Where to hide the comma, which is the only real question
An error of 23.46 cents has to go somewhere. It cannot be removed, for reasons that are arithmetic rather than practical, so the only decision available is where to put it — and that decision, made differently in different centuries, is what separates the tuning systems from each other.
The figure above is the whole argument in one picture, and it repays a slow look. Three systems, twelve notes each, and the disagreements are large: at the major third, just intonation sits fourteen cents below equal temperament and Pythagorean tuning eight cents above it, which is a gap of twenty-two cents between two systems that both claim to be pure.
The unit that makes comparison possible
None of this is discussable without cents, so it is worth being precise about them.
A cent is a hundredth of an equal-tempered semitone, which makes 1200 to the octave. Because pitch perception is logarithmic, cents add where ratios multiply: a pure fifth of 701.955 cents plus a pure fourth of 498.045 cents makes exactly 1200, the octave, which in ratios is .
The sizes worth memorising are small. Two cents is inaudible in isolation. Five cents is the point at which a careful listener notices something. Fourteen cents — the equal-tempered major third’s error — is unmistakable as a quality of sound, though most listeners will describe it as bright or hard rather than as out of tune. Twenty-three cents is a wrong note.
Just intonation: purity, at the cost of moving
The first answer is to refuse the compromise entirely. Tune every note to a simple ratio against the tonic, take the consonances the harmonic series offers, and accept the consequences.
The major scale in just intonation is 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1. Every one of those is a ratio of small integers, and the resulting major triad — 4:5:6 — is as consonant as three notes can be.
The problem arrives the moment the music moves. In C major, D sits at 9/8 above C. But in the chord of G major, the same D must sit at 3/2 above G — and , which agrees. Now take D as the root of a minor triad in the same key, and its fifth, A, must be 3/2 above it: . But A as the sixth degree of C major is 5/3, and . The two disagree by 81/80 — 21.5 cents, the syntonic comma.
So just intonation gives perfect chords in one key and unusable ones nearby. It survives today in barbershop singing, string quartets and unaccompanied choirs — ensembles with continuous pitch control, — which adjust each chord as it arrives and pay for it with the slow sinking the figure above predicts.
Pythagorean tuning: pure fifths, one impossible one
The second answer keeps the fifth pure and lets the thirds fall where they may.
Stack eleven pure fifths from E-flat to G-sharp and every one of them is beatless. The twelfth, from G-sharp back to E-flat, is 23 cents narrow and howls. Its major thirds, built from four stacked fifths, come out at 81/64 — eight-tenths of a semitone… no, precisely 407.8 cents, which is eight cents sharp of equal temperament and twenty-two cents sharp of pure.
That sounds like a fatal objection and was not one, because for several centuries the third was not treated as a consonance. Medieval European music cadences on fifths and octaves; the third is a passing sonority on the way somewhere. Pythagorean tuning is an excellent tuning for music that thinks that, and it was standard until roughly 1450 — which is to say, until composers started wanting the third to be a place to rest.
The lesson generalises: a tuning system is not better or worse in the abstract, but better or worse for a repertoire. Pythagorean tuning did not become wrong. The music changed underneath it.
Meantone: pure thirds, bought with slightly flat fifths
The third answer inverts the priority. If the third is now a consonance, tune it pure and let the fifths take the damage.
Four pure fifths stacked make 81/64. A pure major third is 5/4 = 80/64. The difference is that same syntonic comma of 81/80. Quarter-comma meantone narrows each of the four fifths by a quarter of it — about 5.4 cents each — so that four of them land exactly on 5/4.
The result is remarkable. Every major third in the good keys is exactly pure, which means beatless, which means a quality of stillness that equal temperament cannot produce at all. The fifths are 5.4 cents narrow, which is at the edge of noticeable and entirely acceptable. And one fifth is 36 cents wide, which is a disaster confined to keys nobody used.
Meantone was the European standard from about 1500 to about 1700 — two hundred years of keyboard music written for it, including a great deal that sounds thin and hard on a modern piano for reasons that have nothing to do with the performance.
Equal temperament: nothing pure, everything possible
The fourth answer gives up on purity as a goal. Divide the octave into twelve exactly equal steps of each, which is 100 cents by construction, and the circle closes because it was built to.
What that buys is total transposability. Every key is identical to every other key, every interval is the same size everywhere, a piece can modulate anywhere and return, and an instrument can be built with fixed pitch and no bad regions.
What it costs is that nothing is pure except the octave. The fifth is two cents narrow, which is genuinely negligible. The major third is 13.7 cents sharp, which is not: an equal-tempered major third beats at about ten times a second in the middle of the piano, and that beating is present in every major chord ever played on a modern keyboard.
Most listeners have never heard a pure major third in a chord and have no idea that the roughness is optional.
What the comparison actually shows
Put the four side by side and a pattern appears that the individual descriptions hide.
Every system is exact somewhere and wrong somewhere else, and the systems differ in how the error is distributed, not in how much of it there is. Just intonation concentrates all its error into a few unusable intervals and has none in the rest. Pythagorean puts nearly all of it in one fifth. Meantone puts a little in every fifth and all the rest in one. Equal temperament spreads it perfectly evenly, which means every interval is slightly wrong and none is unusable.
This is the same shape of decision that appears whenever a fixed budget of error has to be allocated, and the extremes are always the same: concentrate it and get perfection plus a catastrophe, or spread it and get uniform mediocrity. Musical history ran that experiment for four hundred years and chose uniform mediocrity, for reasons that were about repertoire rather than about sound. The same trade appears whenever a pattern has to survive being moved.
How the error is heard, as opposed to measured
A chart in cents makes every deviation look like the same kind of thing. The ear does not agree, and the discrepancy between the two is the reason the systems were argued about for centuries rather than settled by measurement.
Two properties of that curve govern everything. The wells are narrow, so small mistunings matter much more than a linear intuition suggests. And they are unequal in depth — the octave and the fifth have deep, steep wells and the thirds have shallow ones — so the same number of cents costs far more at a fifth than at a third.
That asymmetry is exactly why the historical systems look the way they do. Two cents at the fifth, as in equal temperament, is nothing. Fourteen cents at the third is a lot, and it is tolerated because the third’s well is shallow enough that fourteen cents does not fall out of it. Reverse the two — fourteen cents at the fifth — and the result is unusable.
The systems are therefore not arbitrary aesthetic positions. They are allocations of a fixed error budget against a cost function the ear supplies, and the cost function is steeper at the fifth than at the third.
The register nobody mentions
Every figure on this page plots cents, and cents are register-independent by construction. Beating is not.
Two tones fourteen cents apart beat at a rate proportional to their absolute frequency: about 1.6 times a second near the bottom of the treble stave, about 6.4 times a second two octaves up. The same temperament is therefore audibly worse in the treble than in the bass, and a piano tuner spends disproportionate effort on the middle two octaves because that is where the beating is fast enough to be rough and slow enough to be noticed.
It also means the historical arguments were partly arguments about instruments. A temperament tuned for a harpsichord, whose tone decays fast and whose partials are weak above the sixth, hides errors that the same temperament on a modern piano exposes.
The system that is not a compromise
There is one way out that none of the four takes, and it has become practical only recently: change the tuning as the music moves.
An adaptive tuning holds every sounding chord in pure ratios and shifts the reference pitch between chords by whatever the harmony demands. Each chord is beatless; the drift is absorbed silently in the gaps. Software synthesisers do this now, and it produces a sound that no historical keyboard could make.
It also demonstrates what the four classical systems were actually short of, which was not knowledge but mechanism. Every theorist from Zarlino onward knew that pure chords in every key required more than twelve pitches. What was missing was an instrument that could retune between one chord and the next, and it turns out that a good deal of tuning theory was a workaround for a limitation of levers.
What was lost, and whether it mattered
Between meantone and equal temperament sit the well temperaments — Werckmeister, Kirnberger, Vallotti and a dozen others — which spread the comma unevenly on purpose. Every key is playable, and every key is different: the near keys have nearly pure thirds and the far keys have wide ones, so C major and F-sharp major are not transpositions of one another but genuinely distinct sonorities — which makes the distance between keys a matter of sound and not only of notation.
Eighteenth-century writing about key character — D major as triumphant, E-flat as noble, F minor as funereal — was written by people whose instruments had those differences physically present. In equal temperament the differences are gone, and the descriptions became a tradition rather than an observation.
Whether that is a loss is a real question and not a rhetorical one. What was gained was the entire repertoire of nineteenth-century music, which modulates in ways that no unequal temperament tolerates. It was a trade, and the terms were understood at the time.
Where the model stops
Only twelve notes. Every system here divides the octave into twelve. That constraint is the source of the problem, and it is a keyboard’s constraint rather than music’s. Nineteen and thirty-one tone systems both make meantone close properly, and instruments were built for both.
Fixed pitch. The whole difficulty applies to instruments that cannot adjust. Voices, violins and trombones retune continuously and unconsciously, and measurements of good ensembles show that they use none of these systems exactly — they lean toward just intonation on sustained chords and toward Pythagorean tuning on melodic lines, in the same phrase.
Harmonic partials. Purity is defined by partials coinciding, which assumes partials at exact multiples of the fundamental. Real strings are stiff and their partials are progressively sharp, so “pure” on a real piano is a slightly different target than on paper.
The figures plot deviation, not experience. Every chart here shows cents from equal temperament, which makes equal temperament look like the origin and everything else like a departure from it. That framing is an artefact of needing an axis, and it is exactly backwards historically: equal temperament is the newcomer.
The ladder from here
Later rungs: the syntonic comma in full. The wolf and what living with it was like. Meantone’s bargain examined properly. Well temperament, and whether key character was real. Equal temperament as an eighteenth-century decision. Adaptive tuning, and what a computer can do that a keyboard cannot. Measured intonation in real ensembles. Nineteen, thirty-one and fifty-three. And the systems built on other principles entirely, where the question of where to hide the comma does not arise because the comma was never a problem.
Vallotti’s temperament, published in 1779, puts six commas’ worth of narrowing into six fifths and leaves the other six pure. It sounds better than equal temperament in almost every key that eighteenth-century music actually uses, and essentially nobody tunes a piano that way, because the last hundred and fifty years of repertoire would break.