A comma is a polyrhythm that never closes

Every figure in this ladder has a row of little marks along the bottom: the common grid, the subdivision both layers land on. Three against two has a grid of six, four against three has one of twelve, seven against five has one of thirty-five. The grid is what makes a composite possible, and a composite is what all seven previous rungs have been about.

Two cycles have a common grid exactly when the ratio of their periods is a ratio of whole numbers. Take two cycles whose ratio is not, and everything above collapses at once: no grid, no composite, no least common multiple, and no bar in which the pattern repeats.

There is a famous pair of cycles like that, and this site has been writing about it since its first week.

The fifth and the octave are two cycles #

A pure fifth multiplies frequency by 3/2. An octave multiplies it by 2. Stack fifths and the frequency runs 3/2, 9/4, 27/8; stack octaves and it runs 2, 4, 8.

Written as cycles rather than as intervals, that is two clocks: one ticking every log₂(3/2) of an octave — 0.58496 — and one ticking every 1. Asking when the two coincide is asking for whole numbers p and q with p × 0.58496… equal to q, which is asking for 3ᵖ = 2ᵖ⁺𝑞, and by unique factorisation of the integers there is no such pair. Three is not a power of two and never will be.

So the answer to when do these two cycles come back together is never, and it is never for a reason with a proof rather than a reason about precision. It is not that the numbers are awkward. It is that no arrangement of pure fifths and pure octaves lands on the same pitch, at any length of chain, ever.

Twelve against seven drawn as a polyrhythm is what twelve fifths against seven octaves would be if the two were rates rather than ratios — and the difference is the whole point: a polyrhythm’s cycle closes at the least common multiple of two integers, and a chain of fifths has no such multiple because no power of three is a power of two.

What is left over instead #

A cycle that cannot close can still come close, and the record of how close it has come is the figure at the top of this essay.

After 1 fifth the chain is 498 cents from an octave. After 2 it is 204 cents away. After 5, ninety. After 12, 23.46. After 41, 19.84. After 53, 3.62.

Each of those is a record: the chain has never been nearer than that before. And each is the definition of an equal division of the octave, because a chain of n fifths that lands 23 cents from the top is a chain that would land exactly on it if each fifth were flattened by 23/12 of a cent.

That is where twelve comes from. Not from the number of letters in an alphabet, not from anything about the hand, and not from a mystical property of the number itself: twelve is on the list of near-misses, and it is the fifth entry on a list whose earlier entries are too coarse to build a music on.

The list is a continued fraction #

The sequence 1, 2, 5, 12, 41, 53 is not a coincidence and it is not empirical. It is the sequence of convergents of the continued fraction of log₂(3/2), which is the standard machinery for finding the best rational approximations to an irrational number.

The convergents of a number are the fractions that approximate it better than any fraction with a smaller denominator. Run the algorithm on 0.5849625 and it returns 1/2, 3/5, 7/12, 24/41, 31/53 and 179/306 — which read as this many octaves per that many fifths, and whose denominators are exactly the divisions of the octave that have been proposed, argued over and built.

Three hundred and six equal steps has been proposed, incidentally, and by a serious person: the residue there is 1.77 cents, which is well under any threshold at which a listener could tell.

Which is why nineteen is famous for something else #

Nineteen is not on the fifth list. Neither is thirty-one. Both are famous, and running the same machinery on a different interval says why.

The convergents of log₂(5/4) — the pure major third against the octave — are 1/3, 9/28, 19/59 and 47/146. The divisions whose third error sets a new record are 3, 16, 19, 22, 25, 28 and 59, and that list has almost nothing in common with the fifth’s.

So a division of the octave is good at fifths, or good at thirds, or in a very few cases both, and the arithmetic that decides is the same arithmetic run on two different irrational numbers. Nineteen, thirty-one and fifty-three is the essay that works through what each of them decides to make vanish; this one is about why there is a list at all.

The least common multiple was measuring the wrong thing #

The first rung of this ladder put the weight of its explanation on the least common multiple, and the fourth argued that the number of distinct step lengths in the composite is a better statistic for how a polyrhythm is heard.

This rung makes a third point about it, and it is about what the quantity is rather than how well it predicts. The least common multiple is a yes-or-no test wearing a number’s clothing. Nine against two has one, and closes exactly, and closes at eighteen. Forty-one fifths against twenty-four octaves has none at all, and misses by 19.84 cents out of 28,800 — a part in fourteen hundred and fifty.

Nine against two closes exactly at eighteen steps, which is what a rhythm does and a tuning does not: integers have a least common multiple and irrational ratios do not, and every comma in this collection is a measurement of that failure.

By any measure of nearness, forty-one fifths comes closer to closing than a lot of things that close. The least common multiple cannot say so, because it has no value at all for a ratio that is not rational. It answers does this cycle close, and the interesting question about the fifth and the octave is how nearly, which needs the continued fraction instead.

The comma is not small, and it is not a rounding error #

Twenty-three cents is an eighth of a tone. The smallest pitch difference a listener can detect at 500 hertz is about 3.9 cents by the model this site uses, so the comma is six times that — and played as two sustained tones near middle C it is a beat three and a half times a second, which nobody has to be trained to hear.

Four commas against the threshold at which a pitch difference becomes audible put the whole subject in one line: the Pythagorean at 23.5 cents and the syntonic at 21.5 are four times the limen, and the schisma at 1.95 is under it.

This matters because the polyrhythm framing invites an obvious wrong reading: that the fifth and the octave nearly agree, so the difference is a technicality. They do not nearly agree by any standard the ear applies. What is remarkable is not that the chain nearly closes; it is that the twelfth fifth is the first place where it comes near enough that a fudge of two cents per fifth will hide the error, and two cents per fifth is under the threshold while twenty-three cents at the end is well over it.

The whole of equal temperament is that trade: take an error that is audible once and distribute it into twelve errors that are not.

What a rhythm would sound like if it did this #

The parallel can be run the other way, which is a useful test of whether it is a real parallel or a pun.

Set two pulses whose periods stand in the ratio log₂(3/2) — one at 100 beats a minute and one at 170.95 — and they will never coincide again after the first stroke. The pattern of near-coincidences will tighten and loosen; after twelve strokes of the faster it will be close, after forty-one closer, and it will never repeat.

Read as a rate rather than as a ratio, the same 3:2 is a polyrhythm at a few events a second and an interval at a few hundred — one continuum with the comma living at the top of it, where the cycle would have to close in a fraction of a millisecond to be counted at all.

That is a real object and it has a name in the rhythmic literature — a polymetre with an irrational ratio, which is what a tape piece can do and an ensemble cannot. And it is exactly what a chain of pure fifths does, at a few hundred cycles a second instead of two a second.

The two phenomena differ in one respect that is not arithmetic. A drift of twenty-three cents is heard as a mistuning: two notes that should be the same and are not. A rhythmic drift of the same proportion is heard as two things gradually parting, because rhythm is followed by tracking and pitch is not. Same numbers, two mechanisms.

What the picture cannot show #

It cannot show that the fifth is 3/2. Everything here assumes a pure fifth is exactly a 3:2 frequency ratio, which is a definition rather than a measurement. Real strings are stiff, real singers adjust, and what a piano’s octaves actually do is not 2:1 either. The irrationality argument is about the numbers 2 and 3 and it does not survive being told that the intervals are not exactly those numbers.

It cannot show the third. The whole essay is about one chain, generated by one interval, and the section above shows what that costs: the fifth’s convergents predict forty-one, which nobody built, and miss nineteen and thirty-one, which everybody did. Real tuning systems are generated by two intervals, which makes the space two-dimensional and the near-closures a lattice rather than a sequence. The worse-of-two table is a one-dimensional shadow of that lattice and not the lattice itself.

It cannot show which residue matters. The figure plots the distance in cents, treating twenty-three cents at the twelfth fifth and twenty cents at the forty-first as nearly the same. They are not comparable as musical facts: one is spread over twelve notes and one over forty-one, so the per-fifth error is 1.955 cents against 0.484, and the per-fifth number is what a listener hears.

That is not a small correction to the figure; it changes which numbers appear on it. Records on the total distance give 1, 2, 5, 12, 41, 53 — the convergents. Records on the per-fifth error give 1, 2, 3, 5, 7, 12, 29, 41, 53, which is the semiconvergent list the section below is about. The same chain, measured two ways, produces two different lists of famous numbers, and the essay draws the first while arguing from the second.

And it cannot show that anybody ever thought this way. The continued fraction is a nineteenth-century tool applied to a problem people were solving in the fourth century BC and the sixteenth century AD by trial. Nobody arrived at twelve by computing convergents. What the machinery does is explain why the trials converged where they did, which is a different and weaker claim than saying it is how the answer was found.

And the two-interval criterion weights the fifth and the third equally, which nothing justifies. Taking the worse of the two errors is the simplest way to insist that a division be good at both, and it is not the only way: weighting the fifth more heavily — which a tradition building its scale out of a chain of fifths might reasonably do — moves forty-one back onto the list and pushes thirty-one off it. The list in that table is therefore a consequence of an assumption about what a tuning system is for, and the assumption is stated rather than derived. What survives any weighting is the narrower point: one chain cannot produce the list, because the divisions people built are not the ones a single chain ranks highest.

The semiconvergents, and why 29 is on one list and not the other #

The record-holders in the figure at the top of this essay are the convergents: 1, 2, 5, 12, 41, 53. Ask a slightly different question — not how far is the whole chain from closing but how wrong is each individual fifth — and a longer list appears: 2, 3, 5, 7, 12, 29, 41, 53.

Three, seven and twenty-nine are the extra entries, and they are the semiconvergents: the intermediate fractions the continued fraction passes through on its way from one convergent to the next. They are the best approximations by a weaker criterion, and the weaker criterion is the musically relevant one, because a listener hears one fifth at a time and not the sum of forty-one of them.

Twenty-nine equal steps has a fifth 1.49 cents out, against twelve’s 1.96. It is better than twelve at the thing twelve is famous for, it has been proposed more than once, and nobody uses it — because a division is not chosen on its fifth alone. Twenty-nine’s major third is 13.9 cents flat where twelve’s is 13.7 cents sharp: the same size of error in the opposite direction, so the better fifth is bought for nothing at all.

That is the general shape of the trade, and it says that neither list is the one to read. What a division has to be good at is both, so the quantity to take records on is the worse of its two errors — and running that gives a third list, shorter than either and much more recognisable:

That list is the one this collection keeps writing about. Twelve, nineteen, thirty-one and fifty-three all set records on it; the fifth’s convergents alone produce neither nineteen nor thirty-one, and the third’s alone produce neither twelve nor fifty-three. A division is chosen by the interval it handles worst, which is the criterion neither single chain can express.

And forty-one drops off it. Its fifth is superb — 0.48 cents, better than everything below fifty-three — but its major third is 5.83, and thirty-one had already reached 0.78 with ten fewer steps. So forty-one never sets a combined record, and the claim previously made here, that forty-one and fifty-three are the only two numbers below sixty that are near-records on both chains, is wrong in both halves: forty-one is a record on one chain only, and four other numbers below sixty are records on the combined criterion.

That reframes the essay’s own conclusion in a useful direction. The continued fraction of one interval explains why there is a list; it does not explain which list anybody built. Forty-one is the clearest case — it is the best fifth available under fifty-three, it has been proposed, and almost nobody has built an instrument for it, while thirty-one has a four-hundred-year history of instruments and a modern repertoire. The arithmetic that separates them is not in this essay’s figure, because that figure plots one chain.

Twenty-two and thirty-four are the entries that are not famous, and they are informative for the opposite reason. Both are genuine combined records; both are used, sparsely, in modern microtonal practice; and neither was arrived at historically. Which is what would be expected if the criterion is right and history is a poor search algorithm: the divisions that got built are the ones a chain of fifths stumbles onto, and the ones that did not are the ones only a two-interval search finds.

And against a ruler of fifty-three equal steps the just intervals sit almost exactly on divisions, which is the closure figure’s answer restated: 53 is where the chain comes close enough that the leftover stops being a musical quantity.

Where the ladder ends #

Eight rungs. The first four established what a polyrhythm is — two cycles, their coincidences, their composite, and the composite’s properties. The fifth and the sixth asked what a listener does with it: which layer becomes the beat, and what changes when there is one pattern and several beats instead of two patterns. The seventh asked what a performer does, and found that one player producing three against two is running one clock rather than two.

This one removes the assumption all seven rested on. The common grid is not a property of two cycles; it is a property of two cycles whose ratio happens to be rational, and where it fails the whole apparatus fails with it and something else takes over — the continued fraction, and a list of near misses, and a residue with a name.

Which means the ladder has arrived, by a route that started with a drummer’s hands, at the object this site is named after twice over: the gap that will not close, and the essay that opens the whole collection.