The rotation the necklace cannot see

The first rung of this ladder fed Bjorklund’s algorithm three onsets in eight steps and got the tresillo, five in eight and got the cinquillo, and observed that nobody had told it about Cuba.

That result is real and the algorithm was not tuned to produce it. What has grown up around it is a broader claim — that the timelines of the world are Euclidean rhythms — and the broader claim is checkable in a way the narrow one was not. Every named timeline is a specific pattern; every Euclidean pattern with the same onset count in the same number of steps is a specific pattern; and whether one is a rotation of the other is a search over positions.

Running that search does not return what the claim’s usual statement suggests.

The search #

Three of five, then. And the two that fail are not obscure: the son clave is the rhythmic foundation of Cuban son and of most of the popular music descended from it, and the rumba clave is its nearest relative.

The failure is not marginal either. E(5,16) has four gaps of three and one of four; the son clave has two of three, one of four, one of two and one of four. It has a gap of two in it, which the maximally even pattern does not and cannot — evenness is exactly the property of having no gap that small when a larger one is available.

That is a structural refusal rather than a near miss, and it is worth stating as one. Rotation cannot change a pattern’s multiset of gaps, so a pattern with a gap of two can never be a rotation of a pattern without one, at any starting position, in any number of steps. So the search over sixteen rotations was not needed for the two claves: one look at the gaps settles it. What the search does establish is the positive cases — that the bossa-nova and the bell pattern really are rotations rather than merely having the same gap multiset in a different order, which is a thing gaps alone cannot decide.

So the honest form of the claim is: some famous timelines are the maximally even pattern for their onset count, up to rotation, and some are systematically not. That is a more interesting statement than the universal one, because it leaves something to explain — and because a claim that everything fits a model is nearly always a claim that has stopped being tested, which is the failure mode the roughness essays had to be rescued from.

What the exceptions have instead #

The son clave gives up evenness and gets something for it, and the something is identifiable.

A maximally even pattern is, by construction, as close to isochronous as its numbers allow. That makes it a good pulse and a poor marker: every position in it resembles every other, so it does not announce where the cycle begins. E(5,16) drawn as a cycle is nearly a five-against-sixteen pulse, and a listener joining halfway through has little to go on.

That is the trade. Evenness buys a pattern that works as a substrate; asymmetry buys a pattern that works as a reference. A clave is a reference — its whole function in the music is that everything else is positioned relative to it, and musicians speak of a phrase being “in clave” or “crossed”, which is a statement about alignment and therefore about a starting position. It is the rhythmic equivalent of a scale degree that is defined by what it does rather than by where it sits, and it fails for the same reason: the description is of a set, and the content is in an ordering the set does not carry.

A pattern whose job is to locate a listener in the cycle cannot be a pattern all of whose positions look the same.

Which the necklace throws away #

The deeper problem is that the standard description does not merely fail to explain this — it cannot represent it.

A necklace is a cyclic pattern considered up to rotation. It is the standard object in the combinatorial treatment of rhythm here and everywhere: it is what makes counting patterns tractable, and it is exactly what identifies a pattern with all of its own rotations. Under that identification the bembé and the Euclidean E(7,12) are the same object, and the son clave and the bossa-nova are different objects — which is right, and is nothing like enough.

The size of what is thrown away can be measured, using the rhythm ladder’s own machinery.

Eight to one is not a nuance. Syncopation is the standard quantitative handle on what a rhythm feels like against a metre, this collection has a rung about it, and it turns out to be almost entirely a function of the thing a necklace discards.

Which is the sharpest available statement of what a necklace is missing, and it is worth putting the two together. The necklace identifies sixteen patterns as one object. Syncopation separates them into values from one to eight. So the quantity this collection uses to say what a rhythm does against a metre is a quantity the standard combinatorial object cannot see at all — not approximately, not partially: the necklace’s whole equivalence class is a set the measure assigns eight different answers to, and there is no function of the necklace that could recover any of them.

The same computation on the bell pattern is more pointed still. Its twelve rotations score between 0 and 3, and the rotation the tradition plays scores 3 — the maximum. Whatever the tradition selected for, it was not the least syncopated reading of its own necklace.

The bell pattern’s rotation is the maximum #

Taking the second case on its own is worth doing, because it says something the son clave’s does not.

A pattern whose function is to be a reference could plausibly have been chosen to sit with the metre — to make the beat obvious. It is not. It is played at a rotation that maximises the tension between the onsets and the metrical weights, and the result is a pattern that is simultaneously the reference and the thing most in conflict with the pulse.

That is not a paradox once the two functions are separated. What the bell is a reference for is position in the cycle, and a pattern that agreed with the metre everywhere would be a poor position-marker for the same reason an even pattern is. Conflict with the weights is what makes each onset distinguishable from the others, and distinguishable onsets are what let a listener locate themselves.

It is worth being careful, though: the metrical weights being used are a Western theoretical construct, imported here from a model built for European notated music, and applying them to a West African timeline is an assumption rather than a description. Running the twelve rotations under seven different subdivision trees says which half of that caution is needed.

The rotation the tradition plays is the maximum under five of the six trees that discriminate at all, and second under the other two. So the half of the caution about which rotation wins turns out not to be needed: the played rotation is at or one below the top whatever tree is imposed, which is a stronger result than the essay was claiming.

The half that is needed is the other one. The spread does not survive an arbitrary choice — it runs from a factor of five under 3 × 2 × 2 to a range of one under 2 × 6, and to exactly nothing under a bar of twelve with no subdivision, where every rotation scores zero and the measure has no opinion at all. So the eight-to-one the son clave gave and the nought-to-three the bell gave are figures for a particular tree, and a tree with fewer levels flattens them toward nothing.

There is one more thing in that table worth reading. The tree that discriminates most — 3 × 2 × 2, spread one to five — is one of the two where the played rotation is not the maximum. So the weights that separate the rotations best are the weights under which the tradition’s choice looks least like an extreme, which is either a coincidence of one pattern or a hint that the maximum was never what was being selected for. Twelve rotations of one timeline cannot tell those apart.

The same hole, in the other half of the site #

This is a rhythmic instance of a defect this collection has met before in pitch.

The census over subsets cannot tell which note is home: one property tuple covers all eighty-four rotations and transpositions of the diatonic set, so a major scale and its own Aeolian mode are indistinguishable to it. That essay’s conclusion was that the tonic is not in the set, and has to come from somewhere else.

The rhythmic version is exact: the downbeat is not in the necklace, and has to come from somewhere else. And the two are the same defect, because the two objects are the same object — the diatonic set and the bell pattern are both E(7,12), which is the coincidence that first rung was about.

The two objectives are opposed, so there is no best pattern — only a frontier. A necklace has one evenness and many locatabilities, because locatability is a property of where the listener starts and evenness is not, and that is exactly the coordinate the necklace throws away.

So the coincidence has a second half that the first rung did not have room for. The two objects are the same necklace; the tradition using each of them selected a rotation; and neither the algorithm nor the census has any way to produce the rotation, in either domain.

What the algorithm is actually a theory of #

Setting all this out makes it sound like the Euclidean account is a failure, and it is not. It is a theory of a narrower thing than it is usually taken to be, and the narrower thing is worth having.

The claim it supports is: of all the ways to place k onsets in n steps, the ones traditions converge on are disproportionately the maximally even one. That is true, it is surprising, and it does not require every timeline to be Euclidean any more than every scale is maximally even.

The claim it does not support is that the algorithm produces timelines. It produces necklaces, and a timeline is a necklace plus a starting position plus, usually, a specific metre it is played against. Two of the three come from somewhere the algorithm cannot see.

What a starting position costs to specify #

There is an arithmetic way to say how much information a rotation is, and it puts the essay’s claim on a scale.

A necklace of n steps has up to n rotations, so specifying which one is at most log₂ n bits: three for a bar of eight, four for twelve or sixteen. Against that, specifying which five-onset necklace in sixteen is being used is a choice among a few hundred, which is eight or nine bits.

So a starting position is a third to a half as much information as the pattern itself — a large fraction of a small number, for a quantity that a necklace-based description does not have a field for. That is the size of the gap between what the combinatorics enumerates and what a musician has to be told.

Being a rotation is a fact a necklace cannot report and a search can. As necklaces the three are one object and the two are two others; drawn against the Euclidean pattern with the offset computed, the three become a family and the two become genuinely different constructions — which is the whole difference between a set and an arrangement.

Which computation produced the numbers #

The timelines are transcriptions of rhythms, which is the one kind of encoding this site treats as unambiguous — an onset either happens on a step or does not, and there is no equivalent of the interpretive latitude a melodic transcription has.

The rotation is found by search: the Euclidean pattern for the same k and n is generated, all n of its rotations are compared with the timeline, and the first exact match is reported. Where no rotation matches, the figure says so and prints both gap sequences so the reader can see why.

One bug is worth recording because it produced a completely wrong figure that looked plausible. The Euclidean generator returns booleans and the timelines are written as ones and zeros; compared with a strict equality the two never match, and the first version of this figure reported every timeline in the table as not Euclidean — including the tresillo, which is E(3,8) unrotated and is the first rung’s headline result. A figure claiming the Euclidean account fails entirely would have been a much more dramatic finding than the true one, and nothing about its appearance would have given it away.

The syncopation scores use the same function the metre ladder uses, with a bar of twelve read as 2·2·3 and a bar of sixteen as 2·2·2·2.

Whose music, and when #

The son and rumba claves are Cuban, from the Afro-Cuban traditions of the late nineteenth and early twentieth centuries, and their spread through popular music worldwide dates from the twentieth. The bossa-nova pattern is Brazilian and later. The bell pattern in twelve is West African and considerably older than any of them, and it travelled to the Americas with the people who played it.

The Euclidean claim as a general thesis is Godfried Toussaint’s, from the mid-2000s, and it is presented in his own writing with more care than it is usually repeated with — the correspondences are given pattern by pattern, and several are given as rotations. What has hardened in the retelling is the universal version.

It is worth adding that “the standard bell pattern” is a Western label for a family of related patterns rather than one artefact, that transcriptions of it differ, and that the version used here is the most commonly cited one. A different transcription would change which rotation the search reports, though not whether a rotation exists.

What the picture cannot show #

It has no timbre and no dynamics. A clave is played on two sticks with a single sound, but a bell pattern is often played with two distinct tones, and which onsets are high and which low is part of the pattern’s identity and is not in a binary vector.

It has no tempo. A timeline is played at a rate, and the beat a listener induces from it depends on that rate as well as on the pattern. The same onset set at 90 and at 200 to the minute is not heard with the same subdivision as the beat.

It has no microtiming. The deviations from an even grid are systematic and are much of what a groove is, and a quantised transcription has removed them before this essay begins. Two of the “gaps of three” in a played son clave are not the same length.

The metre is imposed. Every syncopation number here needs a metre fixed before it returns anything, and choosing the metre is itself a decision the pattern underdetermines. Rotating the pattern and rotating the metre are the same operation from two sides, and the figures hold one still and move the other.

And syncopation is a model, not a measurement. Longuet-Higgins and Lee’s definition is one of several, it requires a metre to be fixed before it returns anything, and the numbers above would change under a different metrical reading of the same bar. What survives any choice of definition is the spread across rotations, which is the quantity this essay is about.

The ladder from here #

This anchor has two rungs and the second has put a boundary on the first. What is owed is the question the exceptions raise: if the son 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?

That is computable. It is an information-theoretic question about a cyclic sequence, it would apply equally to the pitch case, and it would say whether the timelines that are not Euclidean are optimal for something else or merely arbitrary.