Rhythm and metre

The longest silence is not a third axis

The frontier between evenness and locatability was drawn twice and both times against two quantities, with the third job a timeline does left uncomputed. Computed, neither candidate for it is a third quantity: the longest silence ranks the whole census with the spread of the gaps at 0.94, and syncopation is not a property of a cycle at all — every one of the 273 patterns changes its count when the bar line moves. What the request was actually asking for is a cap rather than an axis, and under the tightest one the census admits thirteen patterns and every named timeline is among them.

16 min read 7 figures Somebody measured it Time is a circle The design was forced

Assumes: The frontier and the ruler · Syncopation is a number about the metre

Every one of the 273 distinct cycles of five onsets in sixteen steps sits on a plane, placed by how uneven its gaps are against how much of the cycle a listener has to hear before knowing where in it they are. The two objectives are opposed, so there is no best pattern, only a frontier of twenty-two — and redrawing that frontier under five different definitions of unevenness left the son clave on four of the five.

Both of those are arguments about two quantities, and a timeline does more than two jobs. The point that survives every ruler makes that obvious: it is the pattern whose gaps run 1, 4, 3, 2, 6, and it is on every frontier because it locates faster than anything else in the census — a cost of 0.709 against the perfectly even pattern’s 1.386. It achieves that by leaving a six-step gap between two of its onsets — five silent steps in a row, and more than a third of the cycle between one stroke and the next.

The longest silence against the spread of the gaps. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how long its longest gap is. The two rank the census together at a Spearman correlation of 0.94, so the longest silence is not a second thing to know about a timeline; it is the same thing measured more crudely. Against the other axis on the frontier — how much of the cycle has to be heard before a listener knows where in it they are — it correlates at -0.02, which is nothing. The named timelines are marked, and all of them sit at 4, the shortest longest-gap any pattern of this size can have. — Fig. 1 Every one of the 273 rotation classes placed by how uneven its gaps are against how long its longest gap is. The two rank the census together at a Spearman correlation of 0.937 — a long gap is most of what being uneven consists of — while against the locating cost the longest silence correlates at −0.017, which is nothing. The three named timelines sit at four steps, the shortest longest-gap any pattern of this size can have, and only two markers are drawn for them: the son and the rumba clave occupy the same point on both axes, which is the finding the last section of this essay is about.

The obvious repair was stated as a request for a third axis: add the longest silence, or add the syncopation, and a frontier becomes a surface. Both requests turn out to be requests for something else, and the interesting part is why.

The pattern the request was about

Start with the pattern that provoked it, because its shape explains what the missing job is.

The fastest-locating cycle of five in sixteen. A rhythm drawn round a circle, one equal arc per step, with the struck steps filled. On a circle the pattern has no beginning, which is why the same necklace of onsets is several different named rhythms depending on where a listener decides bar one is. — Fig. 2 The one pattern of five onsets in sixteen steps that survives every one of the five definitions of unevenness. Its gaps are 1, 4, 3, 2, 6, all different, which is exactly why a fragment of it says so much about where in the cycle it came from: no two windows look alike. The six-step gap is 37 per cent of the cycle, and nothing in the two published axes counts that against it.

All five of its gaps are distinct, which is the property the locating measure rewards: every window of every length is unique, so a listener knows their position almost at once. It is a superb identifier and a poor timeline, because a timeline has to keep a cycle turning and this one goes quiet for five steps out of sixteen.

That is a real third job and it is not measured by either axis. The question is whether adding a measurement of it changes anything, and the answer is available before any arithmetic, from what a Pareto frontier is.

What a third objective can do to a frontier

A frontier holds the patterns nothing else beats on every objective at once. Adding an objective makes beating harder, not easier: a pattern that was dominated because it lost on both of two quantities may now be spared because it wins on the third. Membership can only get cheaper.

Every frontier 5 onsets in 16 can be given, and what it keeps. The same 273 rotation classes under five sets of objectives, all minimised. The bar is how many patterns each frontier holds and the dots say which of the named timelines and the perfectly even pattern survive on it. Adding the longest silence to the two published axes takes the frontier from 22 to 24 and removes nothing, which is what a third objective does: a pattern that is worst on the new axis may still be best on an old one, so membership can only get easier. Adding the syncopation instead takes it to 18 and does remove 11, the son clave among them — and every one of those is a pattern that tied with another on the first two axes and lost the tie on the third. So neither request makes membership mean more, and the one that appears to does it by settling ties rather than by measuring a third job. — Fig. 3 The same 273 patterns under five sets of objectives, all minimised. The two published axes give twenty-two. Adding the longest silence gives twenty-four: two patterns join and none leaves. Adding the syncopation instead gives eighteen — and the eleven it removes, the son clave among them, are removed by tie-breaking rather than by measurement. Replacing the unevenness with the longest silence altogether gives a frontier of eight that holds no named timeline at all.

Which computation produced those numbers is worth setting out, because all of it rests on one enumeration. There are 4,368 ways of putting five onsets in sixteen steps; grouping them by rotation, since a cycle has no beginning, leaves 273 distinct necklaces. The locating cost of each is exact and involves no choice: list the sixteen rotations, take every window of every length from nothing heard to the whole cycle, group the rotations that agree over that window, and average the logarithm of the group sizes. That gives the bits of position still unknown after each amount of hearing, and the cost is the area under the curve. It is the same measure that a colotomic cycle uses to code its position in its instrumentation rather than in its pattern, applied to a single line. A frontier is then whatever survives the domination test over whichever quantities are handed to it, and everything that follows is a question about which quantities those should be.

So the first request cannot do what it was made to do. Adding the longest silence to the two axes takes the frontier from twenty-two of 273 to twenty-four, and it does not remove the wide-gap pattern, because that pattern is still the fastest locator in the census and nothing dominates it. Membership means very slightly less than it did.

Nor is the correlation an accident of these two numbers. Measured the same way over every census from three onsets in eight steps to seven in sixteen, the longest silence and the spread of the gaps rank together between 0.89 and 1.00 — so what makes them one ruler is the quantity rather than the size of the space, and no other census is going to rescue the request.

The son clave surviving a third axis is therefore not the corroboration it looks like. It survives because everything on a two-axis frontier survives a third axis, always, by construction — the only exception being a tie.

Which is the one thing that did happen. Adding the syncopation removes eleven of the twenty-two, and every one of the eleven had a twin: a pattern with exactly the same locating cost and exactly the same unevenness, which the third quantity then separated. The son clave is one of them. It sits at a cost of 0.9248 and an unevenness of 0.7483, and so does the pattern with gaps 4, 3, 3, 2, 4 — which has a minimum syncopation of zero against the clave’s one, and so takes its place. That is a coin toss being settled by a coin, not a third job being measured.

Syncopation is a number about a placement

The deeper trouble with the second request is that the quantity it names does not exist on the objects the census enumerates.

The census counts rotation classes, because a timeline is a cycle and a rotation of it is the same necklace. Syncopation is not a property of a necklace. It is a property of a pattern together with a metre — Longuet-Higgins and Lee price a note on a weak position followed by a rest on a stronger one at the difference in metrical weight — so moving the bar line moves the count.

The same onsets, read from four places, four syncopation counts. A 16-step pattern with 5 onsets, against 4 readings of its bar. A position's weight is zero on the downbeat and one lower at each level down the subdivision tree, drawn here as the depth of the bar hanging beneath it. A note on a weak position followed by a rest on a stronger one costs the difference. from step 1 scores 4 — the note at step 4 against the rest at step 5, costing 2; the note at step 7 against the rest at step 9, costing 2. from step 4 scores 8 — the note at step 4 against the rest at step 5, costing 2; the note at step 8 against the rest at step 9, costing 3; the note at step 10 against the rest at step 13, costing 2; the note at step 14 against the rest at step 15, costing 1. from step 5 scores 1 — the note at step 3 against the rest at step 5, costing 1. from step 12 scores 5 — the note at step 2 against the rest at step 5, costing 2; the note at step 6 against the rest at step 7, costing 1; the note at step 12 against the rest at step 13, costing 2. — Fig. 4 The son clave read from four of the sixteen places its bar line could fall, against a metre of four levels of two. The same five onsets, in the same cyclic order, score 4, 8, 1 and 5. Nothing about the sound changes between these readings; what changes is where the weights are, and the weights are supplied by the listener.

Across the whole census the spread between a pattern’s least and most syncopated rotation runs from three to nine, and not one of the 273 patterns has a spread of zero. There is no value a rotation class could be given. Every possible axis built from this quantity is therefore a choice — the minimum over rotations, the maximum, the mean, the value at some canonical phase — and the first two axes never had to make one, because both are computed over all rotations by construction.

The choice made above was the minimum: the best reading a listener could adopt, which is the charitable one. Taking the maximum instead, or the value at the pattern’s own conventional downbeat, gives a different frontier, and there is no principle in the census that picks between them.

The only rotation-invariant summary is the gaps again

There is one summary that is genuinely a property of the necklace: the total over all sixteen rotations. It is what a listener would accumulate hearing the pattern against every possible metre, and it needs no choice.

Syncopation is a number about a placement, not about a necklace. Longuet-Higgins and Lee's syncopation count for four cyclic patterns of 5 onsets in 16 steps, read from every one of the 16 places the bar line could fall. son clave runs from 1 to 8, rumba clave runs from 2 to 7, the bossa-nova pattern runs from 2 to 7, E(5,16) runs from 2 to 7. Across the whole census of 273 rotation classes the spread runs from 3 to 9 and not one pattern has a spread of zero, so there is no single syncopation value a rotation class could be given. The one summary that is a property of the necklace is the total over all 16 rotations, and every one of these four sits at 79, the largest total in the census — shared with 6 other patterns and attained by exactly those whose every gap is two, three or four steps. — Fig. 5 The syncopation count of four patterns of five onsets in sixteen, read from every one of the sixteen places the bar line could fall. The son clave runs from 1 to 8, the rumba clave and the bossa-nova pattern from 2 to 7, and the perfectly even pattern from 2 to 7. Every one of them totals 79 over the whole cycle, which is the largest total in the census.

Two things fall out of that total and both are surprises.

The first is that it is a function of the gaps and of nothing else. Across the 273 patterns there are thirty-seven distinct multisets of gap lengths, and not one of them carries two different totals: any two patterns with the same five gaps in any cyclic order have the same rotation-total of syncopation. So the third axis, made rotation-invariant in the only way that requires no arbitration, turns out to read exactly the data the evenness rulers read. It is the evenness ruler for the third time — the longest silence was the second.

The second is what maximises it. The census total runs from 23 to 79, and 79 is attained by exactly nine patterns: those whose every gap is two, three or four steps. The son clave, the rumba clave, the bossa-nova pattern and the perfectly even pattern are four of the nine. A gap of one contributes nothing, because an onset immediately followed by another onset has no rest between them to syncopate against; a gap of five or more wastes length, since the extra silence buys at most one more strong position to cross and gives up several that a second onset would have crossed. Between those two failures the contribution per step is flat, which is why the nine tie exactly rather than merely coming close: the total depends on the gaps only through how many steps fall inside gaps of a usable size, and for these nine that is all sixteen of them. The named timelines are at the ceiling of a quantity nobody had computed for them, and they are there by avoiding both extremes of gap length rather than by being uneven in any particular way.

That flatness is also why the quantity separates nothing among them. It says a great deal about which patterns are outside the group of nine and nothing whatever about the ordering inside it — which is the failure the longest silence has, arriving from the other direction, and a second reason to reach for a threshold rather than a ranking.

What was wanted was a constraint

An objective ranks and cannot forbid. That is the whole of the difficulty, and it points straight at the repair: the six-step gap was never a reason to rank a pattern low, it was a reason to rule it out.

The longest silence as a cap rather than as an axis. How many of the 273 rotation classes of 5 onsets in 16 steps have each possible longest gap, from 4 steps to 12. Read as an objective this quantity changed nothing, because a Pareto frontier cannot forbid a pattern for being bad at one job. Read as a constraint it forbids a great many: requiring that no gap exceed 4 steps leaves 13 patterns of 273, and all 3 named timelines of this size are among them — the rumba clave included, which is on none of the five frontiers drawn ruler by ruler. The pattern that locates fastest of all 273 — the one that survives every ruler the frontier has been drawn with — has a gap of 6 steps and is forbidden here, which is what the request for a third axis was actually asking for. — Fig. 6 How many of the 273 patterns have each possible longest gap. Requiring that no gap exceed four steps — a quarter of the cycle — leaves thirteen, and all three named timelines are among them, the rumba clave included. The pattern that survived every one of the five frontiers has a gap of six and is forbidden here, which is what the request for a third axis was actually asking for.

Thirteen of 273. Four is the tightest cap the census can meet, because five gaps summing to sixteen force one of them to at least four, and the thirteen patterns that hit that floor are the ones that never leave the cycle unattended for more than a quarter of its length.

That is a much stronger statement about the named timelines than any frontier made. Membership of the twenty-two was one pattern in twelve; membership of the thirteen is one in twenty-one. And it covers the timeline the frontiers could not: the rumba clave, which is on none of the five published frontiers and is inside the cap. Loosening the cap to five leaves seventy-three patterns and a frontier of eighteen; loosening it to six leaves 147 and restores the original twenty-two exactly, six being the point at which the wide-gap pattern comes back.

So the third dimension the request asked for exists, it does the job it was supposed to do, and it does it by changing the feasible set rather than the ranking. A frontier says which trade-offs are efficient. A cap says which designs are admissible, and the difference is not a technicality: efficiency has nothing to say about a pattern that is superb at one job and unusable at another.

The two claves are one multiset in two orders

One more thing falls out of the finding that so many of these measures read only the gaps, and it settles a question the previous frontier left open.

6 arrangements of the same five gaps. Every rotation class of 5 onsets in 16 steps whose gaps are 2, 3, 3, 4, 4 — the same five durations in a different cyclic order. The spread of the gaps, their range, the entropy of their lengths, the longest of them and the total syncopation over all 16 rotations are identical for all 6, because every one of those reads the multiset and not the order. The locating cost reads the order, and it separates them: 0.925 at the best and 0.972 at the worst. The two the world plays are at the two ends — the son clave among the best and the rumba clave among the worst — which is the whole of why one of them is on four of the five published frontiers and the other is on none. — Fig. 7 The six rotation classes whose gaps are 2, 3, 3, 4 and 4 — the same five durations in six different cyclic orders. The spread of the gaps, their range, the entropy of their lengths, the longest of them and the rotation-total of the syncopation are identical for all six. The locating cost is not, and the two the world plays sit at the two ends of it.

The son clave and the rumba clave differ by one onset moved by one step, and what that move does is reorder the gaps: 3, 3, 4, 2, 4 becomes 3, 4, 3, 2, 4. The multiset is unchanged. Every quantity computed above that reads the multiset — the spread at 0.748, the range at 2, the variety, the longest silence at 4, the rotation-total at 79 — gives the two claves identical numbers, and so cannot tell them apart at all.

The locating cost reads the order, and it separates them: 0.925 for the son against 0.972 for the rumba, with the son needing nine consecutive steps to fix its position and the rumba eleven. Among the six arrangements of these five gaps the son is tied for the best and the rumba tied for the worst, and those are the two the repertoire selected.

That is why one of them is on four frontiers and the other on none: the rumba is dominated by the son at equal unevenness and worse locating cost, which is domination in its purest form. It is also why the frontier was the wrong instrument for the rumba clave — a pattern can be excellent and still be dominated by a near-twin, and the cap, which asks a question about the pattern rather than about its neighbours, admits it without hesitation.

What the picture cannot show

It cannot show a metre the timelines are not played in. Every syncopation count above is taken against four levels of two, which is right for the sixteen-step cycles and wrong for the twelve-step ones. The standard bell pattern lives in twelve and would need a metre of 2 × 2 × 3, and its census is a different one.

It cannot show a gap that is filled quietly. The longest silence here is a run of steps with no onset in the timeline. In performance the cycle is never empty: the bell is one layer of several, and the hole in one part is where another part plays. What the measure captures is the timeline’s own ability to hold a cycle up alone, which is what a solo bell has to do and what an ensemble part does not.

It cannot show why a cap should be a quarter of the cycle. Four steps out of sixteen is the tightest the arithmetic permits, so it is not a stipulation in this census — but the general rule it suggests, that a timeline’s longest silence should not exceed the cycle divided by the number of onsets, rounded up, is a stipulation and would have to be tested on other sizes before it meant anything.

It cannot show a listener who forgets. The locating cost assumes a rotation ruled out at step three stays ruled out, which is why the curve reaches zero and why a pattern has a locating length at all. A memory that decays leaves a floor instead, and the ordering of the census under that floor is not the ordering used here. The cap on the longest gap is untouched by any of that, being a property of the pattern rather than of the listener, which is a further argument for it.

And it cannot show whether any of this is what a listener does. The locating cost is an ideal observer with perfect recall; the syncopation count is a model with published weights; the cap is a designer’s rule. None of the three is a measurement of hearing, and the whole argument is about which computations are well posed rather than about which is true.

Whose timelines these are

The five-in-sixteen census is the space the Cuban claves live in, and the son clave is the one that travelled — out of Cuba and into most of the popular music of the twentieth century, in the way the tresillo did before it. The bossa-nova pattern is Brazilian and is a rotation of the perfectly even one. The rumba clave is the son with one onset moved, and it stayed nearer home.

One thing these cycles are not is long. Sixteen steps at a clave tempo is about two seconds, comfortably inside the psychological present — so a listener holds the whole cycle at once, and the question that breaks the long additive metres, whether the evidence for a cycle arrives before the memory of it runs out, does not arise here at all. Everything above is a question about which of 273 available designs is good, asked of a listener who can hear the whole of any of them.

The claim being made about them is a claim about design, not about history. Nobody in Matanzas computed a Pareto frontier, and nobody capped a longest gap. What the census supports is that the patterns the repertoire kept are the ones that satisfy several constraints at once — a longest silence at the floor, a rotation-total of syncopation at the ceiling, and a locating cost at the good end of their own gap multiset — and that this is a much better description of them than being efficient on any two axes.

The negative result is worth as much. A frontier drawn on two of a design’s objectives is a claim about trade-offs among the patterns that are available, and it will happily hand back a design that fails at a job it was never shown. Adding the job as a third objective does not fix that, because a frontier cannot disqualify. It has to be a constraint, and the constraint has to be stated.

Where this ladder goes next

Five 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; that frontier is a claim about a quantity that had to be chosen, so it has been drawn five times and the son clave is on four; and the third job it left out cannot be added as an axis at all, because an axis cannot forbid and both candidates for it read the gaps that were already being read.

What is owed is the other censuses. Everything above is five onsets in sixteen steps, and every one of the findings has a shape that could be an accident of those two numbers: that four is the floor on the longest gap, that thirteen patterns reach it, that nine patterns share the maximum rotation-total, that thirty-seven multisets carry 273 patterns. The seven-in-twelve census that holds the standard bell pattern, the three-in-eight that holds the tresillo, and the seven-in-sixteen that holds the Brazilian necklace are all enumerable by the same machinery, and the question is whether the named timeline of each sits at the floor of its own longest-gap distribution and at the ceiling of its own rotation-total — which would make those two quantities a description of timelines in general rather than of one census. It is arithmetic, it needs no corpus, and it is the first thing on this ladder that could turn a set of coincidences into a rule.