatic note resolves upward by semitone into the target chord and the target chord then behaves as a tonic for a full phrase, with its own dominant and its own cadence, the process has become a modulation instead, and the label should change accordingly.\nThe distinction matters for analysis because the two devices look identical at the moment of arrival. A V/V and the dominant of a new key are the same chord. What separates them is the cadence that does or does not follow.\n#Part-writing cautions\nThe chromatic note behaves as a leading tone and resolves up by semitone: the F sharp of V/V moves to G. It should not be doubled, and it should not leap away from its resolution. When the secondary dominant carries a seventh, as in V7/V, that seventh resolves down by step, so the C of an open D7 moves to the B of G.\nSpell by function rather than by convenience. V/ii in C major is written with C sharp because the note is the leading tone to D. Written as D flat it would be describing a descent, and the resolution would no longer make sense.\nA secondary dominant is a chord like any other, and the voice-leading rules that govern V govern V/x as well. Because the chords are usually in root position with the bass moving by fifth, parallel octaves between the bass and a doubled root are a standing temptation, and the resolution into the target chord is exactly where they tend to appear.\nDominant function does not require a root. In place of V/ii, the diminished seventh on the raised fourth degree, C sharp, E, G, B flat in C major, supplies the same pull towards D minor without a root position dominant. These substituted chords are common in minor keys and in sequences, where an even chord-to-chord motion is wanted.\n#Hearing the function\nThe raised note is the announcement. In a C major context, an F sharp means G is coming, a C sharp means D is coming, a G sharp means A is coming, and the ear registers this faster than the labels can be written down. Play I–V/V–V–I in C major and stop. Then play I–V/V–vi. The second version builds the same dominant and does not deliver the tonic it promised, and the surprise is audible as a harmonic event rather than as an analysis. That comparison, run in several keys, does more for recognition than any amount of labelling.\nThe ear training studio on this site includes a dominant-function drill for exactly that comparison.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "secondary dominant", "tonicization", "chromatic harmony" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/secondary-dominants.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/seventh-chords-and-resolution/", "url": "https://musics.name/articles/seventh-chords-and-resolution/", "title": "Seventh Chords and How They Resolve in Tonal Harmony", "summary": "The chordal seventh falls, the leading tone rises, and the tritone at the centre of V7 resolves by step. What each diatonic seventh chord is for.", "content_html": "Adding a seventh to a triad changes what the chord is for. A triad can sit still; a seventh chord cannot, because the added note is a dissonance with a destination. That single fact explains the behaviour of every diatonic seventh chord in tonal music, including the ones that sound gentle.
\nStack one more third on each triad of a key and you get the full set.
\n| In major (C) | Notes | Quality | Usual job |
|---|---|---|---|
| Imaj7 | C E G B | major seventh | tonic colour, rarely a resting point |
| ii7 | D F A C | minor seventh | the standard predominant |
| iii7 | E G B D | minor seventh | weak tonic substitute, passing |
| IVmaj7 | F A C E | major seventh | predominant with a softer edge than V7 expects |
| V7 | G B D F | dominant seventh | the dominant, the strongest pull home |
| vi7 | A C E G | minor seventh | tonic substitute, often leading to ii |
| viiø7 | B D F A | half-diminished seventh | dominant function without the root |
| In minor (A) | Notes | Quality | Usual job |
|---|---|---|---|
| i7 | A C E G | minor seventh | tonic colour |
| iiø7 | B D F A | half-diminished seventh | predominant |
| III+maj7 | C E G♯ B | augmented major seventh | tonic area when the leading tone is in use |
| iv7 | D F A C | minor seventh | predominant, heavier than in major |
| V7 | E G♯ B D | dominant seventh | the dominant in minor |
| VImaj7 | F A C E | major seventh | tonic substitute |
| vii°7 | G♯ B D F | fully diminished seventh | dominant function without a root |
Minor is messier because the sixth and seventh scale degrees are flexible. The A minor key uses G♯ when the dominant needs a leading tone and G when the music leans toward the relative major, so a chord like III appears in two forms, C–E–G and C–E–G♯, and the second one takes a major seventh above it.
\nThe seventh of a chord is dissonant against the bass: in a root-position seventh chord it sounds a seventh above the root, which is a second below its octave. Dissonances of that kind resolve by step in the direction the interval leans, and a note that leans against the bass resolves downwards. The chordal seventh therefore falls by step into the third of whatever chord comes next, and composers arrange the next chord so that this resolution is already a chord tone.
\nThat is why the seventh is a compositional convenience rather than an obstacle. Because it must move, it guarantees that the next chord’s third is occupied by a real voice, and because it moves down by step, the motion is smooth. Chains of seventh chords with roots falling by fifth are the clearest result: every seventh resolves into the third of the following chord, and the whole sequence leads somewhere without any chord being static.
\n\nE minor seven, A minor seven, D minor seven, G seven, C. The bass falls by fifth at every step and each seventh sinks one degree into the next chord’s third.
\nThe dominant seventh contains both tendency tones of the key, scale degree 4 and the leading tone, and they form a tritone at the centre of the chord. Scale degree 4 falls to 3, the leading tone rises to 1, and the two voices move in contrary motion into the third of the tonic triad.
\nWhether that tritone appears as a diminished fifth or an augmented fourth depends on the inversion, and the difference matters to a singer reading intervals rather than a theorist reading a symbol. When the leading tone is the lower of the two notes, as in root position and in first inversion, the interval is a diminished fifth. When the seventh is the lower note, as in second and third inversion, it is an augmented fourth. The resolution is the same either way:
\nThat is a third-inversion dominant seventh resolving to a first-inversion tonic. The bass note is the seventh of the chord, so the bass itself makes the stepwise descent from scale degree 4 to 3.
\nTwo rules govern the tritone when it sits between the bass and an upper voice. A diminished fifth above the bass must resolve inward to a third; an augmented fourth above the bass must resolve outward to a sixth. Get that backwards and the interval will still look resolvable on paper while the voices collide in performance.
\nThe dominant seventh has four notes and the tonic triad has three, so one of them usually gives something up. In a strong cadence where the soprano takes the leading tone, the dominant seventh often appears without its fifth, the root doubled in its place. That version resolves to a complete tonic triad with its fifth present.
\nThe alternative is a complete dominant seventh resolving to a tonic with a doubled third and no fifth. Both solutions work, and both avoid the same trap: a seventh chord whose fifth is doubled or whose seventh is doubled will produce parallel octaves or a collision at the resolution, because two voices cannot both resolve the same obligation in the same direction without moving in parallel.
\nThis is also the answer to a question beginners ask about final chords. A tonic seventh chord at the end of a tonal phrase sounds open, because its seventh still has somewhere to go. The stable ending is the triad, not Imaj7.
\nEvery diatonic seventh chord has a habitual next chord, and the pattern is always the same: the seventh resolves, and the destination is chosen so that the resolution is consonant.
\nNone of these are fixed rules of harmony; they are consequences of the resolution. Any chord that contains the note the seventh is moving to will work, which is why hearing the seventh resolve is a more reliable guide to what comes next than memorising a chart.
\nDominant-function chords keep their obligations even when they resolve somewhere unexpected. In a deceptive resolution the dominant seventh moves to vi instead of I, and both tendency tones still do their job: the seventh falls to the fifth of vi and the leading tone rises to its third. The surprise lives entirely in the bass.
\n\nFour errors account for most of them.
\nDoubling the seventh. Two voices on the same dissonant note must both fall, and two voices falling the same distance create parallel motion, usually parallel octaves into the next chord. The seventh is never doubled.
\nLeaving the seventh unresolved. It can move to another chord tone of the next chord only if that tone lies a step below, which is rare. Otherwise the dissonance hangs in the air and the progression sounds unfinished.
\nTreating inversions as interchangeable. V4/2 in particular has a specific destination, the first-inversion tonic, because its bass is the seventh. Used elsewhere it produces a bass line that contradicts its own harmony.
\nForgetting that a diminished fifth and an augmented fourth need opposite treatment. One resolves inward, the other outward, and the two differ by a single inversion.
\nTry a four-chord exercise in C major: ii7, V7, Imaj7, IVmaj7. Play it, then sing the note that changes at each step. The seventh of ii7 becomes the third of V7; the seventh of V7 resolves into the tonic and the leading tone arrives with it; the new seventh on the tonic falls again into the subdominant. Once that chain of descending steps is audible rather than theoretical, seventh chords stop being chords with extra notes and start being chords with directions. The virtual piano is enough to check any of these progressions by ear, and secondary dominants covers what happens when the same mechanism is aimed at a tonic that is not the main one.
", "content_text": "Adding a seventh to a triad changes what the chord is for. A triad can sit still; a seventh chord cannot, because the added note is a dissonance with a destination. That single fact explains the behaviour of every diatonic seventh chord in tonal music, including the ones that sound gentle.\n#The diatonic seventh chords\nStack one more third on each triad of a key and you get the full set.\nIn major (C)NotesQualityUsual jobImaj7C E G Bmajor seventhtonic colour, rarely a resting pointii7D F A Cminor sevenththe standard predominantiii7E G B Dminor seventhweak tonic substitute, passingIVmaj7F A C Emajor seventhpredominant with a softer edge than V7 expectsV7G B D Fdominant sevenththe dominant, the strongest pull homevi7A C E Gminor seventhtonic substitute, often leading to iiviiø7B D F Ahalf-diminished seventhdominant function without the root\nIn minor (A)NotesQualityUsual jobi7A C E Gminor seventhtonic colouriiø7B D F Ahalf-diminished seventhpredominantIII+maj7C E G♯ Baugmented major seventhtonic area when the leading tone is in useiv7D F A Cminor seventhpredominant, heavier than in majorV7E G♯ B Ddominant sevenththe dominant in minorVImaj7F A C Emajor seventhtonic substitutevii°7G♯ B D Ffully diminished seventhdominant function without a root\nMinor is messier because the sixth and seventh scale degrees are flexible. The A minor key uses G♯ when the dominant needs a leading tone and G when the music leans toward the relative major, so a chord like III appears in two forms, C–E–G and C–E–G♯, and the second one takes a major seventh above it.\n#Why the seventh has to fall\nThe seventh of a chord is dissonant against the bass: in a root-position seventh chord it sounds a seventh above the root, which is a second below its octave. Dissonances of that kind resolve by step in the direction the interval leans, and a note that leans against the bass resolves downwards. The chordal seventh therefore falls by step into the third of whatever chord comes next, and composers arrange the next chord so that this resolution is already a chord tone.\nThat is why the seventh is a compositional convenience rather than an obstacle. Because it must move, it guarantees that the next chord’s third is occupied by a real voice, and because it moves down by step, the motion is smooth. Chains of seventh chords with roots falling by fifth are the clearest result: every seventh resolves into the third of the following chord, and the whole sequence leads somewhere without any chord being static.\nE3+B3+D4+G4 A3+E4+G4+C5 D3+A3+C4+F4 G3+D4+F4+B4 C4+E4+G4+C5\nE minor seven, A minor seven, D minor seven, G seven, C. The bass falls by fifth at every step and each seventh sinks one degree into the next chord’s third.\n#The tritone in the dominant seventh\nThe dominant seventh contains both tendency tones of the key, scale degree 4 and the leading tone, and they form a tritone at the centre of the chord. Scale degree 4 falls to 3, the leading tone rises to 1, and the two voices move in contrary motion into the third of the tonic triad.\nWhether that tritone appears as a diminished fifth or an augmented fourth depends on the inversion, and the difference matters to a singer reading intervals rather than a theorist reading a symbol. When the leading tone is the lower of the two notes, as in root position and in first inversion, the interval is a diminished fifth. When the seventh is the lower note, as in second and third inversion, it is an augmented fourth. The resolution is the same either way:\nroot position V7 resolves to I, with the tritone closing into the third of the tonicfirst inversion V6/5 resolves to I, with the leading tone in the bass rising to the tonicsecond inversion V4/3 resolves to I, with the tritone opening outwardthird inversion V4/2 resolves to I6, because the bass is the seventh and it must fall\nF3+G3+B3+D4 E3+G3+C4\nThat is a third-inversion dominant seventh resolving to a first-inversion tonic. The bass note is the seventh of the chord, so the bass itself makes the stepwise descent from scale degree 4 to 3.\nTwo rules govern the tritone when it sits between the bass and an upper voice. A diminished fifth above the bass must resolve inward to a third; an augmented fourth above the bass must resolve outward to a sixth. Get that backwards and the interval will still look resolvable on paper while the voices collide in performance.\n#Incomplete dominants and complete tonics\nThe dominant seventh has four notes and the tonic triad has three, so one of them usually gives something up. In a strong cadence where the soprano takes the leading tone, the dominant seventh often appears without its fifth, the root doubled in its place. That version resolves to a complete tonic triad with its fifth present.\nThe alternative is a complete dominant seventh resolving to a tonic with a doubled third and no fifth. Both solutions work, and both avoid the same trap: a seventh chord whose fifth is doubled or whose seventh is doubled will produce parallel octaves or a collision at the resolution, because two voices cannot both resolve the same obligation in the same direction without moving in parallel.\nThis is also the answer to a question beginners ask about final chords. A tonic seventh chord at the end of a tonal phrase sounds open, because its seventh still has somewhere to go. The stable ending is the triad, not Imaj7.\n#The other sevenths and their destinations\nEvery diatonic seventh chord has a habitual next chord, and the pattern is always the same: the seventh resolves, and the destination is chosen so that the resolution is consonant.\nii7 goes to V, and its seventh is scale degree 1 falling to the leading tone, which is the third of the dominantviiø7 and vii°7 go to I, with scale degree 6 in the seventh falling to scale degree 5IVmaj7 goes to V, with scale degree 3 falling to the fifth of the dominantvi7 usually goes to ii or IV, with scale degree 5 falling to the third of the new chordImaj7 goes to IV, with the leading tone falling to the third of the subdominant\nNone of these are fixed rules of harmony; they are consequences of the resolution. Any chord that contains the note the seventh is moving to will work, which is why hearing the seventh resolve is a more reliable guide to what comes next than memorising a chart.\nDominant-function chords keep their obligations even when they resolve somewhere unexpected. In a deceptive resolution the dominant seventh moves to vi instead of I, and both tendency tones still do their job: the seventh falls to the fifth of vi and the leading tone rises to its third. The surprise lives entirely in the bass.\nG3+B3+D4+F4 A3+C4+E4+C5\n#Where seventh chords go wrong\nFour errors account for most of them.\nDoubling the seventh. Two voices on the same dissonant note must both fall, and two voices falling the same distance create parallel motion, usually parallel octaves into the next chord. The seventh is never doubled.\nLeaving the seventh unresolved. It can move to another chord tone of the next chord only if that tone lies a step below, which is rare. Otherwise the dissonance hangs in the air and the progression sounds unfinished.\nTreating inversions as interchangeable. V4/2 in particular has a specific destination, the first-inversion tonic, because its bass is the seventh. Used elsewhere it produces a bass line that contradicts its own harmony.\nForgetting that a diminished fifth and an augmented fourth need opposite treatment. One resolves inward, the other outward, and the two differ by a single inversion.\n#Working with them\nTry a four-chord exercise in C major: ii7, V7, Imaj7, IVmaj7. Play it, then sing the note that changes at each step. The seventh of ii7 becomes the third of V7; the seventh of V7 resolves into the tonic and the leading tone arrives with it; the new seventh on the tonic falls again into the subdominant. Once that chain of descending steps is audible rather than theoretical, seventh chords stop being chords with extra notes and start being chords with directions. The virtual piano is enough to check any of these progressions by ear, and secondary dominants covers what happens when the same mechanism is aimed at a tonic that is not the main one.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "seventh chords", "dominant seventh", "resolution" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/seventh-chords-and-resolution.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/sonata-form/", "url": "https://musics.name/articles/sonata-form/", "title": "Sonata Form: The Tonal Plot", "summary": "Sonata form is a tonal argument, not a set of labelled boxes. What matters is which key you are in, and how the music gets home again.", "content_html": "Ask someone to explain sonata form and you will get a diagram: exposition, development, recapitulation, with an optional introduction and coda, and a list of themes labelled first and second. The diagram is not wrong, but it describes the surface. What holds a sonata movement together is a much older and simpler idea: leave the tonic, come back, and make the return mean something.
\nOnce that plot is clear, the sections stop being boxes to fill and become problems to solve. The transition has a job because the music has to leave. The development has a job because the music has been left. The recapitulation has a job because the material written for another key has to be brought home, and some of it does not fit.
\nThe essential scheme is a two-part design: a stretch of music that establishes the tonic and then leaves it, and a stretch that returns to the tonic and stays. That polarity is older than the sonata, older even than the classical era; the binary dance forms of the Baroque already present the outline in miniature, with the first half modulating to the dominant and the second half working its way back.
\nWhat the eighteenth century added was scale and weight. A binary dance form travels to the dominant and back in a few bars. A sonata movement does the same thing over several minutes, and the trip is long enough to need architecture: a clear departure, a middle section that keeps the destination uncertain, and a return that reveals how the opening material and the closing material relate.
\nThe name itself is a later invention. The theorist Adolf Bernhard Marx is usually credited with fixing the term in the nineteenth century, and most of the modern vocabulary (first theme, transition, secondary theme) comes from nineteenth- and twentieth-century analysis rather than from the composers' own descriptions. Charles Rosen’s Sonata Forms (1980) is still the clearest argument that the form is best understood as a family of solutions to the tonal problem rather than a template.
\nThe vocabulary is still worth having, because it names the jobs:
\n| Section | Textbook job | Key area | What it often actually does |
|---|---|---|---|
| First theme | state the main idea | tonic | presents a motive, a texture, or a figure the movement will use |
| Transition | modulate | tonic → new key | develops the first idea and modulates, or briefly links the areas |
| Secondary theme | provide contrast | V, III or VI | often reuses the opening motive rather than contrasting with it |
| Closing area | confirm the new key | new key | repeats cadential figures until the new key is unambiguous |
| Development | work the material | unstable | fragments, sequences and combines motives, avoiding the tonic |
| Recapitulation | return | tonic | restates the exposition with the second area transposed |
| Coda | finish | tonic | sometimes long enough to be a second development |
The exposition’s first job is to make the tonic unmistakable. That is why so many movements open with an assertive tonic statement: a triadic figure, a cadential progression, a repeated rhythmic motive. The tonic has to be established before it can be left, and the leaving has to sound like an intention rather than an accident.
\nThe transition does the leaving. It may be a long developmental passage that seizes on the first theme’s motive and pushes it toward the new key, or it may be nothing more than three or four bars of common-chord pivot. Its one measurable obligation is arrival: the transition must prepare a cadence that closes in a new key and stops, so that the ear registers the arrival as a place rather than a waypoint. Recent sonata theory calls that moment the medial caesura, and the term earns its place because the pause is what makes the second key area audible as a distinct space.
\nThe secondary theme’s key is chosen from a small set of options. In a major-key movement it is normally the dominant, occasionally the relative minor or the mediant. In a minor-key movement it is normally the relative major, sometimes the minor dominant. Beethoven’s op. 2 no. 1, in F minor, moves to A♭ major for its second area; Mozart’s K. 545, in C major, moves to G. The choice is not free: the new key has to be far enough from the tonic to sound like a journey and close enough to be recoverable.
\n\nTwo cadences, one after the other: a dominant-to-tonic close in C major, then the same shape closing in G major. That is the exposition’s tonal plot in six chords: establish, then establish somewhere else. In a minor-key movement the second close would land on the relative major or the minor dominant instead.
\nThe secondary and closing areas then confirm the new key with cadences, which is a repetitive-sounding job and often is repetitive in performance. It is also the part of the exposition the recapitulation will have to solve.
\nThe development is where the movement spends the tonality it accumulated. Its characteristic procedures are recognizable across two centuries.
\nFragmentation takes a motive apart and repeats a piece of it: four bars become two, then one, then half a bar, until the figure is a rhythmic fingerprint rather than a tune. Sequence reuses that fragment at descending or ascending pitch levels, which produces motion without committing to a key. Contrapuntal combination sets two exposition ideas against each other, one of them inverted, extended or placed in a different register, which demonstrates that they belong to the same piece. Registral and textural shifts keep the ear oriented while the harmony wanders, and many developments build toward a dominant pedal that holds the dominant for as long as the tension can be sustained.
\nThe dominant pedal is the standard preparation for the return, and the moment when the pedal finally resolves is the retransition. Occasionally a development will fake the return, arriving on the tonic’s own dominant-functioning chord in a way that sounds like home until the harmony refuses to settle; this false recapitulation is a favourite of Haydn’s and a reliable way to extend a movement without adding material.
\nThe one thing a development avoids is the tonic in root position with a cadence. If the tonic arrives fully prepared in the middle of the movement, the movement’s argument is over.
\nRestating the exposition in the tonic looks mechanical and is not. The first theme comes back unchanged, because it was already in the tonic. The transition cannot come back unchanged, because its job was to modulate, and there is nowhere left to go; it has to be rewritten so that the secondary material arrives without leaving the key. Sometimes the transition is shortened to a few chords, sometimes it is extended into a passage that touches several keys before settling.
\nThe secondary theme is transposed into the tonic, and that transposition is the movement’s proof. The material that sounded like a destination in the exposition sounds like a resolution in the recapitulation, and the listener hears the difference even without noticing the key change.
\n\nThat is the sound of the retransition resolving: a dominant seventh holding on, then the tonic. Every recapitulation is some version of this moment, and the smaller the gap between the preparation and the arrival, the stronger the effect.
\nThe coda is the section the textbooks append last and the composers treat most freely. It may be a handful of cadences, and it may be a second development that revisits the movement’s material in a new light. The first movement of Beethoven’s Eroica has a coda long and eventful enough that analysts have argued about whether it counts as a third development section.
\nHaydn wrote expositions that state one theme and then return to it in the new key rather than introducing a contrasting second theme. That single habit breaks the diagram’s most basic assumption, that the two key areas are carried by two different ideas, while leaving the tonal plot untouched. The movement still leaves and returns; only the labelling fails.
\nMozart varies the proportions instead. Some of his second themes are long and lyrical; others are a brief cadential gesture that arrives almost before the transition has finished. His development sections range from a few transitional bars to a substantial working-out. Counting sections tells you very little about what a Mozart movement is doing.
\nBeethoven expands the coda and the development together, so that the movement’s weight shifts toward the end. Late in the tradition, the sonata principle survives in movements that no diagram can hold: the sonata-rondo, where a recurring tonic refrain alternates with episodes; the slow-movement form that presents two key areas but no development; the concerto first movement, where the exposition happens twice, once for the orchestra and once for the soloist, and the cadenza turns the moment before the coda into a soloist’s recapitulation.
\nThe most useful correction is to stop counting sections and start tracking keys. Cadences are the markers: each one tells you where the music thinks it is. When a movement’s cadences stop confirming the tonic and start confirming its dominant, you are in the exposition. When no key holds for more than a phrase, you are in the development. When the material that used to close in the dominant closes in the tonic instead, the recapitulation is under way.
\nTake a movement you know and mark only where the music lands. Write the key next to every strong cadence, from the first bar to the last, and the shape of the movement appears without any need to name themes. The circle of fifths is a useful reference for the keys involved, because the relationships that sonata form exploits (dominant, relative major, subdominant, mediant) are exactly the relationships that the circle lays out.
\nThen listen again with a single question in mind: what is the music doing at the moment it leaves the tonic? Sometimes the answer is a modulatory sequence, sometimes a single pivot chord, and sometimes a cadence that refuses to resolve until the new key is already established. The technique varies from movement to movement, and the plot does not.
", "content_text": "Ask someone to explain sonata form and you will get a diagram: exposition, development, recapitulation, with an optional introduction and coda, and a list of themes labelled first and second. The diagram is not wrong, but it describes the surface. What holds a sonata movement together is a much older and simpler idea: leave the tonic, come back, and make the return mean something.\nOnce that plot is clear, the sections stop being boxes to fill and become problems to solve. The transition has a job because the music has to leave. The development has a job because the music has been left. The recapitulation has a job because the material written for another key has to be brought home, and some of it does not fit.\n#The plot, not the sections\nThe essential scheme is a two-part design: a stretch of music that establishes the tonic and then leaves it, and a stretch that returns to the tonic and stays. That polarity is older than the sonata, older even than the classical era; the binary dance forms of the Baroque already present the outline in miniature, with the first half modulating to the dominant and the second half working its way back.\nWhat the eighteenth century added was scale and weight. A binary dance form travels to the dominant and back in a few bars. A sonata movement does the same thing over several minutes, and the trip is long enough to need architecture: a clear departure, a middle section that keeps the destination uncertain, and a return that reveals how the opening material and the closing material relate.\nThe name itself is a later invention. The theorist Adolf Bernhard Marx is usually credited with fixing the term in the nineteenth century, and most of the modern vocabulary (first theme, transition, secondary theme) comes from nineteenth- and twentieth-century analysis rather than from the composers' own descriptions. Charles Rosen’s Sonata Forms (1980) is still the clearest argument that the form is best understood as a family of solutions to the tonal problem rather than a template.\nThe vocabulary is still worth having, because it names the jobs:\nSectionTextbook jobKey areaWhat it often actually doesFirst themestate the main ideatonicpresents a motive, a texture, or a figure the movement will useTransitionmodulatetonic → new keydevelops the first idea and modulates, or briefly links the areasSecondary themeprovide contrastV, III or VIoften reuses the opening motive rather than contrasting with itClosing areaconfirm the new keynew keyrepeats cadential figures until the new key is unambiguousDevelopmentwork the materialunstablefragments, sequences and combines motives, avoiding the tonicRecapitulationreturntonicrestates the exposition with the second area transposedCodafinishtonicsometimes long enough to be a second development\n#The exposition: leaving home\nThe exposition’s first job is to make the tonic unmistakable. That is why so many movements open with an assertive tonic statement: a triadic figure, a cadential progression, a repeated rhythmic motive. The tonic has to be established before it can be left, and the leaving has to sound like an intention rather than an accident.\nThe transition does the leaving. It may be a long developmental passage that seizes on the first theme’s motive and pushes it toward the new key, or it may be nothing more than three or four bars of common-chord pivot. Its one measurable obligation is arrival: the transition must prepare a cadence that closes in a new key and stops, so that the ear registers the arrival as a place rather than a waypoint. Recent sonata theory calls that moment the medial caesura, and the term earns its place because the pause is what makes the second key area audible as a distinct space.\nThe secondary theme’s key is chosen from a small set of options. In a major-key movement it is normally the dominant, occasionally the relative minor or the mediant. In a minor-key movement it is normally the relative major, sometimes the minor dominant. Beethoven’s op. 2 no. 1, in F minor, moves to A♭ major for its second area; Mozart’s K. 545, in C major, moves to G. The choice is not free: the new key has to be far enough from the tonic to sound like a journey and close enough to be recoverable.\nG4+B4+D5 C4+E4+G4 D4+F#4+A4 G4+B4+D5\nTwo cadences, one after the other: a dominant-to-tonic close in C major, then the same shape closing in G major. That is the exposition’s tonal plot in six chords: establish, then establish somewhere else. In a minor-key movement the second close would land on the relative major or the minor dominant instead.\nThe secondary and closing areas then confirm the new key with cadences, which is a repetitive-sounding job and often is repetitive in performance. It is also the part of the exposition the recapitulation will have to solve.\n#The development: what working the material means\nThe development is where the movement spends the tonality it accumulated. Its characteristic procedures are recognizable across two centuries.\nFragmentation takes a motive apart and repeats a piece of it: four bars become two, then one, then half a bar, until the figure is a rhythmic fingerprint rather than a tune. Sequence reuses that fragment at descending or ascending pitch levels, which produces motion without committing to a key. Contrapuntal combination sets two exposition ideas against each other, one of them inverted, extended or placed in a different register, which demonstrates that they belong to the same piece. Registral and textural shifts keep the ear oriented while the harmony wanders, and many developments build toward a dominant pedal that holds the dominant for as long as the tension can be sustained.\nThe dominant pedal is the standard preparation for the return, and the moment when the pedal finally resolves is the retransition. Occasionally a development will fake the return, arriving on the tonic’s own dominant-functioning chord in a way that sounds like home until the harmony refuses to settle; this false recapitulation is a favourite of Haydn’s and a reliable way to extend a movement without adding material.\nThe one thing a development avoids is the tonic in root position with a cadence. If the tonic arrives fully prepared in the middle of the movement, the movement’s argument is over.\n#The recapitulation: the same music, rewritten\nRestating the exposition in the tonic looks mechanical and is not. The first theme comes back unchanged, because it was already in the tonic. The transition cannot come back unchanged, because its job was to modulate, and there is nowhere left to go; it has to be rewritten so that the secondary material arrives without leaving the key. Sometimes the transition is shortened to a few chords, sometimes it is extended into a passage that touches several keys before settling.\nThe secondary theme is transposed into the tonic, and that transposition is the movement’s proof. The material that sounded like a destination in the exposition sounds like a resolution in the recapitulation, and the listener hears the difference even without noticing the key change.\nG3+B3+D4+F4 C4+E4+G4\nThat is the sound of the retransition resolving: a dominant seventh holding on, then the tonic. Every recapitulation is some version of this moment, and the smaller the gap between the preparation and the arrival, the stronger the effect.\nThe coda is the section the textbooks append last and the composers treat most freely. It may be a handful of cadences, and it may be a second development that revisits the movement’s material in a new light. The first movement of Beethoven’s Eroica has a coda long and eventful enough that analysts have argued about whether it counts as a third development section.\n#Why the textbook diagram fails\nHaydn wrote expositions that state one theme and then return to it in the new key rather than introducing a contrasting second theme. That single habit breaks the diagram’s most basic assumption, that the two key areas are carried by two different ideas, while leaving the tonal plot untouched. The movement still leaves and returns; only the labelling fails.\nMozart varies the proportions instead. Some of his second themes are long and lyrical; others are a brief cadential gesture that arrives almost before the transition has finished. His development sections range from a few transitional bars to a substantial working-out. Counting sections tells you very little about what a Mozart movement is doing.\nBeethoven expands the coda and the development together, so that the movement’s weight shifts toward the end. Late in the tradition, the sonata principle survives in movements that no diagram can hold: the sonata-rondo, where a recurring tonic refrain alternates with episodes; the slow-movement form that presents two key areas but no development; the concerto first movement, where the exposition happens twice, once for the orchestra and once for the soloist, and the cadenza turns the moment before the coda into a soloist’s recapitulation.\nThe most useful correction is to stop counting sections and start tracking keys. Cadences are the markers: each one tells you where the music thinks it is. When a movement’s cadences stop confirming the tonic and start confirming its dominant, you are in the exposition. When no key holds for more than a phrase, you are in the development. When the material that used to close in the dominant closes in the tonic instead, the recapitulation is under way.\n#How to hear it\nTake a movement you know and mark only where the music lands. Write the key next to every strong cadence, from the first bar to the last, and the shape of the movement appears without any need to name themes. The circle of fifths is a useful reference for the keys involved, because the relationships that sonata form exploits (dominant, relative major, subdominant, mediant) are exactly the relationships that the circle lays out.\nThen listen again with a single question in mind: what is the music doing at the moment it leaves the tonic? Sometimes the answer is a modulatory sequence, sometimes a single pivot chord, and sometimes a cadence that refuses to resolve until the new key is already established. The technique varies from movement to movement, and the plot does not.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "sonata form", "form", "tonality" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/sonata-form.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/string-quartet-voicing/", "url": "https://musics.name/articles/string-quartet-voicing/", "title": "Voicing a String Quartet: Registers, Doublings, Weight", "summary": "The quartet is four registers, not four violins. What each instrument can sustain, where the texture muddies, and which double stops lie under the hand.", "content_html": "A string quartet has no mixer, no conductor, and nowhere to hide. Four instruments sit close enough to hear each other’s bow noise, and every vertical sonority is the direct result of four players deciding how loudly to play. That makes the quartet the most exposing ensemble in the standard repertoire and a good place to learn what a texture actually costs, because nothing you write can be fixed in post.
\nThe four parts are not interchangeable versions of the same voice. Each occupies a register band with its own behaviour, and the character of a line changes when it crosses into a different band.
\n| Part | Sounding range | Comfortable tessitura | Behaviour in the texture |
|---|---|---|---|
| Violin I | G3 to about E7 | G3 to E6 | Projects through anything; the top octave of its range is brilliant and carries a melody over a full four-part texture |
| Violin II | G3 to about E7 | G3 to C6 | The same instrument in the tenor-of-the-treble role; useful as a second melodic voice because it can match the first violin’s colour but not its penetration |
| Viola | C3 to about E6 | C3 to A5 | Covered and slightly hoarse on the C and G strings, nasal and penetrating on the A string; the middle of a quartet texture usually lives here |
| Cello | C2 to about G5 | C2 to A4 | The floor of the ensemble; above A4 it becomes a tense, thin voice, not a louder version of the same thing |
Two consequences follow immediately. First, the cello’s high register is a colour, not a range: writing the bass line above the staff changes its function as well as its pitch. Second, the viola’s lower strings cannot be made to shout, so any texture that requires the inner voice to cut through is asking for a different instrument.
\nThe general rule is the one orchestrators repeat: wide spacing at the bottom, close spacing at the top. Low intervals are thick because their partials collide in a region where the ear resolves them poorly. A third or a second in the cello’s lower octave does not sound like a third; it sounds like a rumble with identity problems.
\n\nA close-position C minor triad around middle C. On strings this is the expensive version: the cello and viola are crowded, the third sits low where it beats, and the voicing needs two instruments in a narrow band.
\n\nThe same chord with the third moved up an octave. The bottom is now a fifth and a fourth apart, the minor third sits in the viola’s speaking register, and the chord is recognisably the same harmony with twice the clarity. Neither example is played by a quartet here, but the spacing problem is the same one strings have, only worse, because the cello sustains and the ear has time to notice the beating.
\nThe upper limit of the spacing rule matters less than the lower. A wide gap between the first and second violin is a normal texture, and it is how a quartet sounds transparent; a wide gap in the low half of the ensemble leaves the harmony hollow, because a fifth with nothing between it and the melody is a chord with its middle missing.
\nThe quartet’s reputation for difficulty comes partly from repertoire in which all four parts matter. In Haydn’s six quartets op. 76 (1797 to 1798) the material is handed around the group, and the second violin and viola are given statements of the main idea rather than held notes. In Beethoven’s op. 130 (1825 to 1826) the texture often splits into two independent duos, and the first movement of op. 131 (1826) begins with a fugue, which makes the point structurally instead of decoratively.
\nThat distribution has a practical effect on voicing. If the second violin has the melody, the first violin cannot double it without erasing the exchange, and if the viola carries the tune, the inner harmony has to be thinned so that the tune is not competing with a chord tone in its own register. The dull quartet is usually not badly written harmony; it is four parts all doing the same job.
\nThere is one asymmetry worth keeping in mind. The ear reads the top voice as the melody and the bottom voice as the harmonic foundation, and both are heard more clearly than anything between them. Inner parts can therefore move faster, cross, and trade material without confusion, while the outer voices need care about register and articulation. The second violin and the viola are the parts with the most freedom in the texture and the least room for filler material.
\nA string player can sound two notes at once, which tempts a composer into writing chordal textures no section could sustain. What actually works is defined by the shape of the hand over four strings.
\nOpen strings are the other idiom to know. G, D, A, and E on the violin (C, G, D and A on viola and cello) ring with a resonance stopped notes cannot match, and a melody that keeps touching them sounds open and in tune. The cost is that an open string has no dynamic shape: it is whatever the bow gives it, so a phrase that needs a swell on an open note forces the player to substitute a stopped note or to live with the flat result. The same applies to pizzicato and to the mute, which changes colour instantly and quietly, which is why quartets use it for transitions.
\nViola parts are written in alto clef, with middle C on the middle line. That is not a historical curiosity: the viola’s range sits exactly where treble and bass clefs would need too many ledger lines, and the alto clef has been the convention since the classical period. When a composer sketches at the piano and then assigns the inner lines, the clef is where the transfer usually breaks, because a pianist reads a chord shape and the viola player needs a single line.
\nThe deeper problem with the piano sketch is idiomatic gravity. A pianist’s hands move in ways a string player’s do not: close voicings, repeated chords, big leaps in the left hand, and wide left-hand intervals that no cello can manage at speed. If the sketch is well made, the fix is not to transcribe it but to redistribute it. A chord that the left hand holds for a bar might become a walking line in the cello with two notes in the viola; a piano leap might become a registral hand-off between instruments.
\n\nThe same triad descending through the four registers. Individually each note is unremarkable; assigned to first violin, second violin, viola, and cello in turn, the figure is a hand-off that no single instrument could colour as specifically.
\nFour practical questions catch most of the problems before rehearsal does.
\nThe ensemble’s balance is acoustic, so the answer to a texture that is too thick is always a change of register or a reduction in the number of notes, never a request to play softer. A quartet that is told to play quietly all the time loses its forte, and a composer who writes four-part chords continuously has no way to build a climax at all.
", "content_text": "A string quartet has no mixer, no conductor, and nowhere to hide. Four instruments sit close enough to hear each other’s bow noise, and every vertical sonority is the direct result of four players deciding how loudly to play. That makes the quartet the most exposing ensemble in the standard repertoire and a good place to learn what a texture actually costs, because nothing you write can be fixed in post.\n#Four instruments, four registers\nThe four parts are not interchangeable versions of the same voice. Each occupies a register band with its own behaviour, and the character of a line changes when it crosses into a different band.\nPartSounding rangeComfortable tessituraBehaviour in the textureViolin IG3 to about E7G3 to E6Projects through anything; the top octave of its range is brilliant and carries a melody over a full four-part textureViolin IIG3 to about E7G3 to C6The same instrument in the tenor-of-the-treble role; useful as a second melodic voice because it can match the first violin’s colour but not its penetrationViolaC3 to about E6C3 to A5Covered and slightly hoarse on the C and G strings, nasal and penetrating on the A string; the middle of a quartet texture usually lives hereCelloC2 to about G5C2 to A4The floor of the ensemble; above A4 it becomes a tense, thin voice, not a louder version of the same thing\nTwo consequences follow immediately. First, the cello’s high register is a colour, not a range: writing the bass line above the staff changes its function as well as its pitch. Second, the viola’s lower strings cannot be made to shout, so any texture that requires the inner voice to cut through is asking for a different instrument.\n#Spacing and weight: where a quartet chord goes wrong\nThe general rule is the one orchestrators repeat: wide spacing at the bottom, close spacing at the top. Low intervals are thick because their partials collide in a region where the ear resolves them poorly. A third or a second in the cello’s lower octave does not sound like a third; it sounds like a rumble with identity problems.\nC3+Eb3+G3+C4\nA close-position C minor triad around middle C. On strings this is the expensive version: the cello and viola are crowded, the third sits low where it beats, and the voicing needs two instruments in a narrow band.\nC3+G3+C4+Eb4\nThe same chord with the third moved up an octave. The bottom is now a fifth and a fourth apart, the minor third sits in the viola’s speaking register, and the chord is recognisably the same harmony with twice the clarity. Neither example is played by a quartet here, but the spacing problem is the same one strings have, only worse, because the cello sustains and the ear has time to notice the beating.\nThe upper limit of the spacing rule matters less than the lower. A wide gap between the first and second violin is a normal texture, and it is how a quartet sounds transparent; a wide gap in the low half of the ensemble leaves the harmony hollow, because a fifth with nothing between it and the melody is a chord with its middle missing.\n#The roles are not a melody and three accompanists\nThe quartet’s reputation for difficulty comes partly from repertoire in which all four parts matter. In Haydn’s six quartets op. 76 (1797 to 1798) the material is handed around the group, and the second violin and viola are given statements of the main idea rather than held notes. In Beethoven’s op. 130 (1825 to 1826) the texture often splits into two independent duos, and the first movement of op. 131 (1826) begins with a fugue, which makes the point structurally instead of decoratively.\nThat distribution has a practical effect on voicing. If the second violin has the melody, the first violin cannot double it without erasing the exchange, and if the viola carries the tune, the inner harmony has to be thinned so that the tune is not competing with a chord tone in its own register. The dull quartet is usually not badly written harmony; it is four parts all doing the same job.\nThere is one asymmetry worth keeping in mind. The ear reads the top voice as the melody and the bottom voice as the harmonic foundation, and both are heard more clearly than anything between them. Inner parts can therefore move faster, cross, and trade material without confusion, while the outer voices need care about register and articulation. The second violin and the viola are the parts with the most freedom in the texture and the least room for filler material.\n#Double stops, open strings, and the limits of the hand\nA string player can sound two notes at once, which tempts a composer into writing chordal textures no section could sustain. What actually works is defined by the shape of the hand over four strings.\nThirds and sixths lie comfortably under the fingers in most positions and can be sustained. They are the safe double stops.Octaves need one finger across two strings, which is playable and tiring; a long passage of octaves is a different proposition from a short one.Fifths are playable but intonation-critical, since the same finger stops both strings in the same position.Tenths require a large stretch across strings in a high position. They exist in the repertoire as effects, not as texture.Four-note chords are normally rolled, not sustained, unless the notes happen to fall under one hand shape.\nOpen strings are the other idiom to know. G, D, A, and E on the violin (C, G, D and A on viola and cello) ring with a resonance stopped notes cannot match, and a melody that keeps touching them sounds open and in tune. The cost is that an open string has no dynamic shape: it is whatever the bow gives it, so a phrase that needs a swell on an open note forces the player to substitute a stopped note or to live with the flat result. The same applies to pizzicato and to the mute, which changes colour instantly and quietly, which is why quartets use it for transitions.\n#The viola’s clef, and why piano sketches mislead\nViola parts are written in alto clef, with middle C on the middle line. That is not a historical curiosity: the viola’s range sits exactly where treble and bass clefs would need too many ledger lines, and the alto clef has been the convention since the classical period. When a composer sketches at the piano and then assigns the inner lines, the clef is where the transfer usually breaks, because a pianist reads a chord shape and the viola player needs a single line.\nThe deeper problem with the piano sketch is idiomatic gravity. A pianist’s hands move in ways a string player’s do not: close voicings, repeated chords, big leaps in the left hand, and wide left-hand intervals that no cello can manage at speed. If the sketch is well made, the fix is not to transcribe it but to redistribute it. A chord that the left hand holds for a bar might become a walking line in the cello with two notes in the viola; a piano leap might become a registral hand-off between instruments.\nC5 G4 E4 C4\nThe same triad descending through the four registers. Individually each note is unremarkable; assigned to first violin, second violin, viola, and cello in turn, the figure is a hand-off that no single instrument could colour as specifically.\n#Designing a quartet texture\nFour practical questions catch most of the problems before rehearsal does.\nWhich part has the melody, and is that obvious from the texture alone? If two parts are equally prominent, one of them should be marked differently, or the doubling should go.What is the lowest sounding interval in this bar? If it is smaller than a fifth or so below C3, thin the voicing or move the chord up.Can the inner parts be sung by one player each, or are they written as a piano’s chord? Lines that leap a seventh in a bar will be played as such by every listener.What does the cello do while the melody is playing? A cello that only holds whole notes for three minutes is a wasted part, and one that plays the melody for two bars and then nothing is a soloist interruptus.\nThe ensemble’s balance is acoustic, so the answer to a texture that is too thick is always a change of register or a reduction in the number of notes, never a request to play softer. A quartet that is told to play quietly all the time loses its forte, and a composer who writes four-part chords continuously has no way to build a climax at all.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "string quartet", "voicing", "orchestration" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/string-quartet-voicing.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/swing-timing/", "url": "https://musics.name/articles/swing-timing/", "title": "Swing Timing: What the Notation Does Not Say", "summary": "Jazz notation asks for swing and leaves the ratio to the players, because the timing is tempo-dependent and a fixed written value would be wrong at most speeds.", "content_html": "A written instruction like swing or medium swing asks a player to change the placement of notes that are, on the page, perfectly even. That looks like an instruction with missing information, and it is one, deliberately. The missing information is the ratio, and the reason it is left out is that the ratio is not a constant.
\nA jazz chart in common time usually writes eighth notes as eighth notes and prints swing once at the top, or above the first line, or in the part. What the player is expected to know is that each pair of eighths inside a beat becomes unequal: the first note takes more than half of the beat, the second less.
\nWhere the feel has to be exact, scores write it out with the one symbol standard notation already has for dividing a beat in three, the triplet bracket. A written triplet pair with its first two notes tied gives the model most often taught: the first eighth occupies two thirds of the beat, the second one third. Some educational material uses a dotted eighth followed by a sixteenth instead, which is a different and more literal figure, and reading it as swing notation is a source of confusion rather than a shorthand worth adopting.
\nThere is a matching instruction in the other direction. Charts that want the pairs played equally, mostly in Latin and funk contexts and in passages taken from them, mark the music even eighths or straight so that the swing default is cancelled.
\nRecordings are the evidence here, and they agree on two things. The first is that the pair is genuinely unequal, which is why swing is not a myth of notation. The second is that the inequality is rarely the written two-to-one: the first note commonly takes somewhere between roughly 55 and 70 per cent of the beat, and the figure varies with tempo, with era and style, with the player, and with the position of the phrase.
\nThe variation inside one performance matters as much as the average. A soloist can sit close to even for four bars and widen the pairs at a phrase ending, where the extra weight lands on the first note of a pair. Two players in the same band can disagree by a few per cent without anyone calling it out of time, because what they are sharing is a placement of the beat rather than a length of the eighth. The lengthening of an individual note is not the only mechanism either: the first note of a pair tends to be the accented one and the second tends to be lighter and shorter, so the feel is carried by weight as much as by ratio.
\nTempo is the variable with the clearest effect. As the beat gets faster the pairs flatten towards even, because the time available for inequality is disappearing, and at the fastest tempi a bebop line notated in plain eighths is played close to straight. The lilt does not vanish; it moves into the accent pattern, the articulation and the phrasing across the barline.
\n| Approximate tempo | Typical first note of the pair | What the pairs sound like |
|---|---|---|
| 60–90 bpm | near or beyond two thirds | a rolling limp, the basis of a shuffle |
| 100–160 bpm | roughly two thirds to three fifths | the familiar swing feel, both notes clearly present |
| 170–220 bpm | narrowing towards even | the lilt survives in the accents while the pairs even out |
| 230 bpm and above | close to even | the eighth line reads almost straight; swing lives in phrasing |
The boundaries are approximate and move with style. A medium-tempo blues and a medium-tempo bebop head will not place the pairs identically even when a metronome agrees about the tempo, which is part of why the same written line can sound like two different styles.
\nSwing eighths are only one case of a general habit, and knowing which level is in play keeps the analysis honest.
\nAt medium tempo the pair sits inside the beat, and the beat is divided into two unequal halves. When the music wants the division to be explicitly three-part, as in a shuffle or some blues feels, the triplet model becomes literal and can be written. At fast tempo the eighth is no longer the expressive unit: pairs of sixteenths inside the eighth carry the inequality, and the eighths themselves are close to even. In all three cases the underlying question is the same, which unit is being divided and how unevenly.
\nConfusing the levels produces a specific and recognisable error. A line of fast eighths played with a wide medium-tempo ratio sounds staggered rather than fast, because the ratio is being applied where the ear is not subdividing. The polyrhythm article describes the neighbouring case, where the division is not merely uneven but genuinely conflicting.
\n\nEight pitches. Written as eighths they are equidistant on the page, and the swing instruction is a claim about where each pair sits inside its beat, not about the pitches.
\n\nThree notes that fill one beat evenly are a triplet. This is the shape the written model uses, and it is why the model is teachable: it converts a timing judgement into a countable figure that players can agree on before they have developed the judgement.
\nSequencers offer to do the work automatically. Many express swing as a percentage of the pair, with an even division at 50 per cent and the triplet model near 67, and applying a value shifts every second eighth later in the beat.
\nThree things are missing from that operation. The first is tempo dependence: the same percentage at 80 and at 240 bpm produces two different feels, and only one of them is idiomatic. The second is accent and articulation, since a played pair is unequal in weight as well as in length, and moving a note without changing how it is attacked leaves the shape without its emphasis. The third, and the largest, is the reference. A player places the pair against something: the ride pattern, the bass line, the click of the hi-hat on two and four. The ratio is a by-product of leaning against that reference, and quantising to a global percentage removes the leaning while keeping the arithmetic.
\nThe practical consequence for anyone writing or programming the music is that the ratio is the wrong thing to specify. What a chart needs to make clear is where the beat is and which subdivision is in play; what a player needs is a shared reference to place the pairs against, which is why practising with a metronome on two and four rather than on every beat produces a different and more useful result.
\nFor a composer, three options are honest. Mark the feel and write even eighths, which trusts the idiom and keeps the page readable. Write triplets where the feel must be exact, and mark the passage even eighths where it must not swing. Or, if the music is going to be performed by a machine, express the feel as a groove parameter and accept that it will be a rendering choice rather than a specification, since the same passage played by a person will place the same notes differently on a different evening. The distinction between a written ratio and a played one is the same distinction that separates notated meter from heard meter: the page holds the category, and the performance supplies the value.
", "content_text": "A written instruction like swing or medium swing asks a player to change the placement of notes that are, on the page, perfectly even. That looks like an instruction with missing information, and it is one, deliberately. The missing information is the ratio, and the reason it is left out is that the ratio is not a constant.\n#What the notation specifies\nA jazz chart in common time usually writes eighth notes as eighth notes and prints swing once at the top, or above the first line, or in the part. What the player is expected to know is that each pair of eighths inside a beat becomes unequal: the first note takes more than half of the beat, the second less.\nWhere the feel has to be exact, scores write it out with the one symbol standard notation already has for dividing a beat in three, the triplet bracket. A written triplet pair with its first two notes tied gives the model most often taught: the first eighth occupies two thirds of the beat, the second one third. Some educational material uses a dotted eighth followed by a sixteenth instead, which is a different and more literal figure, and reading it as swing notation is a source of confusion rather than a shorthand worth adopting.\nThere is a matching instruction in the other direction. Charts that want the pairs played equally, mostly in Latin and funk contexts and in passages taken from them, mark the music even eighths or straight so that the swing default is cancelled.\n#What performance does\nRecordings are the evidence here, and they agree on two things. The first is that the pair is genuinely unequal, which is why swing is not a myth of notation. The second is that the inequality is rarely the written two-to-one: the first note commonly takes somewhere between roughly 55 and 70 per cent of the beat, and the figure varies with tempo, with era and style, with the player, and with the position of the phrase.\nThe variation inside one performance matters as much as the average. A soloist can sit close to even for four bars and widen the pairs at a phrase ending, where the extra weight lands on the first note of a pair. Two players in the same band can disagree by a few per cent without anyone calling it out of time, because what they are sharing is a placement of the beat rather than a length of the eighth. The lengthening of an individual note is not the only mechanism either: the first note of a pair tends to be the accented one and the second tends to be lighter and shorter, so the feel is carried by weight as much as by ratio.\n#Where the ratio moves\nTempo is the variable with the clearest effect. As the beat gets faster the pairs flatten towards even, because the time available for inequality is disappearing, and at the fastest tempi a bebop line notated in plain eighths is played close to straight. The lilt does not vanish; it moves into the accent pattern, the articulation and the phrasing across the barline.\nApproximate tempoTypical first note of the pairWhat the pairs sound like60–90 bpmnear or beyond two thirdsa rolling limp, the basis of a shuffle100–160 bpmroughly two thirds to three fifthsthe familiar swing feel, both notes clearly present170–220 bpmnarrowing towards eventhe lilt survives in the accents while the pairs even out230 bpm and aboveclose to eventhe eighth line reads almost straight; swing lives in phrasing\nThe boundaries are approximate and move with style. A medium-tempo blues and a medium-tempo bebop head will not place the pairs identically even when a metronome agrees about the tempo, which is part of why the same written line can sound like two different styles.\n#Three levels of division\nSwing eighths are only one case of a general habit, and knowing which level is in play keeps the analysis honest.\nAt medium tempo the pair sits inside the beat, and the beat is divided into two unequal halves. When the music wants the division to be explicitly three-part, as in a shuffle or some blues feels, the triplet model becomes literal and can be written. At fast tempo the eighth is no longer the expressive unit: pairs of sixteenths inside the eighth carry the inequality, and the eighths themselves are close to even. In all three cases the underlying question is the same, which unit is being divided and how unevenly.\nConfusing the levels produces a specific and recognisable error. A line of fast eighths played with a wide medium-tempo ratio sounds staggered rather than fast, because the ratio is being applied where the ear is not subdividing. The polyrhythm article describes the neighbouring case, where the division is not merely uneven but genuinely conflicting.\nC4 D4 E4 F4 G4 F4 E4 D4\nEight pitches. Written as eighths they are equidistant on the page, and the swing instruction is a claim about where each pair sits inside its beat, not about the pitches.\nG3 B3 D4\nThree notes that fill one beat evenly are a triplet. This is the shape the written model uses, and it is why the model is teachable: it converts a timing judgement into a countable figure that players can agree on before they have developed the judgement.\n#Why a fixed offset sounds mechanical\nSequencers offer to do the work automatically. Many express swing as a percentage of the pair, with an even division at 50 per cent and the triplet model near 67, and applying a value shifts every second eighth later in the beat.\nThree things are missing from that operation. The first is tempo dependence: the same percentage at 80 and at 240 bpm produces two different feels, and only one of them is idiomatic. The second is accent and articulation, since a played pair is unequal in weight as well as in length, and moving a note without changing how it is attacked leaves the shape without its emphasis. The third, and the largest, is the reference. A player places the pair against something: the ride pattern, the bass line, the click of the hi-hat on two and four. The ratio is a by-product of leaning against that reference, and quantising to a global percentage removes the leaning while keeping the arithmetic.\n#The reference, not the ratio\nThe practical consequence for anyone writing or programming the music is that the ratio is the wrong thing to specify. What a chart needs to make clear is where the beat is and which subdivision is in play; what a player needs is a shared reference to place the pairs against, which is why practising with a metronome on two and four rather than on every beat produces a different and more useful result.\nFor a composer, three options are honest. Mark the feel and write even eighths, which trusts the idiom and keeps the page readable. Write triplets where the feel must be exact, and mark the passage even eighths where it must not swing. Or, if the music is going to be performed by a machine, express the feel as a groove parameter and accept that it will be a rendering choice rather than a specification, since the same passage played by a person will place the same notes differently on a different evening. The distinction between a written ratio and a played one is the same distinction that separates notated meter from heard meter: the page holds the category, and the performance supplies the value.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "swing", "jazz", "rhythm" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/swing-timing.md", "mime_type": "text/markdown", "title": "Markdown source" } ] }, { "id": "https://musics.name/articles/voice-leading-in-four-parts/", "url": "https://musics.name/articles/voice-leading-in-four-parts/", "title": "Voice Leading: Why the Rules Exist in Four-Part Writing", "summary": "Parallel fifths, tendency tones, spacing and doubling, read as solutions to one problem: keeping four lines heard as four lines.", "content_html": "Part-writing rules usually arrive as a list of prohibitions, which makes them feel arbitrary and easy to resent. They are neither. Almost all of them follow from a single aim: four voices that behave like four voices. Once you can hear what a broken rule does to a texture, the list stops being a set of prohibitions and becomes a set of diagnostics.
\nThe rules describe a style. In four-part choral writing from roughly 1650 to 1900, each voice is meant to have its own contour and its own register, so that a listener can follow the bass, or the top line, or the inner parts, and hear lines rather than a chord with notes stacked on it.
\nEvery rule on the list protects that independence, and every rule can be broken when independence is not the point. Parallel fifths in a Debussy piano piece or a distorted guitar part sound deliberate because the texture is not claiming four independent lines. The rules are not physics; they are the grammar of one musical language, and knowing the grammar is what allows you to break it on purpose rather than by accident.
\nFour-part writing starts from the bass, because the bass note decides which chord is in root position and which is inverted, and inversion is where a chord’s weight comes from. Once the bass is fixed, the upper three voices are arranged above it.
\nSpacing works on a simple principle: the closer two adjacent upper voices sit to each other, the more they sound like part of one harmonious mass, and the further apart they sit, the more the upper voice separates from the harmony underneath. Standard practice keeps soprano, alto and tenor within an octave of their neighbours and lets the bass sit further below, as much as a twelfth under the tenor in choral writing. Wider gaps between upper voices do not create clarity. They create a hole that the ear fills by hearing the middle voice as detached.
\nTwo voices can move in similar motion, in parallel motion (similar motion by the same interval), in contrary motion, or obliquely, where one stays and the other moves. The choice between them is most of what part writing is.
\nContrary motion between the outer voices is the default answer when you do not know what to do next, because the two ends of the texture pulling apart is what keeps the frame of the music open. Similar motion into a perfect fifth or octave between the outer voices, especially when the soprano gets there by leap, is the case called direct or hidden fifths and octaves, and it is discouraged for the same reason as the parallel kind.
\nParallel perfect fifths and octaves are the famous prohibition, and the reason is that they stop the texture from growing. Two voices an octave apart are the same pitch class, so the second voice is not adding a line at all. Two voices a fifth apart and moving in parallel keep the same relationship, and because a perfect fifth above the bass reinforces a frequency already present in the bass, the upper voice merges into the sound of the lower one rather than standing beside it. Two explanations, the contrapuntal and the acoustic, describe the same effect: the voices stop being independent.
\n\nThose are two parallel fifths. The upper voice is not wrong as a note; it simply has no independent life.
\nTwo notes in tonal harmony come with an obligation attached.
\nThe leading tone rises to the tonic. It sits a semitone below its destination, which is the smallest melodic step available and the most directed, and the style offers it no equally stable alternative. In an outer voice, where the ear follows it, it rises. In an inner voice it may fall, usually into a complete tonic chord filling in the fifth, and the texture softens rather than breaks.
\nThe chordal seventh falls by step. It is the note that makes a seventh chord a seventh chord, and it is formed as a dissonance against the bass, so it cannot sit still. In a sequence of seventh chords, the resolution of one seventh into the third of the next chord is what makes the chain sound inevitable.
\n\nHere the two tendency tones are alone and unaccompanied: the leading tone climbs to the tonic while the seventh falls to the third of the tonic chord. The interval between them is a diminished fifth, which resolves into a third.
\n\nThe same two notes with the leading tone falling instead. The result is consonant and the phrase is not fully closed, because the obligation was refused.
\nDoubling is where most unnoticed errors live, and each convention has a plain reason.
\n| Practice | Why |
|---|---|
| Double the root of a root-position triad | the root is the most stable and least constrained note in the chord |
| Do not double the leading tone | it must resolve upward, and two copies cannot both resolve without creating parallel octaves |
| Do not double the chordal seventh | it must resolve down, with the same problem in reverse |
| Double the third in a diminished triad | the root and the fifth form a tritone and both are pulled toward resolution |
| Do not double the third of a major triad | it is the character note of the chord, and doubling it makes an unstable sonority louder |
| Keep adjacent upper voices within an octave | wider spacing thins the chord and detaches the inner voice |
| Prefer contrary motion in the outer voices | the frame of the texture stays open and each line keeps its own contour |
| Avoid voice crossing and overlap | a line that passes through its neighbour’s territory stops being a separate line |
The rule against doubling the leading tone explains an apparent exception in cadences. In a perfect authentic cadence where the soprano takes the leading tone, the dominant seventh is often written without its fifth, with the root doubled instead. That spare note becomes the fifth of the final tonic chord, so the closing chord is complete: the texture gives up one doubling in the middle of the dominant in order to have a full tonic at the end. The same logic drives the alternative solution, a complete dominant seventh resolving to a tonic with a doubled third.
\nFour-part choral writing is one texture among many, and each texture sets its own priorities. In two-voice counterpoint the concern shifts almost entirely to the intervals you land on and the motions that get you there, which is the subject of first-species counterpoint. In orchestration, spacing rules depend on which instruments are involved, because a tenth between two clarinets does not sound like a tenth between a flute and a cello. In a keyboard texture the hands cover a wider, more open arrangement and the inner-voice rules apply loosely or not at all.
\nThe useful habit is to treat the rules as a description of what happens when a texture holds together. Write the passage you hear, then ask which rule you broke and what the break costs. If the answer is that two voices merged, or the cadence stopped pressing, you have found a real problem. If the answer is that you wanted the two voices to merge, you have made a decision instead of a mistake.
\nThe fastest way to internalise tendency tones is to sing the resolution rather than compute it: play a dominant seventh, sing the leading tone and resolve it upward, then sing the seventh and resolve it downward until the motion feels obligatory rather than chosen. After that, harmonise a chorale melody in four parts and check the result against the table above, in this order: bass motion, tendency tones, parallels, spacing, doubling. Errors in the first three are audible in performance; errors in the last two usually are not, which is exactly why they survive so long in a draft. To hear the difference between a resolution and a refusal, the ear training studio will play both at once.
", "content_text": "Part-writing rules usually arrive as a list of prohibitions, which makes them feel arbitrary and easy to resent. They are neither. Almost all of them follow from a single aim: four voices that behave like four voices. Once you can hear what a broken rule does to a texture, the list stops being a set of prohibitions and becomes a set of diagnostics.\n#The aim: independence\nThe rules describe a style. In four-part choral writing from roughly 1650 to 1900, each voice is meant to have its own contour and its own register, so that a listener can follow the bass, or the top line, or the inner parts, and hear lines rather than a chord with notes stacked on it.\nEvery rule on the list protects that independence, and every rule can be broken when independence is not the point. Parallel fifths in a Debussy piano piece or a distorted guitar part sound deliberate because the texture is not claiming four independent lines. The rules are not physics; they are the grammar of one musical language, and knowing the grammar is what allows you to break it on purpose rather than by accident.\n#The bass leads\nFour-part writing starts from the bass, because the bass note decides which chord is in root position and which is inverted, and inversion is where a chord’s weight comes from. Once the bass is fixed, the upper three voices are arranged above it.\nSpacing works on a simple principle: the closer two adjacent upper voices sit to each other, the more they sound like part of one harmonious mass, and the further apart they sit, the more the upper voice separates from the harmony underneath. Standard practice keeps soprano, alto and tenor within an octave of their neighbours and lets the bass sit further below, as much as a twelfth under the tenor in choral writing. Wider gaps between upper voices do not create clarity. They create a hole that the ear fills by hearing the middle voice as detached.\n#Motion between voices\nTwo voices can move in similar motion, in parallel motion (similar motion by the same interval), in contrary motion, or obliquely, where one stays and the other moves. The choice between them is most of what part writing is.\nContrary motion between the outer voices is the default answer when you do not know what to do next, because the two ends of the texture pulling apart is what keeps the frame of the music open. Similar motion into a perfect fifth or octave between the outer voices, especially when the soprano gets there by leap, is the case called direct or hidden fifths and octaves, and it is discouraged for the same reason as the parallel kind.\nParallel perfect fifths and octaves are the famous prohibition, and the reason is that they stop the texture from growing. Two voices an octave apart are the same pitch class, so the second voice is not adding a line at all. Two voices a fifth apart and moving in parallel keep the same relationship, and because a perfect fifth above the bass reinforces a frequency already present in the bass, the upper voice merges into the sound of the lower one rather than standing beside it. Two explanations, the contrapuntal and the acoustic, describe the same effect: the voices stop being independent.\nC4+G4 D4+A4\nThose are two parallel fifths. The upper voice is not wrong as a note; it simply has no independent life.\n#Tendency tones\nTwo notes in tonal harmony come with an obligation attached.\nThe leading tone rises to the tonic. It sits a semitone below its destination, which is the smallest melodic step available and the most directed, and the style offers it no equally stable alternative. In an outer voice, where the ear follows it, it rises. In an inner voice it may fall, usually into a complete tonic chord filling in the fifth, and the texture softens rather than breaks.\nThe chordal seventh falls by step. It is the note that makes a seventh chord a seventh chord, and it is formed as a dissonance against the bass, so it cannot sit still. In a sequence of seventh chords, the resolution of one seventh into the third of the next chord is what makes the chain sound inevitable.\nB4+F5 C5+E5\nHere the two tendency tones are alone and unaccompanied: the leading tone climbs to the tonic while the seventh falls to the third of the tonic chord. The interval between them is a diminished fifth, which resolves into a third.\nB4+F5 G4+E5\nThe same two notes with the leading tone falling instead. The result is consonant and the phrase is not fully closed, because the obligation was refused.\n#Doubling and spacing\nDoubling is where most unnoticed errors live, and each convention has a plain reason.\nPracticeWhyDouble the root of a root-position triadthe root is the most stable and least constrained note in the chordDo not double the leading toneit must resolve upward, and two copies cannot both resolve without creating parallel octavesDo not double the chordal seventhit must resolve down, with the same problem in reverseDouble the third in a diminished triadthe root and the fifth form a tritone and both are pulled toward resolutionDo not double the third of a major triadit is the character note of the chord, and doubling it makes an unstable sonority louderKeep adjacent upper voices within an octavewider spacing thins the chord and detaches the inner voicePrefer contrary motion in the outer voicesthe frame of the texture stays open and each line keeps its own contourAvoid voice crossing and overlapa line that passes through its neighbour’s territory stops being a separate line\nThe rule against doubling the leading tone explains an apparent exception in cadences. In a perfect authentic cadence where the soprano takes the leading tone, the dominant seventh is often written without its fifth, with the root doubled instead. That spare note becomes the fifth of the final tonic chord, so the closing chord is complete: the texture gives up one doubling in the middle of the dominant in order to have a full tonic at the end. The same logic drives the alternative solution, a complete dominant seventh resolving to a tonic with a doubled third.\n#Where the rules stop\nFour-part choral writing is one texture among many, and each texture sets its own priorities. In two-voice counterpoint the concern shifts almost entirely to the intervals you land on and the motions that get you there, which is the subject of first-species counterpoint. In orchestration, spacing rules depend on which instruments are involved, because a tenth between two clarinets does not sound like a tenth between a flute and a cello. In a keyboard texture the hands cover a wider, more open arrangement and the inner-voice rules apply loosely or not at all.\nThe useful habit is to treat the rules as a description of what happens when a texture holds together. Write the passage you hear, then ask which rule you broke and what the break costs. If the answer is that two voices merged, or the cadence stopped pressing, you have found a real problem. If the answer is that you wanted the two voices to merge, you have made a decision instead of a mistake.\n#Practising it\nThe fastest way to internalise tendency tones is to sing the resolution rather than compute it: play a dominant seventh, sing the leading tone and resolve it upward, then sing the seventh and resolve it downward until the motion feels obligatory rather than chosen. After that, harmonise a chorale melody in four parts and check the result against the table above, in this order: bass motion, tendency tones, parallels, spacing, doubling. Errors in the first three are audible in performance; errors in the last two usually are not, which is exactly why they survive so long in a draft. To hear the difference between a resolution and a refusal, the ear training studio will play both at once.", "date_published": "2026-09-25T12:00:00+00:00", "date_modified": "2026-09-25T12:00:00+00:00", "authors": [ { "name": "Yunus Emre Vurgun", "url": "https://yunusemrevurgun.com" } ], "tags": [ "voice leading", "part writing", "counterpoint" ], "language": "en", "attachments": [ { "url": "https://musics.name/articles/voice-leading-in-four-parts.md", "mime_type": "text/markdown", "title": "Markdown source" } ] } ] }