What the clave buys with its unevenness

The previous rung took the usual claim about Euclidean rhythms — that Bjorklund’s algorithm produces the world’s timelines — and found the truthful version of it. The tresillo is E(3,8) exactly; the bossa-nova pattern and the standard bell pattern are rotations of theirs; and the son and rumba claves, the two best-known timelines in the world, are not Euclidean at any rotation whatever — a result about the one thing a timeline is for, which is a starting position.

It ended with a question rather than a rung. If the clave gives up evenness to be a marker, is there a quantity it maximises the way E(5,16) maximises evenness — some measure of how unambiguously a fragment of the pattern identifies its own position in the cycle?

There is, it is exact, and the answer it gives is neither of the two obvious ones.

The measure #

A timeline repeats. A player or a dancer coming in partway through hears some number of consecutive steps and has to work out not what the pattern is — everybody knows that — but where in it they are.

That is an exact question. A cycle of length n has n rotations. After hearing w consecutive steps, the rotations still consistent with what has been heard are the positions at which that window occurs, and the listener’s remaining uncertainty is the logarithm of how many there are. Average over all n starting points and the result is the residual entropy at window w, in bits, falling from log₂n when nothing has been heard to zero at the first length that tells every rotation apart.

Two numbers come out of it. The locating length is the shortest window at which every rotation is distinguishable, which is the worst case and the one a player who has just come in cares about. The cost is the area under the entropy curve, which is the average case.

The son clave reaches zero at nine steps of sixteen. The bossa-nova pattern reaches zero at fifteen. The bossa is a rotation of E(5,16), and a rotation has the same entropy curve as what it is a rotation of, so that number is the Euclidean pattern’s: the most even pattern of five in sixteen requires a listener to hear almost the entire cycle before its position is unambiguous.

That is the answer to the previous rung’s question, and it is a large effect. Nine against fifteen; an average cost of 0.93 bits against 1.39.

Why the even pattern is the worst one #

The reason is not a coincidence of this particular size, and stating it makes the rest of the essay inevitable.

A maximally even pattern is as close as an integer cycle allows to a pattern with a rotational symmetry, and a pattern with a rotational symmetry cannot be located at all: two of its rotations are literally the same sequence, so no amount of listening distinguishes them. Four onsets in sixteen is the clean case — E(4,16) is a strike every four steps, its rotations collapse to four distinct sequences, and two bits of position are unknowable however long anybody listens.

Five in sixteen has no exact symmetry, so the even pattern can eventually be located; but it is nearly symmetric, its gaps are 3, 3, 3, 3, 4, and every window shorter than the whole cycle looks like several different places in it.

Evenness and locatability are opposed by construction. A pattern is easy to locate when it has a distinctive feature, and a feature is a departure from regularity. The perfectly regular pattern has none.

That is the shape of the relation and it is not quite the strength of it. Running the same census at six other sizes, the most even pattern is the worst locator at some of them and not at all of them:

Where it fails it fails narrowly, so the opposition is real everywhere and exact only at some sizes. What decides it is whether the most even pattern of that size is also the nearest to a rotational symmetry, and at 5 in 12 or 7 in 16 some less even pattern happens to be nearer.

The frontier’s size is worth carrying too. It runs from six of seven cycles at 3 in 8 — where being on it means almost nothing — to fifty-three of 715 at 7 in 16, and twenty-two of 273 at the size this essay is about. So the son clave’s membership is a real constraint at 5 in 16 and the tresillo’s at 3 in 8 is nearly vacuous, which is worth knowing before either is offered as evidence of design.

The bembe is not on its frontier either #

Placing the collection’s other timelines on their own frontiers gives a mixed answer rather than the clean one this section was heading toward.

The tresillo is on the 3-in-8 frontier, along with five of the other six cycles at that size. The son clave and the bossa are on the 5-in-16 frontier, at its two ends. The rumba clave is not, as the section below establishes — and neither is the bembe, the twelve-step bell pattern, which sits at a locating length of eleven of twelve where the frontier at its own unevenness offers ten.

So of the five timelines this collection carries, two are Pareto-optimal in a way that means something, one is optimal in a way that means very little, and two are dominated. That is a weaker result than a story about optimal design and it is the one the arithmetic gives, and the honest reading is that the frontier describes what a locatable timeline can be rather than what these traditions were selecting for.

So there is no best, only a frontier #

Which means the previous rung’s question was slightly the wrong question. It asked whether the non-Euclidean timelines are optimal for something else or merely arbitrary, and the answer is that they cannot be optimal for locatability alone, because the patterns that are best at locating are the clustered ones — five strikes in a row and eleven rests — and a clustered pattern is useless as a timeline for a reason that has nothing to do with information. A cycle cannot cadence, so what it has instead of an ending is a continuous texture, and eleven consecutive rests is not one.

There are two objectives, they conflict, and what a pattern can be is Pareto-optimal: not improvable on one without being made worse on the other.

Of the 273 distinct cycles of five onsets in sixteen steps, twenty-two are on that frontier. The son clave is one of them. So is the bossa, at the extreme even end where the frontier begins.

And the son clave is the frontier’s second point — the least uneven pattern that improves on maximal evenness at all. Moving from the bossa to the son costs 0.35 in the spread of the gaps and buys a fall in locating length from fifteen steps to nine and in average cost from 1.39 bits to 0.93. It is the first step off the even pattern, and it is an optimal step.

The rumba clave is not on it #

The two claves differ by one onset moved by one step, and the measure separates them.

The rumba clave is dominated. It has exactly the same unevenness as the son — the same spread of gaps, 0.75 — and a worse locating cost, 0.97 against 0.93, with a locating length of eleven steps against nine. There is a pattern with its evenness and better locatability, and that pattern is the son clave.

That is a real asymmetry between two patterns usually described as variants of one another, and it should be read carefully rather than triumphantly. It does not say the rumba clave is worse music. It says that on this one measure, with this one definition of evenness, the son is on the frontier and the rumba is just inside it — and that if a tradition were selecting for these two things and nothing else, the son is what it would arrive at.

The gap of two, and what it costs elsewhere #

The mechanism is small enough to point at. The son clave’s gaps are 3, 3, 4, 2, 4; the bossa’s are 3, 3, 4, 3, 3. The son has one gap of two and nothing else in the cycle is a gap of two, so a listener who hears two adjacent onsets a step apart knows exactly where they are, and everything else in the entropy curve is about how long it takes for that gap to arrive.

That is why the locating length is nine rather than five. Five steps would be enough if the window were guaranteed to contain the short gap; nine is what it takes for every starting point to have seen enough, and the worst starting point is the one that begins just after it.

The same feature has a cost in a different currency, and this collection has already measured it. Syncopation is a property of a pattern and a metre together, priced as a note on a weak position followed by a rest on a stronger one — and the son clave’s short gap puts an onset on a weak sixteenth immediately before a strong position that is empty. The feature that makes the pattern locatable is the feature that makes it syncopated, and they are not two properties but one, read against two different things: against the cycle’s rotations, and against the metre.

What it is for, in the tradition’s own terms #

The measure was built to answer a formal question and it lands on a practical one.

A clave is not a rhythm a piece contains; it is a frame the whole ensemble is in, and being on the wrong side of it — playing 3–2 material over a 2–3 clave — is a specific, named, correctable error. The measure above says exactly how much of the cycle a player has to hear before that error becomes impossible: nine steps of sixteen for the son, which is a little over half a bar.

For the even pattern it is fifteen steps, which is to say a whole cycle. As evenly as possible is therefore not the design brief it looks like. A maximally even timeline could not support the practice, because there would be no partial hearing from which the side could be inferred, and a player joining would have to be told rather than able to hear.

Rhythm is a circle and the bar line is a choice made the point that the same pattern is shared across continents and differs in where somebody decided to start counting. This rung adds the consequence: when a pattern is even, the decision cannot be recovered from the sound, and when it is not, it can.

Which computation produced the numbers #

The entropy is counted rather than estimated. For a cycle of length n, all n rotations are listed; for each window length w from 0 to n, every rotation’s first w steps are taken and identical windows are grouped; the residual entropy of a rotation is the log₂ of the size of its group, and the reported figure is the mean over the n rotations. The locating length is the first w at which every group has one member.

The unevenness is the standard deviation of the inter-onset gaps around the cycle, which is a plain measure and not the site’s maximallyEven predicate — the predicate is a yes-or-no and the frontier needs a quantity.

The census enumerates every subset of five positions in sixteen, reduces each to one representative of its rotation class by taking the lexicographically smallest rotation, and scores the 273 that remain. A timeline is a cycle, so its rotations are the same object and counting them separately would put 4,368 points on a figure that has 273 things in it.

One thing about the measure is worth flagging as a choice rather than a fact. The window is taken as consecutive steps from a starting point, which models a listener who begins hearing at an arbitrary moment and keeps listening. A listener who has heard a scattered subset of the cycle, or who is weighting recent steps more heavily, is a different model and would give a different frontier.

Whose music, and when #

The son clave is Cuban and its ubiquity in twentieth-century popular music runs through son, mambo, salsa and everything downstream. The rumba clave is the same five onsets with one moved and belongs to the rumba complex. The bossa-nova pattern is Brazilian. The standard bell pattern is West African and the oldest of them.

The claim here is not that anybody computed anything. It is that a practice which requires players to locate themselves in a cycle by ear will be under pressure to use patterns that make that possible, and that the pattern which became the most widespread of them all is, on this measure, an optimal first step away from evenness.

That is a claim about selection and it is not testable from one pattern. What would test it is the rest of the world’s timelines — a corpus of them, scored the same way, to see whether they cluster on the frontier or scatter over the 273. This collection has five timelines and the frontier has twenty-two points on it, so the sample cannot distinguish selection from coincidence, and this is the fourth time in a row that a rhythm result here has ended at exactly that boundary.

What the picture cannot show #

Five onsets in sixteen is one size. The frontier’s shape at other sizes has not been computed, and there is no reason to expect the named timelines of other sizes — the bell pattern’s seven in twelve — to sit in the same relative place.

Unevenness has more than one definition. The spread of the gaps is one; the sum of squared deviations from the perfectly even positions is another and ranks patterns differently; the site’s own maximal-evenness predicate is a third and is binary. The son clave’s position on the frontier is stable under the first two and the frontier’s shape is not.

The listener is a perfect memory with no noise. Every step heard is heard correctly and remembered exactly, which is why the entropy reaches zero at all. A listener with any confusion between a gap of two and a gap of three would find the son clave harder to locate than nine steps, and by an amount nothing here computes.

And nothing here is heard. Whether a real player locates a clave in nine steps, or in three because they are using timbre and dynamics and the rest of the ensemble, is a listening question. What is computed is what is available in the onset pattern alone, which is the same restriction every rung of this ladder has worked under — and which the metre ladder found to be the binding one when it varied the input rather than the model.

Where this ladder goes next #

Three rungs. Bjorklund’s algorithm produces patterns the world plays; the match holds only up to rotation, and the two most famous timelines are not matches at all; and the thing they are good at instead is telling a listener where in the cycle they are, at a cost in evenness that puts the son clave on the frontier between the two.

The rung after it is the one the last section names and this site cannot yet write, and it is the same shape as the debt the additive-metre ladder recorded. The frontier is drawn and there are five points on it from the real world. What decides whether the frontier is a description of a selection pressure or a pattern in twenty-two coincidences is a corpus — the timelines of West Africa and the Caribbean and Brazil and Indonesia, scored the same way — and it is a corpus of a few hundred short binary strings — the same shape of debt the melody ladder recorded, which is the smallest corpus this collection has ever needed and does not have. The aksak metres owe the same debt and it is the same kind of counting.