The ratio that stops being two

Eight rungs of this ladder have assumed a polyrhythm is something a listener can hold. Two clocks at once; a resultant nobody played; which of the two is the beat; what one player can do that two cannot. Every one of them takes a ratio of small whole numbers and asks what follows.

Nothing so far has asked how large the numbers may be. It is the last question this model has, and answering it closes the ladder.

The obvious bound, which is not one #

The first place to look is the length of the pattern. Two streams at p and q pulses to the bar have a resultant that repeats after the least common multiple of p and q subdivisions, and lcm(11, 7) is seventy-seven against lcm(3, 2)'s six. Seventy-seven of anything sounds like too many.

It is not a bound, and the reason is one line. The subdivision is the bar divided by the lcm, so the cycle is lcm times bar-over-lcm, which is the bar. A three-against-two and an eleven-against-seven have identical cycle lengths in seconds, at every tempo, always. Both close after one bar and both fit inside the window in which a listener can still hear a repetition at any usable speed.

So the arithmetic that looks like the constraint is a constant. The lcm counts subdivisions, and subdivisions get shorter exactly as fast as they get more numerous.

What does separate them #

If the cycle is the same length, what differs is what is inside it: how many events, and how close together the closest two get.

The second of those has an exact answer and it does not need a search. The onsets of the two streams fall at i/p and j/q of the bar, so the distance between two of them is |iq − jp| over pq — and the smallest non-zero value the numerator can take is the greatest common divisor of p and q. The closest approach is therefore

gcd(p, q) / pq of the bar, which is 1 / lcm(p, q).

For every ratio in this essay that is the same as 1/(pq), because every one of them has p and q coprime and a coprime pair’s lcm is its product. It is not the same in general, and the difference is worth stating because of which quantity turns out to be doing the work.

The lcm was dismissed two sections ago and it is the constraint after all. It was dismissed as the cycle length, where it genuinely cancels — the bar is lcm subdivisions of length bar-over-lcm, so it disappears. It comes back as the closest approach, where nothing cancels it. The same number is a constant in one place and the whole answer in the other, which is a good reason not to conclude that a quantity is irrelevant from one appearance of it.

Where it matters is the ratios nobody writes in lowest terms. Six against four has a product of 24 and an lcm of 12, so its closest approach is a twelfth of the bar and not a twenty-fourth — twice what the product rule predicts. Twelve against eight, product 96, is also a twenty-fourth. A ratio’s difficulty is set by its reduced form and its density, not by the two numbers as written, and ranking by product would put 12:8 among the impossible ones when it is a three-against-two played four times as fast.

The numbers fall away fast because the product grows fast:

The bound is where those numbers cross the order threshold. Two onsets less than about twenty milliseconds apart are not heard as two events in sequence; they are heard as one event with a thickened attack, and which of them came first cannot be reported. Between twenty and forty the answer is unreliable. Above forty they are securely two.

So thirteen against eleven is not a rhythm a listener fails to follow. It is a rhythm that, at an ordinary tempo, contains events that are not two events.

Which is a bound with a tempo in it #

The threshold is a fixed number of milliseconds, and the closest approach is a fraction of the bar, so the constraint rearranges into a statement about speed.

A ratio p against q needs a bar of at least lcm(p, q) times forty milliseconds for its onsets to be securely separate — which is p·q times forty for the coprime ratios below and for every ratio anybody names.

Read down that column and the repertoire appears. Three against two and four against three are unconditional — they work at any speed anybody plays, which is why they are in every tradition on earth. Five against four is unconditional in practice, since nothing is played at three hundred to the minute in four. Seven against five requires a moderate tempo, which is where it is in fact found: in slow movements, in the rubato-heavy Romantic piano repertoire, and in art music with a notated ratio rather than a played one.

Past that the requirement is a slow bar, and at a slow bar each individual stream is running into the other bound.

The rates at which a periodic series can be held as a pulse run from about a tenth of a second to two, so at a hundred to the minute a bar, a beat and a half-beat are all inside the window and the finer subdivisions are not. That window is a fact about the listener and does not move when the ratio does.

The two bounds squeeze from opposite ends. The product bound says the bar must be long; the tempo window says the subdivisions must not be too fast, which for large p in a long bar is the same thing said backwards. What is left between them is small, and it is roughly the set of ratios that music actually uses.

The count of events, which moves the same way #

There is a third quantity that tracks the product and is worth having, because it is what a performer would say the difficulty is.

A p-against-q bar has p + q − gcd(p, q) onsets in it, and the number of distinct inter-onset intervals is exactly min(p, q) / gcd(p, q) — not “grows with the smaller of the two”, which is what this rung previously recorded, but equal to it once the common factor is taken out. Both are exact on every ratio checked. Three against two gives four onsets and two distinct gaps — long, short, short, long, which is a shape with a name in every tradition that uses it. Eleven against seven gives seventeen onsets and seven distinct gaps, which is not a shape at all; it is a list.

So the same gcd governs all three quantities. It divides out of the gap count, it multiplies into the closest approach, and it subtracts from the onset count — which is another way of saying that a polyrhythm has a shape, given by the reduced ratio, and a density, given by the factor removed. Twelve against eight has the shape of three against two and four times the density: two distinct gaps, sixteen onsets, and a closest approach four times tighter than 3:2’s.

A pattern with two distinct gaps is learnable as a shape. A pattern with seven is learnable as two independent streams or not at all, which is why the standard teaching device for large ratios is to stop thinking of the composite and count each part separately. That instruction is an admission that the resultant has ceased to be an object.

Two ways of failing, and they sound different #

It is worth separating the two failures, because they are not degrees of the same thing.

Fusion is what happens when the closest approach is under the threshold. Two onsets become one event with a flam. The pattern is still perfectly rhythmic — it just has fewer events in it than were notated, and the ones it has are the wrong shape. Nothing is confusing about it; it is simply not the rhythm on the page.

Loss of stream separation is different and is the same boundary the melody ladder runs into: whether the events group into two lines or one. A polyrhythm is two streams only if the listener can segregate them, and segregation needs a cue — different pitch, different timbre, different spatial position, different player. This is why polyrhythms are almost always scored across instruments and why two hands on one piano is the hard case that has a rung of its own.

Scored against three candidate metres, a composite of this kind supports more than one reading and none strongly — which is the second way a cross-rhythm fails, and it sounds different from the first: one collapses into a single line and the other stays two layers that a listener cannot place a beat in.

Why the anchor closes here #

The model this ladder has been about is precise: two isochronous streams at a rational period ratio, sounding together. Nine rungs have now bounded it in every dimension the model has.

The ratio itself — what two clocks at a ratio produce. Its resultant — the third pattern, which nobody plays and everybody hears. Which stream is the beat — decided by rate and by fit, and changing hands at a computable tempo. The limit as the rate rises — where a polyrhythm becomes a chord. The limit as it is generalised — a process piece whose form is the orbit of one pattern. Whether one performer or two — and what changes. A case where the word is misapplied — the bell pattern, which is not one. The irrational limit — a comma, which is a polyrhythm that never closes. And now the bound on the numbers, which is the last free parameter.

The test the previous closures were made against was whether every dimension of the model has been bounded, and it is checkable here in the same way: name a variable this model has, and it is on that list. What is not on the list is not a property of two streams at a ratio — how a polyrhythm is learned, how it is notated, what it means in a particular repertoire — and each of those belongs somewhere else.

The one that has somewhere obvious to go is microtiming, which is what happens when the streams are not isochronous. That is a different model, not a further rung of this one: it is about deviations from a grid, and this ladder has been about the grid.

What happens if the threshold is wrong #

The whole bound rests on one quoted number, so it is worth asking how much depends on it.

Not much, structurally, and a great deal numerically. The order threshold is a range rather than a value; published figures for judging the order of two sounds run from about twenty milliseconds up to fifty or more, and they depend strongly on whether the sounds are distinguishable, on training, and on whether the task is order or mere separation.

The rate axis this ladder has used since its second rung puts the whole question in place: below about twenty events a second a series is a rhythm, above it a texture, and further up a pitch. Everything here is in the first band, and the bound this essay computes is not that boundary but a much lower one — the rate at which two onsets stop being two.

That is the useful property of a bound whose form is a product: the ratios are ranked by p·q regardless of where the threshold is put, so the order in which ratios become unusable is fixed even though the tempo at which each one does is not. The ranking is the claim; the tempi are an illustration of it at one threshold.

Which computation produced the numbers #

The closest approach is computed twice and the two are compared. Once as the identity — the bar divided by the least common multiple — and once by generating both streams’ onset positions, sorting them and taking the smallest gap. The figure prints both, and the fact that they agree is the check that the identity is right rather than merely plausible.

That check is the reason the correction above was available. Both computations were already in the figure and both were already agreeing, because every ratio the figure draws is coprime and the product and the lcm are the same number there. Running the same pair on a ratio with a common factor is what separates them, and it takes one line: 6 against 4 measures a twelfth of a bar where the product rule says a twenty-fourth. A check that compares a formula against a search only tests the formula on the inputs it is given, and a table of coprime ratios cannot distinguish a rule that depends on the product from one that depends on the lcm.

The onset count and the distinct-gap count are exact rather than sampled: both streams’ positions are generated, deduplicated, sorted, and the gaps counted to nine decimal places. The two identities — p + q − gcd onsets, and min(p, q)/gcd distinct gaps — hold on every ratio tried, coprime and not.

The order threshold of twenty and forty milliseconds is quoted from the psychophysical literature on temporal order judgement, not measured here. It is broad, it depends heavily on what the sounds are, and two clearly distinct timbres are separable at shorter intervals than two identical clicks. The bound moves with it and the shape of the argument does not.

The tempo window is the same one the beat-rate rung uses, and the bar lengths are the threshold multiplied by the product, which is the constraint rearranged.

And read as a beat rather than as rates, the same axis says which layer of a cross-rhythm is taken as the pulse: whichever falls nearest the preferred tapping rate, which is a property of the tempo rather than of the ratio.

Whose music, and when #

Three against two and four against three are effectively universal — West and Central African drumming, Cuban and Brazilian traditions, Indian tala, European art music from the fourteenth century, and every popular idiom descended from any of those.

Five against four and seven against five are far rarer and their distribution is informative: they appear in notated art music, in a few virtuoso instrumental traditions, and in electronic music, and hardly at all in music that is learned by ear and played in ensembles. That is what the product bound predicts, because the ratios that survive at speed are exactly the ones with small products, and music learned in ensembles is played at dance tempi.

The extreme cases — ratios with products in the hundreds — appear in twentieth-century notated music, where they are written and are not what is heard. That is not a criticism of the notation. A performer given eleven against seven produces something with the right shape and the right proportions, and the fact that the listener does not receive two clocks does not mean they receive nothing.

What the picture cannot show #

Every onset here is a point. Real onsets have rise times of tens of milliseconds, and the effective separation between two events depends on their envelopes as much as on their nominal positions. The first fifty milliseconds of a note are exactly the interval this essay’s threshold lives in.

The streams are identical. The model has no timbre, so it cannot represent the one thing that most affects whether two streams stay two. A polyrhythm played on a bell and a drum is a different perceptual object from the same ratio on two identical woodblocks, and the difference is larger than anything computed here.

And it has no performer in it. A real player does not place onsets at exact fractions; the deviations are systematic and are the groove. A polyrhythm played by people has its closest approaches moved by tens of milliseconds in whichever direction the players find comfortable, which for the tight ratios is very likely away from fusion.

And the ranking is by lcm, which for the ratios anybody plays is the product. Every ratio in every table here is coprime, so nothing in the numbers changes; what changes is what the rule would say about a ratio written unreduced. That is not a hypothetical case — a score asking for six against four, or twelve against eight, is asking for a three-against-two at a stated density, and a rule that ranked it by 24 or 96 would call it hard when it is the easiest ratio there is.

Where this leaves the ladder #

polyrhythm closes at nine rungs, which is the fifth anchor in this collection to close and the first in the rhythm field.

The bound it closes on is not a rhythmic one. It is the interval at which two events stop being two, which is a fact about the auditory system that has nothing to do with music, and it turns out to set the boundary of what the rhythmic material can be. That is the same shape as the melody ladder’s first rung, where the largest usable melodic leap turned out to be set by a streaming boundary measured on synthetic tones by somebody not asking about music at all.