Rhythm and metre

The other censuses keep evenness, not locating

Five onsets in sixteen suggested two rules: a named timeline's longest gap is the shortest its census allows, and its syncopation totalled over every rotation is the largest. Nine of ten named timelines across five censuses meet both — but so does the evenly spaced pattern, in all twenty-one censuses from two onsets in eight to eight in sixteen, and in four of those five censuses the named timeline is the evenly spaced pattern. What the rules describe is evenness. The quantity the frontier was built on points the other way: the evenly spaced pattern is the slowest of its census to locate in nineteen of twenty-one, the standard bell pattern is the slowest of all sixty-six patterns of seven in twelve, and among the orders of its own gaps a named timeline is usually both the most even and the slowest to locate. The son clave, whose most even order is also the fastest, is the exception the locating story was told about.

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

Assumes: The longest silence is not a third axis · What the clave buys with its unevenness

The longest silence is not a third axis worked through every cyclic pattern of five onsets in sixteen steps and found two facts about the ones the world plays. Their longest gap is four steps, the shortest longest-gap any of the 273 patterns can have. And their syncopation, counted against the metre from every one of the sixteen places the bar line could fall and totalled, is 79 — the largest total in the census, shared by only nine patterns. The son clave, the rumba clave and the bossa-nova pattern were at both.

That essay said plainly what it could not tell: whether those were facts about timelines or about the numbers five and sixteen. Four being the floor, thirteen patterns reaching it, nine sharing the ceiling — any of those could be an accident of one census. The test is to run the same arithmetic on the censuses that hold the other named timelines and see whether each sits at its own floor and its own ceiling.

5 censuses: the floor of the longest gap, the ceiling of the syncopation total, and how few patterns reach both. For each census of k onsets in n steps, counted by rotation class: 3 in 8: patterns 7, floor of longest gap 3, at the floor 1, ceiling of total 14, at the ceiling 2, at both 1; 5 in 8: patterns 7, floor of longest gap 2, at the floor 2, ceiling of total 12, at the ceiling 2, at both 2; 5 in 12: patterns 66, floor of longest gap 3, at the floor 6, ceiling of total 28, at the ceiling 6, at both 6; 7 in 12: patterns 66, floor of longest gap 2, at the floor 3, ceiling of total 20, at the ceiling 38, at both 3; 5 in 16: patterns 273, floor of longest gap 4, at the floor 13, ceiling of total 79, at the ceiling 9, at both 9. The rotation-total of syncopation is taken against a metre of 2 × 2 × 2 for 8 steps, 2 × 2 × 3 for 12 steps, 2 × 2 × 2 × 2 for 16 steps. In every census drawn the evenly spaced pattern is at the floor and at the ceiling. — Fig. 1 Five censuses, each counted by rotation class: how many patterns there are, the floor of the longest gap and how many reach it, the ceiling of the syncopation total and how many reach it, and how many reach both. Both rules together admit one pattern of seven in three in eight and nine of 273 in five in sixteen.

Nine of ten timelines meet both rules

The named timelines the test can be run on are ten. The tresillo is three onsets in eight and the cinquillo five in eight; the fume-fume is five in twelve and the standard bell pattern seven; and six are five in sixteen — the son clave, the rumba clave, the bossa-nova pattern, and the shiko, the soukous and the gahu, which are the other three of the six timelines Godfried Toussaint set side by side in his comparison of rhythmic similarity measures.

10 timelines against the rules of their own censuses: 9 meet both. Each named timeline against the census of its own size: son clave (5 in 16, gaps 3 3 4 2 4): longest gap 4 against a floor of 4, rotation-total 79 against a ceiling of 79, not the evenly spaced pattern, 6 cyclic orders of its gaps, of which it is the fastest to locate and the most evenly placed; rumba clave (5 in 16, gaps 3 4 3 2 4): longest gap 4 against a floor of 4, rotation-total 79 against a ceiling of 79, not the evenly spaced pattern, 6 cyclic orders of its gaps, of which it is the slowest to locate and not the most evenly placed; the bossa-nova pattern (5 in 16, gaps 3 3 4 3 3): longest gap 4 against a floor of 4, rotation-total 79 against a ceiling of 79, the evenly spaced pattern itself, 1 cyclic order of its gaps, of which it is the only one; the standard bell pattern (7 in 12, gaps 2 2 1 2 2 2 1): longest gap 2 against a floor of 2, rotation-total 20 against a ceiling of 20, the evenly spaced pattern itself, 3 cyclic orders of its gaps, of which it is the slowest to locate and the most evenly placed; tresillo (3 in 8, gaps 3 3 2): longest gap 3 against a floor of 3, rotation-total 14 against a ceiling of 14, the evenly spaced pattern itself, 1 cyclic order of its gaps, of which it is the only one; cinquillo (5 in 8, gaps 2 1 2 1 2): longest gap 2 against a floor of 2, rotation-total 12 against a ceiling of 12, the evenly spaced pattern itself, 2 cyclic orders of its gaps, of which it is the slowest to locate and the most evenly placed; fume-fume (5 in 12, gaps 2 2 3 2 3): longest gap 3 against a floor of 3, rotation-total 28 against a ceiling of 28, the evenly spaced pattern itself, 2 cyclic orders of its gaps, of which it is the slowest to locate and the most evenly placed; shiko (5 in 16, gaps 4 2 4 2 4): longest gap 4 against a floor of 4, rotation-total 79 against a ceiling of 79, not the evenly spaced pattern, 2 cyclic orders of its gaps, of which it is the slowest to locate and the most evenly placed; soukous (5 in 16, gaps 3 3 4 1 5): longest gap 5 against a floor of 4, rotation-total 69 against a ceiling of 79, not the evenly spaced pattern, 12 cyclic orders of its gaps, of which it is the slowest to locate and the most evenly placed; gahu (5 in 16, gaps 3 3 4 4 2): longest gap 4 against a floor of 4, rotation-total 79 against a ceiling of 79, not the evenly spaced pattern, 6 cyclic orders of its gaps, of which it is the fastest to locate and not the most evenly placed. — Fig. 2 Ten named timelines, each against the census of its own size: its gaps, its longest gap against that census’s floor, its syncopation total against that census’s ceiling, how many cyclic orders its gaps can be put in, and where it stands among them on locating and on evenness. Nine are at both the floor and the ceiling; the soukous is at neither.

Nine of the ten are at both. The tresillo’s longest gap is three, the floor for three in eight, and its total is fourteen, the ceiling. The cinquillo is at two and twelve, the fume-fume at three and twenty-eight, the standard bell pattern at two and twenty. In five in sixteen the son clave, the rumba clave, the bossa-nova pattern, the shiko and the gahu all have a longest gap of four and a total of 79.

The one that fails is the soukous. Its gaps are 3, 3, 4, 1 and 5. The five is past the floor, and the one-step gap puts two onsets side by side with no rest between them to syncopate against, so its total is 69 against the ceiling of 79. It fails both rules for one reason, which is that it is the least even of the ten: a gap of one and a gap of five in the same short cycle.

Nine of ten looks like a rule confirmed. The next table says what it confirms.

The evenly spaced pattern meets both rules everywhere

The evenly spaced pattern is the slowest to locate in 19 of 21 censuses. For each census of k onsets in n steps, counted by rotation class: 2 in 8: patterns 4, floor of longest gap 4, ceiling of total 12, at both 1, even pattern at both yes, even pattern's locating slowest; 3 in 8: patterns 7, floor of longest gap 3, ceiling of total 14, at both 1, even pattern at both yes, even pattern's locating slowest; 4 in 8: patterns 10, floor of longest gap 2, ceiling of total 16, at both 1, even pattern at both yes, even pattern's locating slowest; 5 in 8: patterns 7, floor of longest gap 2, ceiling of total 12, at both 2, even pattern at both yes, even pattern's locating slowest; 6 in 8: patterns 4, floor of longest gap 2, ceiling of total 8, at both 3, even pattern at both yes, even pattern's locating slowest; 2 in 12: patterns 6, floor of longest gap 6, ceiling of total 22, at both 1, even pattern at both yes, even pattern's locating slowest; 3 in 12: patterns 19, floor of longest gap 4, ceiling of total 27, at both 1, even pattern at both yes, even pattern's locating slowest; 4 in 12: patterns 43, floor of longest gap 3, ceiling of total 32, at both 1, even pattern at both yes, even pattern's locating slowest; 5 in 12: patterns 66, floor of longest gap 3, ceiling of total 28, at both 6, even pattern at both yes, even pattern's locating slowest; 6 in 12: patterns 80, floor of longest gap 2, ceiling of total 24, at both 1, even pattern at both yes, even pattern's locating slowest; 7 in 12: patterns 66, floor of longest gap 2, ceiling of total 20, at both 3, even pattern at both yes, even pattern's locating slowest; 8 in 12: patterns 43, floor of longest gap 2, ceiling of total 16, at both 10, even pattern at both yes, even pattern's locating slowest; 9 in 12: patterns 19, floor of longest gap 2, ceiling of total 12, at both 10, even pattern at both yes, even pattern's locating slowest; 10 in 12: patterns 6, floor of longest gap 2, ceiling of total 8, at both 5, even pattern at both yes, even pattern's locating slowest; 2 in 16: patterns 8, floor of longest gap 8, ceiling of total 46, at both 1, even pattern at both yes, even pattern's locating slowest; 3 in 16: patterns 35, floor of longest gap 6, ceiling of total 61, at both 2, even pattern at both yes, even pattern's locating 2nd slowest; 4 in 16: patterns 116, floor of longest gap 4, ceiling of total 76, at both 1, even pattern at both yes, even pattern's locating slowest; 5 in 16: patterns 273, floor of longest gap 4, ceiling of total 79, at both 9, even pattern at both yes, even pattern's locating slowest; 6 in 16: patterns 504, floor of longest gap 3, ceiling of total 82, at both 3, even pattern at both yes, even pattern's locating slowest; 7 in 16: patterns 715, floor of longest gap 3, ceiling of total 85, at both 3, even pattern at both yes, even pattern's locating 2nd slowest; 8 in 16: patterns 810, floor of longest gap 2, ceiling of total 88, at both 1, even pattern at both yes, even pattern's locating slowest. The rotation-total of syncopation is taken against a metre of 2 × 2 × 2 for 8 steps, 2 × 2 × 3 for 12 steps, 2 × 2 × 2 × 2 for 16 steps. In every census drawn the evenly spaced pattern is at the floor and at the ceiling. — Fig. 3 Every census from two onsets in eight to eight in sixteen. In all twenty-one the evenly spaced pattern is at the floor of the longest gap and at the ceiling of the syncopation total, and in nineteen it is the slowest pattern of its census to locate; in three in sixteen and seven in sixteen it is the second slowest.

Run over every census from two onsets in eight to eight in sixteen, the evenly spaced pattern is at the floor and at the ceiling in all twenty-one. The first half of that is not a finding at all. As evenly as possible is the account of Bjorklund’s algorithm, whose pattern has gaps of only two sizes, the cycle over the onsets rounded down and rounded up; its longest gap is therefore the cycle over the onsets rounded up, and no pattern can do better, because gaps that sum to the cycle cannot all be shorter than their average. The second half is checked census by census rather than proved. The total depends only on which gap lengths a pattern uses — no set of gaps carries two different totals in any of the twenty-one censuses — and in every one of them the even pattern’s two lengths turn out to be among the combinations that score highest, including the censuses where one of those lengths is a single step.

So a named timeline at both rules has shown one thing: that its gaps are as even, in the two respects the rules read, as the Euclidean pattern’s. And four of the five censuses with a named timeline in them do not even test that. The tresillo, the cinquillo, the fume-fume and the standard bell pattern are each the evenly spaced pattern of their own census, up to rotation, so they meet the floor by construction and the ceiling as the even pattern does in every census checked. The rules are tested only in five in sixteen, where four of the six named timelines are not the even pattern. There the shiko, the son clave, the rumba clave and the gahu pass and the soukous fails.

The rules also differ in how much they exclude. In seven in twelve the floor admits three of sixty-six patterns and the ceiling thirty-eight; in five in sixteen the floor admits thirteen of 273 and the ceiling nine, every one of which is also at the floor. Seven in sixteen, which holds no named timeline here, admits three patterns of 715 at both. Together they are a strong filter where the onsets are sparse. Where they are dense it barely filters at all: in eight in twelve ten of forty-three patterns pass both, in nine in twelve ten of nineteen, and in ten in twelve five of six. And the two rules are not equally about evenness in every census. In seven in twelve the ceiling is reached by three sets of gaps — five twos and two ones, which is the bell pattern’s, but also three ones, three twos and a three, and four ones, a two and two threes — so a pattern with four single-step gaps in a row of seven can sit at the ceiling there. In that census the floor does all the excluding.

So the filter is a description of evenness, and every census’s evenly spaced pattern passes it.

The evenly spaced pattern is the slowest to locate

The same table has a second column, and it is the one that changes the picture. What the clave buys with its unevenness measured a second job a timeline does — how quickly a listener who has heard part of the cycle knows where in it they are — and found the son clave trading evenness for it. The locating cost is exact: list the cycle’s rotations, and after each amount of hearing count how many rotations still agree with what was heard.

In nineteen of the twenty-one censuses the evenly spaced pattern is the slowest of all its patterns to locate, and in the other two it is the second slowest. That is not surprising once it is said. A perfectly even pattern looks nearly the same from many of its starting points — the tresillo’s three gaps are 3, 3 and 2, so two of its three onsets are followed by an identical gap — and a fragment that looks the same from several positions cannot say which one it came from.

7 onsets in 12, by evenness against locatability. Every one of the 66 rotation classes of 7 onsets in 12 steps, placed by how uneven its gaps are against how much of the cycle has to be heard on average before its position is known. The two objectives are opposed — the perfectly even pattern is the hardest to locate and the most clustered ones are the easiest — so there is no best pattern, only a frontier, and 11 of the 66 are on it. None of the named timelines is among them. The named timelines are marked. — Fig. 4 All sixty-six patterns of seven onsets in twelve steps, by how uneven their gaps are against how long they take to locate, with the frontier between the two drawn. Eleven patterns are on it and the standard bell pattern is not: it is at the even end and the slow end together.

What it means for the timelines is the part that matters. Seven in twelve holds the standard bell pattern, the timeline of a great deal of West African and diaspora music, and the bell pattern is its census’s evenly spaced pattern. So it is the slowest of all sixty-six patterns of seven in twelve to locate, and it is not on the frontier between evenness and locating: eleven patterns are, and the bell pattern is not, because another order of its own gaps is exactly as even by the spread of the gaps and locates faster. The same is true of the fume-fume, slowest of sixty-six, of the cinquillo and the tresillo, slowest of seven each, and of the bossa-nova pattern, slowest of 273.

Whether the bell pattern is on the frontier at all depends on the ruler, which is the point the frontier and the ruler made about the son clave. The frontier above measures unevenness by the spread of the gaps, and the spread cannot tell the bell pattern from the other orders of its own gaps; one of them locates faster, so the bell pattern is dominated. Measure unevenness instead by how far the onsets sit from perfectly even places, or by the chord-length deficit, and the bell pattern is more even than those orders, is dominated by nothing, and is back on the frontier — as are the fume-fume and the cinquillo under both of those rulers. The shiko is on none of the five frontiers. So the even timelines are not inefficient designs; they are the even end of the trade, and whether a frontier shows it depends on whether its ruler can see placement.

How much of a cycle has to be heard before its position is known. A listener who has heard w consecutive steps of a repeating pattern knows the pattern and not the phase, and the number of rotations still consistent with what has been heard is the uncertainty left. Each curve falls from log₂n bits at nothing heard to zero at the length that first tells every rotation apart. The son clave reaches zero at 9 of its sixteen steps and the bossa-nova pattern, which is a rotation of the Euclidean one, at 15. The dashed lines are the Euclidean patterns with the same onset counts, and the even pattern is the slowest to locate in every case. — Fig. 5 How much uncertainty about position is left after each number of steps heard, for four named timelines, with the Euclidean pattern of the same size dashed. The tresillo, the standard bell pattern and the bossa-nova pattern lie on their dashed lines because they are those patterns. The son clave reaches certainty after nine of its sixteen steps; the bossa-nova pattern needs fifteen.

Five of the ten named timelines are the evenly spaced pattern of their census, and all five are the slowest to locate there. The trade the clave was said to make is not a trade the timelines as a group make. Half of them sit at the extreme the clave was said to have left.

Among the orders of its own gaps

The rules read which gap lengths a pattern uses and the locating cost reads the order they come in, so the sharpest comparison holds the gaps fixed and varies only the order. Every named timeline whose gaps can be arranged in more than one cyclic order can be set against the other arrangements of exactly the same durations.

The 3 cyclic orders of the gaps 1 1 2 2 2 2 2, and where the standard bell pattern sits. Every rotation class of 7 onsets in 12 steps whose gaps are 1, 1, 2, 2, 2, 2, 2, in a different cyclic order. The longest gap, the spread of the gaps and the syncopation total over every rotation are identical for all of them. .x.x.x.xx.xx: locating cost 1.070, located after 8 steps, 0.352 steps from even on average; .x.x.x.x.xxx: locating cost 1.181, located after 10 steps, 0.455 steps from even on average; .x.x.xx.x.xx (the standard bell pattern): locating cost 1.237, located after 11 steps, 0.286 steps from even on average. The standard bell pattern is the most evenly placed order and among the slowest to locate. — Fig. 6 The three cyclic orders of five gaps of two steps and two of one, in twelve. They share their longest gap, their spread and their syncopation total. The standard bell pattern is the order whose onsets sit nearest to perfectly even, at 0.286 steps on average, and the slowest of the three to locate, needing eleven steps where another order needs eight.

The standard bell pattern’s gaps are five twos and two ones. There are three ways to arrange them round the cycle, and they are identical in every quantity that reads only the gaps. They are not identical in how evenly their onsets sit — the bell pattern keeps its two short gaps as far apart as they can go, and its onsets are 0.286 steps from perfect evenness on average against 0.352 and 0.455 for the others — and they are not identical in locating. The bell pattern is the most even order of its gaps and the slowest to locate, eleven steps of twelve against eight for the order that puts its two short gaps closer together.

Eight of the ten named timelines have gaps that can be put in more than one order; the tresillo and the bossa-nova pattern have one order each. Of the eight, five are both the most evenly placed order of their own gaps and the slowest to locate: the bell pattern, the cinquillo, the fume-fume, the shiko and the soukous. The cinquillo needs seven steps of eight where its other order needs six, the fume-fume eleven of twelve against eight, and the shiko fourteen of sixteen against twelve.

The 6 cyclic orders of the gaps 2 3 3 4 4, and where son clave, gahu and rumba clave sit. Every rotation class of 5 onsets in 16 steps whose gaps are 2, 3, 3, 4, 4, in a different cyclic order. The longest gap, the spread of the gaps and the syncopation total over every rotation are identical for all of them. ...x..x..x...x.x (son clave): locating cost 0.925, located after 9 steps, 0.403 steps from even on average; ...x...x.x..x..x (gahu): locating cost 0.925, located after 9 steps, 0.568 steps from even on average; ...x...x..x..x.x: locating cost 0.925, located after 9 steps, 0.568 steps from even on average; ...x...x..x.x..x: locating cost 0.940, located after 10 steps, 0.632 steps from even on average; ...x..x...x..x.x (rumba clave): locating cost 0.972, located after 11 steps, 0.490 steps from even on average; ...x..x...x.x..x: locating cost 0.972, located after 11 steps, 0.492 steps from even on average. Son clave is the most evenly placed order and among the fastest to locate; gahu is not the most evenly placed order and among the fastest to locate; rumba clave is not the most evenly placed order and among the slowest to locate. — Fig. 7 The six cyclic orders of gaps 2, 3, 3, 4 and 4 in sixteen. The son clave is the most evenly placed of the six and tied for the fastest to locate, at nine steps. The gahu is tied with it on locating and less even. The rumba clave is tied for the slowest, at eleven.

Three are not, and all three share one set of gaps. The son clave, the rumba clave and the gahu are three of the six orders of 2, 3, 3, 4 and 4. The son clave is the most evenly placed of the six, at 0.403 steps from even, and tied for the fastest to locate, at nine steps. The gahu is tied with it on locating and is less even, at 0.568. The rumba clave is tied for the slowest to locate, at eleven steps, and is neither the most nor the least even.

The exception the locating story was told about

So the son clave is the only named timeline whose most even order is also its fastest to locate. It is the one pattern in which the two jobs pull the same way, and it is the pattern the frontier essays were about. Told from the son clave, evenness and locatability looked like a trade the repertoire had settled well. Told from the whole table, what the repertoire keeps with remarkable consistency is evenness of placement — at the census level, where five of ten timelines are the even pattern, and within each set of gaps, where six of eight are the most even order — and locating is what that usually costs.

That leaves a question the arithmetic cannot answer by itself, which is how a listener finds their place in a timeline that is built to be hard to locate. A cycle that says where it is gave one answer for a gong cycle: the position is coded in the instrumentation rather than in the pattern, since different instruments mark different points. A bell pattern in an ensemble is seldom heard alone. The drums, the dancers and the song all carry the position, and the bell can be as even and as hard to locate as it likes, because it is not the part responsible for saying where the cycle is.

The arithmetic

A census is every placement of k onsets in n steps, grouped by rotation into necklaces. The floor is the shortest longest-gap in the census and the ceiling is the largest total, over all n rotations, of Longuet-Higgins and Lee’s syncopation count — a number about the metre rather than about the pattern, which is why only its total is a property of a necklace — taken against a metre of four levels of two for sixteen steps, three levels of two for eight, and 2 × 2 × 3 for twelve. The locating cost is the average, over every amount of hearing from nothing to the whole cycle, of the bits of position still unknown, which is the measure every frontier on this subject has used. Distance from even is the root mean square distance of the onsets from a perfectly even set of the same size, at whichever starting offset makes it smallest. The censuses run from two onsets to n − 2 in eight and twelve steps, and from two to eight in sixteen.

What ten timelines cannot show

The named timelines are ten, and ten is a handful. Nine of ten at both rules and five of eight at the most even and slowest order are counts of examples rather than rates, and a corpus of a few hundred timelines scored the same way is the only thing that would turn either into a statement about what traditions select.

The transcriptions are one author’s. The shiko, soukous, gahu, cinquillo and fume-fume are as Toussaint writes them, and every one of these patterns is played with variants. A timeline moved by one step is, as the two claves show, a different order of the same gaps with a different locating cost.

The metre for twelve is an assumption. A cycle of twelve can be heard as four groups of three or as three groups of four, and the syncopation totals — though not the gaps, the floor or the locating cost — depend on which. The bell pattern is at the ceiling under four groups of three.

A reflection is a different order. The soukous ties on locating with its own mirror image, which is the same pattern played backwards; the census counts the two separately, as a listener would, since a timeline has a direction.

And locating assumes perfect recall. A rotation ruled out stays ruled out, which is exactly what the cycle that outruns the memory showed a forgetting listener does not do.

Whose timelines

The tresillo, the cinquillo, the son clave and the rumba clave are Cuban, the bossa-nova pattern Brazilian, the fume-fume Ghanaian, the gahu an Ewe dance-drumming pattern, the soukous from Congolese dance music, and the standard bell pattern the timeline of a large part of West African music and its diaspora. The rotation the necklace cannot see found that the Euclidean algorithm reproduces several of them only up to a starting point, and that is the other half of the same story: the traditions keep the even necklace and fix where it starts, which is exactly the information a perfectly even pattern cannot supply by itself.

Still open: whether a forgetting listener still finds the even order slowest

Every locating number above is an ideal observer who remembers everything heard. With a memory that decays, the uncertainty never reaches zero and settles at a floor, and the order of patterns by that floor need not be the order by the ideal cost. The decaying measure exists and is arithmetic on the same censuses. What it would settle is whether the five timelines that are the most even and slowest order of their gaps stay the slowest for a listener whose memory spans only part of the cycle — or whether, once recall is limited, the even order becomes the easiest to hold, and evenness and locating stop pulling apart at all.