Nineteen, thirty-one and fifty-three

Twelve equal steps to the octave is such a familiar arrangement that it takes an effort to see it as a choice. It is one, and the reasons for it are not the reasons usually given. It is not that twelve produces especially good intervals — its major third is 13.7 cents sharp of pure, which is a great deal. It is that twelve is small, that it puts a seven-note scale under one hand, and that it makes every problem in the arithmetic of tuning disappear at once.

Other numbers make other problems disappear. Here is what happens to the octave when it is divided differently.

The picture has structure. Most numbers are bad at one interval or the other or both. A handful — 12, 19, 31, 41, 53 — are good at both, and they are exactly the divisions with a literature, a repertoire and, in several cases, surviving instruments.

The test that decides what vanishes #

The interesting question about a division is not how accurate it is. It is which distinctions it refuses to make.

An equal division has a fixed number of steps, so any two pitches that round to the same step become the same pitch. When a comma rounds to zero steps, that comma has been tempered out: the division has decided the two notes it separates are one note. This is not an approximation error. It is a structural decision with consequences that run all the way up into what chord progressions are possible.

Which commas vanish can be worked out without listening to anything.

The method is short. Every interval in five-limit tuning is a product of powers of 2, 3 and 5, so it can be written as three exponents. The Pythagorean comma is $3^{12}/2^{19}$, which is $(-19, 12, 0)$. The syntonic comma is $81/80 = 3^4/(2^4 \cdot 5)$, which is $(-4, 4, -1)$.

A division of $n$ steps has its own best approximation to each prime: $n$ steps for the octave, $\operatorname{round}(n\log_2 3)$ for the twelfth, $\operatorname{round}(n \log_2 5)$ for the seventeenth. Push the exponents through those and the comma comes out as a whole number of steps.

For twelve steps the approximations are 12, 19 and 28. The syntonic comma is then

$$4(19) - 4(12) - 1(28) = 76 - 48 - 28 = 0.$$

Zero. Twelve-tone equal temperament does not distinguish $9/8$ from $10/9$, does not distinguish the Pythagorean third from the just third, and cannot represent the syntonic comma at all. That is why the comma pump never happens on a piano: the pitch that should have drifted is the same key.

What each of them decides #

Run the same test on the others and the divisions separate into families.

Nineteen tempers out the syntonic comma and does not temper out the Pythagorean one. Its fifth is 7.2 cents narrow, which is a lot — noticeably flat, though usable — and its major third is 7.4 cents flat, which is much better than twelve’s. It is a meantone system: four of its fifths make exactly its major third. It has separate notes for C♯ and D♭, which is why it has nineteen of them rather than twelve, and in that system a sharp is lower than the corresponding flat.

Thirty-one tempers out the syntonic comma as well, and does it far better. Its fifth is 5.2 cents narrow and its major third is 0.8 cents sharp — audibly pure. Thirty-one equal is very nearly quarter-comma meantone with the chain closed, which is what makes it historically important rather than merely arithmetically neat.

Fifty-three does something different. It does not temper out the syntonic comma — that stays as one step — and it does not close the chain of fifths in the Pythagorean sense either. What it tempers out is the schisma, the two-cent gap between the two commas. In exchange it gets a fifth that is 0.068 cents from pure.

That number deserves reading twice. Sixty-eight thousandths of a cent. A chain of fifty-three-tone fifths and a chain of genuinely pure fifths stay together for dozens of steps before the difference reaches anything a laboratory could measure, let alone an ear.

Which computation produced the number #

The fifth’s error in $n$-tone equal temperament is the distance from $1200\log_2(3/2) = 701.955$ cents to the nearest multiple of $1200/n$:

$$e(n) = \left\lfloor \frac{701.955\,n}{1200} \right\rceil \frac{1200}{n} - 701.955.$$

For $n = 53$ the nearest multiple is the 31st step: $31 \times 22.6415 = 701.887$ cents, and the error is $-0.068$. For $n = 12$ the nearest is the 7th: $700$ exactly, error $-1.955$ — which is the Pythagorean comma divided by twelve, as it has to be.

The scatter in the first figure is that function evaluated at every integer from 5 to 60, twice, once for the fifth and once for the third. No division was singled out beforehand. The five that get labels are labelled because both of their errors are small, and that criterion was applied after the curve existed.

Everything follows from the size of the fifth #

There is a shorter way to say all of the above, and it explains why the divisions fall into families rather than scattering.

Pick a fifth. Every other interval in the system is then determined, because a chain of fifths generates all twelve — or nineteen, or thirty-one — note names, and the major third is four fifths up with two octaves taken off. So the size of the third is a strictly decreasing function of the size of the fifth: narrow the fifth by one cent and the third narrows by four.

A pure fifth of 701.955 cents gives a third of $4 \times 701.955 - 1200 = 1607.82$, which folds to 407.82 — the Pythagorean third, 21.5 cents sharp. To reach a pure third of 386.31 the fifth has to come down to 696.58, which is 5.4 cents narrow and is exactly quarter-comma meantone. Everything between those two fifth sizes is a compromise, and the whole history of temperament is a walk along that one line.

That is why the divisions group as they do. Nineteen’s fifth is 694.7 and thirty-one’s is 696.8, so both sit in the meantone region and both temper out the syntonic comma; nineteen simply overshoots. Twelve’s is 700, at the sharp end of the same region. Fifty-three’s is 701.89, essentially pure, so it lands in the Pythagorean region and keeps the syntonic comma as a real distinction.

What the extra notes are called #

A nineteen-note octave has seven more notes than a twelve-note one, and where they come from is not obvious until the chain is drawn.

They are the notes the chain of fifths produces and twelve equal temperament folds together. Extend the chain past twelve links and the thirteenth note is not a repeat of the first: in a meantone system it lands a little below it. So a nineteen-note system has a separate C sharp and D flat, a separate D sharp and E flat, and so on for five pairs, plus separate E sharp and F flat and B sharp and C flat — which comes to nineteen.

The consequence startles people who meet it first on a keyboard. In nineteen equal temperament, and in meantone generally, a sharp is lower than the enharmonic flat. C sharp sits at 1 step of 19 and D flat at 2. That is the opposite of the relationship pianists learn from the circle of fifths, where the two are simply the same key, and the opposite again of Pythagorean tuning, where sharps run high.

The direction is decided by the size of the fifth and by nothing else. In a system whose fifth is narrower than 700 cents, seven fifths up — a sharp — lands lower than five fifths down — a flat. Above 700 it goes the other way. Twelve equal temperament sits exactly on the crossing point, which is precisely why it can merge them, and merging them is the single largest thing it buys.

What is given up #

The reason none of these displaced twelve is not that they sound worse. Several of them sound better. The reason is that they are larger.

And a keyboard is the whole of the objection. Twelve notes to the octave puts a scale under a hand and a chord under three fingers, and the black keys are what make an octave’s worth of them findable without looking. A nineteen-note octave requires a different instrument; thirty-one requires a considerably different one; fifty-three requires a rethink of what a keyboard is. It also means notation with more accidentals than the five in use, fretting schemes with frets a millimetre apart in the upper positions, and an ensemble whose instruments all agree on the system. Every one of those is a coordination problem rather than an acoustic one, and coordination problems are what actually decide these things.

A thirty-one-note octave means a keyboard with thirty-one keys per octave, or a generalised layout that no player already knows, or a fretting scheme with frets a millimetre apart in the upper positions. It means notation with more accidentals than the five in use. It means an ensemble whose instruments all agree on the system. Every one of those is a coordination problem rather than an acoustic one, and coordination problems are what actually decide these things.

Twelve is small enough to be cheap and accurate enough to be tolerable, and the combination is what won. The major third being fourteen cents sharp is the price, and by the nineteenth century, when the fixed-pitch instrument became the centre of European musical life, it was a price everyone had already agreed to pay.

The seventh partial, added #

The scatter plot scores each division on the fifth and the major third and stops there, and the seventh partial is audibly present in every brass and reed instrument. Adding it to the score is the same arithmetic once more, and it rearranges the table considerably.

Thirty-one rises to second and nineteen falls to thirty-eighth. Thirty-one’s seventh partial is out by just over a cent, which is a better approximation than its own fifth manages; nineteen’s is out by twenty-one, which is more than a syntonic comma and the worst of any division in the top twenty on the two-interval score.

The larger movement is twelve’s. Its 7/4 is 31.17 cents sharp — a third of a semitone, the largest error in the table by half again, and it drops the division from twenty-third to forty-third of the fifty-six examined. That is a real fact about the system every reader of this essay is listening in, and it is why the dominant seventh chord is the one place where twelve-tone equal temperament and the harmonic series most conspicuously part company: the chord’s own seventh is being asked to stand for a partial a third of a semitone away.

Two divisions also drop out of the top five on the way. Thirty-four and fifty-six are second and third on the two-interval score and neither survives into the top three when the seventh is added, so the shortlist the hero figure produces is not stable under the criterion being widened by one interval.

What does not move is fifty-three, which is first on both. The division whose fifth is essentially exact is also the best of the fifty-six on a three-interval average, and it remains the one nobody composes in — which sharpens the essay’s closing observation rather than softening it. Accuracy is not what decided this, and adding a third interval to the measurement makes the gap between what the arithmetic prefers and what the instruments do larger rather than smaller.

Where the model stops #

The patent val is not the only val. The prime approximations used above take the nearest step for each prime independently, which is the obvious choice and not always the best one. Some divisions are more useful if a prime is deliberately mapped to the second-nearest step; 17 and 22 are the usual examples. The table above uses the obvious mapping throughout and would give different answers for some divisions under a different one, which is a real limitation and not a rounding detail.

Two intervals is not a complete criterion, and a section above adds the third one and reports what it does to the ordering rather than leaving it as a guess.

Three intervals is not a complete criterion either, and the point generalises past the one extension made above. Adding the eleventh partial would move the ordering again, and a division that is excellent on the first four odd harmonics may be poor on a ratio some tradition happens to care about. What the seventh-partial table establishes is not a better shortlist but that the shortlist moves — a two-interval score is a choice of what to measure, and two of its top five do not survive one more interval being added to it.

Accuracy is not the same as usefulness. Fifty-three’s fifth is essentially perfect, and fifty-three is nonetheless almost unplayable as a harmonic system, because a step of 22.6 cents is smaller than any interval anyone uses melodically and the notes come too thickly to be spelled. Its actual historical use is as a measuring system — a ruler for describing other tunings — rather than as a set of pitches to compose in.

Equal steps are themselves an assumption. Everything above assumes the octave is divided into identical parts. There is no acoustic requirement for that. Systems built from an unequal generator, and systems that do not use the octave as their period at all, are perfectly coherent and are ruled out here by the framing rather than by any argument.

Nothing here says anything about melody. The whole comparison is conducted on vertical intervals — how well two simultaneous notes approximate a ratio. A division could score badly on that and be excellent for melodic purposes, and the question of what makes a good melodic step size is not one this method addresses at all.

Whose music, and when #

The alternatives are not a modern curiosity. They have a longer documented history than equal temperament does.

Nicola Vicentino built the archicembalo in 1555, a keyboard with thirty-one notes to the octave arranged over six manuals, and wrote music for it. His stated purpose was to recover the chromatic and enharmonic genera of ancient Greek theory; what he actually achieved was a working extended-meantone instrument two centuries before anyone had the arithmetic to explain why it worked so well.

Christiaan Huygens worked out the thirty-one-tone division mathematically in 1661, noticed that it reproduced meantone almost exactly, and published on it. The Dutch connection persisted: Adriaan Fokker built a thirty-one-tone organ in the 1950s, it is still playable in Haarlem, and a small repertoire exists for it.

Fifty-three was arrived at independently at least three times. Jing Fang, in China in the first century BC, computed a chain of fifty-three fifths and observed that it very nearly closed. Nicholas Mercator described the division in the seventeenth century. And Turkish theory, in the twentieth-century Arel–Ezgi–Uzdilek codification, uses fifty-three commas as the framework in which maqam intervals are specified — which is the one place where a fifty-three-fold division is in ordinary professional use rather than in a museum.

Nineteen has the shortest history and the liveliest present. Costeley wrote a chanson in 1558 explicitly for a nineteen-note keyboard; the division was then largely ignored until the twentieth century, when guitarists discovered that nineteen frets to the octave is physically buildable and that its flat fifth and good third suit some kinds of writing very well.

The pattern across all of these is worth stating plainly. Every one of them was invented by somebody who wanted purer thirds than a twelve-note keyboard could give, and every one of them failed to spread for the same reason: it needed an instrument nobody had. That is a fact about manufacturing and about musical labour, not about acoustics, and it is the same reason equal temperament itself took three centuries to become standard after the arithmetic was published.

The ladder from here #

Later rungs on this anchor: what a chain of fifths does in a system that has no octave-based closure at all. Stretched octaves and the piano tuned deliberately wrong, which is a case where twelve equal steps are not even what the instrument has. The seventh partial, and the intervals that go missing when a system only has room for threes and fives. And the systems outside Europe that met the same arithmetic and made different decisions about it, of which maqam and raga are the two with the largest surviving theory.

Fifty-three equal steps get the fifth right to seven hundredths of a cent, and nobody plays it. Twelve gets the third wrong by fourteen, and everybody does.