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.
\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.
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