Interval Inversion and Compound Intervals Explained

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To invert an interval, move its lower note up an octave, or its upper note down. The two numbers always add to nine: a third inverts to a sixth, and a fourth inverts to a fifth. Qualities flip, so major becomes minor and augmented becomes diminished, while perfect stays perfect. A compound interval is wider than an octave, and it takes its name from the simple interval left over when the octave is removed.

#Moving one note by an octave inverts the pair

Take E♭3 and G3. The pair is a major third, four semitones. Move the lower note up an octave, and the pair becomes G3 and E♭4, a minor sixth of eight semitones. Four and eight add to twelve, and that holds for every inversion.

The first chord is the third and the second is its inversion. The top note of the first chord, G3, becomes the bottom note of the second. Nothing about the pitch classes changes, only which note is at the bottom. The same move works for chords. C4, E4 and G4 becomes E4, G4 and C5 when the lowest note moves up, and the bass changes from C to E, which is the first inversion of the C major triad.

#The numbers add to nine because the letters wrap around

Count the letter names from the bottom note up to the top note, including both ends. A third covers three letters, C, D and E, and a sixth covers six. Now take the inversion, which runs from the top note up to the bottom note by the octave. Its letters, counted in the same way, are the other half of the cycle. The two paths together take seven letter steps around the cycle of A to G, and each path counts its starting letter once. That gives n + m = 7 + 2 = 9, where n and m are the two interval numbers.

The same argument explains why the rule works for every pair. It depends only on the seven letter names, not on the accidentals, which is why the numbers can be found from the score before the accidentals are checked.

#Twelve simple intervals and their inversions

The table lists the twelve simple intervals within an octave, each with its inversion. In every row the two semitone counts add to twelve, and the two interval numbers add to nine.

Read any row in either direction. To find an inversion quickly, count the semitones of the simple interval and subtract them from twelve.

#Quality flips, but perfect intervals stay perfect

Major and minor swap. Perfect intervals stay perfect: a fourth inverts to a fifth, a fifth to a fourth, and a unison to an octave. Augmented and diminished swap as well. C up to F♯ is an augmented fourth of six semitones, and F♯ up to C is a diminished fifth of six semitones, so the two belong together.

A perfect fifth shows the same pattern. A♭3 up to E♭4 is a perfect fifth of seven semitones, and moving A♭ up an octave gives E♭ up to A♭, a perfect fourth of five.

#Letter names set the number, semitones set the quality

The number of an interval is a letter count, and the quality comes from the semitone count. C up to G is a fifth because it spans the letters C through G, and its seven semitones make it perfect. C up to G♭ is still a fifth by letters, but six semitones make it diminished. The score’s spelling therefore determines the name, and the sound alone does not.

This is why a spelled interval can differ from its sounding twin. The major third and the diminished fourth, for example, cannot be told apart by sound alone in equal temperament, yet the letters name them differently. The reading intervals at sight article practises naming from the spelled page.

#Chords and counterpoint use the same inversions

The first inversion of C major is E, G, C. Above its bass E, the chord contains a minor third to G and a minor sixth to C. The minor sixth from E to C is the inversion of the major third from C to E, which is why a first-inversion triad carries a third and a sixth above the bass. The triads and inversions article covers the chord side of the same idea.

In invertible counterpoint, the two voices can swap places. A third becomes a sixth, a fifth becomes a fourth, and an octave stays an octave. Composers used this to write lines that work in either voice, and the imitation and invertible counterpoint article shows how the swap is built.

#Reading a falling interval as a rising one

A falling interval can be checked by its inversion. E5 falls to G♯4, which is a minor sixth: six letters down (E, D, C, B, A, G) and eight semitones. Move the lower note, G♯4, up an octave, and the pair becomes E5 and G♯5, a rising major third of four semitones from E5. Both descriptions name the same two pitch classes, with one note moved by an octave. A reader who sees a falling sixth can check it by rising a third from the lower note, and the habit reduces the arithmetic needed in sight-reading.

#Compound intervals are octaves plus a simple interval

A compound interval is an octave plus a simple interval. A ninth is an octave plus a second, an eleventh is an octave plus a fourth, and a thirteenth is an octave plus a sixth. To name one, subtract seven from its number: 9 leaves 2, 11 leaves 4 and 13 leaves 6. The quality is the quality of the simple interval, so a major ninth is a major second with an octave added. In semitones, a major ninth is 14, a perfect eleventh is 17 and a major thirteenth is 21.

Each pair starts on C3 and reaches a note a compound interval above it: a major ninth, a perfect eleventh and a major thirteenth. Jazz players use the same numbers for chord extensions, which the chord extensions article covers.

#Where this misleads people

Interval numbers count letter names, not semitones. C to E♯ spans the letters C, D and E, so it is a third. It is augmented, because it covers five semitones, and it sounds the same as a perfect fourth, C to F. The number comes from the letters and the quality from the semitones.

A compound interval has to be reduced before it is inverted. A ninth reduces to a second, and its inversion is a seventh, since 2 and 7 add to nine. A ninth does not invert to a second.

Inversion can also be forgotten at the extremes. A unison inverts to an octave, not to a fifth, and an octave inverts to a unison. Players who learn the table for thirds and sixths often forget that the rule covers the unison and the octave too.

#What to do with it

Choose a starting note and, for each of the twelve simple intervals up from it, play the pair and name it aloud before you check. Then move the lower note up an octave and name the inversion. The two names should add to nine. For compound intervals, play a note and then the ninth, eleventh and thirteenth above it, and name each by subtracting seven. Repeat the drill with the pair spelled both ways, such as C to D♭ and C to C♯, and confirm that the letters change the name even though the sound does not. The ear training method explains how to name these pairs by sound.

Questions

How do you invert an interval?

Move the lower note up an octave, or the upper note down. The new interval's number is nine minus the original number, so a third becomes a sixth and a fourth becomes a fifth. Major becomes minor, augmented becomes diminished, and perfect stays perfect.

What is a compound interval?

A compound interval is wider than an octave. To name it, subtract seven from its number: a ninth becomes a second, an eleventh becomes a fourth and a thirteenth becomes a sixth. Its quality is the quality of that simple interval.

What is the inversion of a major third?

A minor sixth. E♭ up to G is a major third of four semitones. Moving E♭ up an octave gives G up to E♭, a minor sixth of eight semitones. Four and eight add to twelve, as every inversion does.

What is the inversion of an augmented fourth?

A diminished fifth. C up to F♯ is an augmented fourth of six semitones. Moving C up an octave gives F♯ up to C, a diminished fifth, also six semitones.