The frontier and the ruler

What the clave buys with its unevenness put every one of the 273 distinct cycles of five onsets in sixteen steps on a plane. One axis is how much of the cycle a listener has to hear before knowing where in it they are — a question that exists at all because a cycle has no beginning; the other is how uneven the pattern’s gaps are. The two objectives are opposed, so there is no best pattern — only a frontier of twenty-two, and the son clave is on it.

Half of that result is exact. The locating cost is a count: list the sixteen rotations, take every window of every length, group the identical ones, average the log of the group sizes. Nothing about it is a choice.

The other half is a choice, and its own last section says so:

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.

A frontier is a statement about two quantities. If one of them is a choice, then so is the frontier, and the question is which of the findings survive the choice being made differently.

Five rulers, and two of them are one #

The five are not arbitrary. Each is in use somewhere in the literature on rhythm or on scales, and they fall into groups.

Two read the gaps between successive onsets. Their standard deviation is the one the published frontier uses. The range — the longest gap minus the shortest — is the crudest possible version of the same idea and is what the maximal-evenness predicate is built on: a pattern is maximally even exactly when that range is one.

Two read the positions rather than the gaps. The root-mean-square distance between each onset and where a perfectly even onset would fall is the obvious one. The other is the sum of the chord lengths between every pair of onsets placed on a circle, which a maximally even set provably maximises — so the deficit against that maximum is a measure of unevenness, and it is the one the theory of maximally even sets was originally stated with.

The fifth reads the variety of gap lengths: the entropy of the distribution of gaps, which is zero when every gap is the same and rises as more distinct lengths appear.

Ranked against one another over all 273 patterns, the first four agree closely. No two of them have a rank correlation below 0.888, and the two position-based ones agree at 0.998 — which is to say they are one measure written twice, and finding that out is worth the cost of computing both.

The fifth is anti-correlated with the other four at −0.26.

The ruler that is not a ruler #

That is not a subtlety and it is not a disagreement about emphasis. The variety measure is not measuring unevenness at all, and the reason is visible in one example.

A pattern of five onsets in sixteen with gaps 3, 3, 4, 3, 3 uses two distinct gap lengths. A pattern with gaps 1, 1, 1, 1, 12 also uses two distinct gap lengths. The entropy of the gap distribution is higher for the first, because its two lengths are split four-to-one against the second’s — so on this ruler the near-perfectly-even pattern is the more uneven of the two, and a pattern that puts four onsets in a row and then waits three-quarters of a cycle is the evener.

The lesson generalises past this figure. A measure of unevenness has to be sensitive to how large the departures are and not only to how many kinds of departure there are. Every count-based or category-based statistic on an interval distribution has this failure available to it, and the normalised pairwise variability measures used on speech rhythm and on musical timing are close enough relatives to be worth checking.

So the honest sweep is four rulers rather than five, and the fifth is a null with a reason rather than a disagreement to be split.

What survives, and it survives harder #

With the variety measure set aside, the published result does not merely survive; it is understated.

The son clave is on the frontier under every one of the four. So is the bossa-nova pattern, which is a rotation of the maximally even pattern and sits at the frontier’s even end. Neither result depends on the ruler.

What does depend on the ruler is how much that membership is worth. The published frontier holds twenty-two of 273 patterns, or eight per cent, and the essay that drew it was careful to say that eight per cent is a real constraint but not an overwhelming one. Under the range measure the frontier holds twelve; under the chord-length deficit, eight; under the onset displacement, seven.

The reason the frontier shrinks is worth stating because it is not that the stricter ruler is somehow better. It is that the gap-based measures are coarse: gaps in a sixteen-step cycle are small integers, so the standard deviation of five of them takes few distinct values, many patterns tie, and a tie cannot be dominated. The position-based measures are continuous and almost nothing ties, so almost everything that can be dominated is.

That coarseness is doing something else as well, and it is the sharpest correction here.

The two claves are not equally uneven #

The published account observes that the son and rumba claves have exactly the same unevenness — a gap spread of 0.748 apiece — and that the son therefore dominates the rumba outright, having the same evenness and a better locating cost.

The first half of that is an artefact of the ruler. The two patterns’ gaps are 3, 3, 4, 2, 4 and 3, 4, 3, 2, 4: the same five numbers in a different order. Any measure that reads only the multiset of gaps must call them identical, and the spread, the range and the variety all do.

A measure that reads the positions does not. The son’s onsets sit 0.403 steps from perfectly even in the root-mean-square and the rumba’s sit 0.490 — twenty-two per cent further. On the chord-length deficit the gap is wider still, 0.021 against 0.057.

The conclusion is unchanged and the reasoning has to change. On the gap-based rulers the son dominates the rumba by having equal evenness and better locatability. On the position-based rulers it dominates by being both more even and more locatable. Either way the rumba is on none of the four frontiers, which is the strongest form the original claim could have taken.

That is the shape of a robust result: the verdict holds under every ruler and the reason for it does not.

The bembe stops being dominated #

Not everything holds. The published account places the collection’s other timelines on their own frontiers and reports that the standard bell pattern — seven onsets in twelve steps — is not on its, sitting at a locating length of eleven of twelve where the frontier at its own unevenness offers ten.

The mechanism is the same coarseness working the other way. The bell pattern is the maximally even pattern of seven in twelve; its gaps are 2, 2, 1, 2, 2, 1, 2 and its range is one, which is the definition of maximal evenness. At this size the gap-based rulers put several other patterns at the same spread, and one of them locates a listener slightly faster, so the bell pattern is dominated by a tie. The position-based rulers separate them, the bell pattern comes out strictly the evenest thing there is, and nothing can dominate the extreme point of an axis.

So one of the five conclusions in the published account is ruler-dependent and the sweep reverses it. The count now reads: of the timelines this collection carries, the tresillo is on four of five frontiers at its size, the son clave and the bossa on four of five at theirs, the bell pattern on two of five, and the rumba clave on none. That is a mixed result and it was a mixed result before; what has changed is which of the mixture is a fact about the music.

A tie is what decides a frontier #

Two of the three corrections above came from ties rather than from disagreements, and that is the general lesson of the sweep rather than an accident of two sizes.

A pattern is on a frontier when nothing beats it on both axes at once. Beating it requires being strictly better on one and no worse on the other, so the patterns that survive are the ones nothing ties with — and a coarse ruler manufactures ties. Five cyclic gaps in a sixteen-step cycle are five small integers summing to sixteen, and their standard deviation takes only a few dozen distinct values across 273 patterns. Many patterns share a value; a shared value cannot be improved on without being strictly bettered on the other axis; so the frontier fills up.

That is why the gap-based frontier holds twenty-two points and the positional one holds seven, and it is why the two disagree about the bell pattern rather than about the son clave. The son clave is separated from everything near it on both kinds of ruler. The bell pattern is not: at seven onsets in twelve steps it ties with several patterns on gap spread and one of them locates faster, so a coarse ruler shows it dominated and a fine one shows it extreme.

The same thing happens at five onsets in twelve, where no named timeline exists to argue about. The maximally even pattern there is on the positional frontier and off the gap-based one, for exactly the reason above and with nothing at stake. So the disagreement is systematic and it is about resolution, which means it can be predicted rather than discovered: expect the two families to differ wherever the cycle is short enough that the gaps are few and small, and to agree wherever it is long enough that they are not.

The pattern that satisfies everybody #

One more number falls out of running all five and it is a useful sanity check on the whole enterprise.

Of the 273 cycles of five in sixteen, exactly one is on all five frontiers at once — including the variety measure’s. Its gaps are 1, 4, 3, 2, 6, its locating cost is 0.709 against the son clave’s 0.925, and it locates a listener in five steps against the son’s nine. Nobody plays it.

That is the right outcome and it is worth saying why. The frontier describes what is available, not what is chosen, and the patterns at the locatable end of any of these frontiers are clustered — several onsets close together and then a long silence. A cycle has no ending and what it has instead is a continuous texture, so a pattern with a six-step hole in it cannot do a timeline’s job whatever its information content. The universal frontier point is the arithmetic recording that only one of a timeline’s several jobs has been measured.

Which computation produced the numbers #

The census is unchanged: every subset of the given size, reduced to one representative of its rotation class by taking the lexicographically smallest rotation, scored by the residual-entropy measure. The locating cost is the area under that entropy curve.

The five unevenness measures are computed on the same representatives. The gap spread and range are read off the cyclic gaps; the onset displacement minimises the root-mean-square distance from a perfectly even set over all alignments of that set, in eighths of a step, because the even positions of five onsets in sixteen do not fall on integers; the chord-length deficit is the sum of the sines of the half-angles between every pair of onsets, subtracted from the same sum for the maximally even pattern of that size, so that it is zero for the even pattern and positive for everything else; and the variety measure is the Shannon entropy of the gap-length distribution.

A frontier is the set of patterns not dominated on both axes at once, and the rank correlations are Spearman’s over the whole census rather than over the frontier — a correlation computed on frontier members only would be a correlation among the survivors of the thing being tested.

The comparison across rulers is a comparison of sets, not of positions. Two rulers can rank the census identically and produce different frontiers, because a frontier is decided by ties and by extremes rather than by the middle of an order.

What the picture cannot show #

Four rulers is not all rulers. The four that survive here are the ones in circulation; a fifth that reads something else about a pattern could put a different set on the frontier, and nothing rules that out. What can be said is that four independent choices, two of which are the standard ones in the theory of maximally even sets, agree on the two conclusions this collection has drawn from the frontier.

The locating cost has its own choices in it. The window is a run of consecutive steps, the listener’s memory is perfect and their hearing is exact. A listener whose memory fades has a different ranking altogether, and that is a second axis with the same problem as the first: it has been swept and the sweep is on a neighbouring ladder rather than this one.

Five onsets in sixteen is one size, and the sweep across sizes is small. Four sizes appear here and the four rulers agree least at seven in twelve — 0.60 rather than 0.89 — which is a hint that the agreement is a property of large censuses and not of the measures. Sixty-six patterns is not many.

And nothing here says anybody chose anything. The tresillo, the son clave and the bell pattern are what traditions arrived at, by paths nobody recorded, under pressures that included being danceable and teachable and playable on the instruments to hand. That two of them are Pareto-optimal on two computable quantities is a fact about the patterns, and it becomes evidence about the process only with a corpus. The same debt stands on the additive-metre ladder, where the arithmetic likewise bounds the space and the tradition picks inside it.

Whose music, and when #

The son and rumba claves are Cuban, the bossa-nova pattern Brazilian, the standard bell pattern West African and the oldest of the four. The theory of maximally even sets is from the pitch literature of the late 1980s, where it was invented to explain the diatonic scale, and the same construction produces both — which is why the chord-length measure used here on a rhythm is the one originally written for a scale.

The measures of rhythmic evenness in circulation come from at least three fields that do not read each other: music theory, corpus phonetics and the computational-geometry work that named the Euclidean rhythms. The finding that one of them is anti-correlated with the others is a fact about that fragmentation rather than about music.

Where this ladder goes next #

Four 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; what 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; and that frontier is a claim about a quantity that had to be chosen, so it has now been drawn four times and the son clave is on all four.

What is owed is the third axis. Two of a timeline’s jobs are now measured — how even it is and how quickly it locates — and the frontier’s universal point is a pattern with a six-step hole in it, which fails at a job nothing here computes. Syncopation against a metre is computable and this collection already computes it; so is the length of the longest silence, which is the crudest possible stand-in for whether a pattern can hold a cycle up. Adding either as a third axis turns a frontier into a surface and would say whether the named timelines are optimal in three dimensions or merely in the two that were easy — and it needs no corpus, only the census this ladder already enumerates.