Euclidean rhythm — the series · Acoustics & Music Theory

Six rhythms, all from the same construction. Euclidean rhythms generated by Bjorklund's algorithm, which spreads a number of onsets as evenly as a number of steps allows. The patterns were not transcribed from recordings; they are what the algorithm produces, and the names beside them are where each one is already in use.

As evenly as possible, which turns out to be a famous rhythm

Ask an algorithm to space five strikes over eight beats as evenly as it can. It produces the cinquillo. Ask for three over eight and it produces the tresillo. Nobody told it about Cuba.

part 1 · rhythm

The rotation the necklace cannot see

Ask Bjorklund's algorithm for the world's timelines and the usual answer is that it produces them. Search every rotation of each Euclidean pattern for a match and the answer is more interesting: the tresillo is E(3,8) exactly, the bossa-nova 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. Where the match does hold it holds up to a starting position, and a starting position is the one thing a timeline is.

part 2 · rhythm

5 onsets in 16, by evenness against locatability. Every one of the 273 rotation classes of 5 onsets in 16 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 22 of the 273 are on it. son clave and the bossa-nova pattern are among them. The named timelines are marked.

What the clave buys with its unevenness

The essay before this one found that the two best-known timelines in the world are not Euclidean at any rotation, and asked whether they maximise something else. They do: how quickly a fragment of the cycle says where in the cycle it is. The son clave locates itself in nine of its sixteen steps where the even pattern needs fifteen — and of two hundred and seventy-three patterns, twenty-two are on the frontier between the two objectives and the son clave is one of them.

part 3 · rhythm

Five rulers on 5 onsets in 16, and what each frontier keeps. The same 273 rotation classes and the same locating cost, with the frontier recomputed against five different definitions of unevenness. The frontier's size varies from 7 to 22 of 273, so how exclusive membership is depends on the ruler. Of the named timelines of this size, son clave is on 4 of the five, rumba clave is on 0 of the five, the bossa-nova pattern is on 4 of the five. The four rulers that measure how large a pattern's departure from even is rank the whole census together, at no worse than 0.89 between any two of them; the one that counts how many distinct gap lengths a pattern uses rather than how large its departures are is anti-correlated with them at -0.26, and its frontier holds none of the named timelines and does not hold the perfectly even pattern.

The frontier and the ruler

A Pareto frontier is a claim about two quantities and only one of them was measured. The locating cost is exact; the unevenness it was traded against was one choice among several. Under four different rulers the son clave stays on the frontier and the frontier shrinks from twenty-two of 273 patterns to seven, so its membership means more rather than less — but the bembe stops being dominated, the two claves stop being equally uneven, and one plausible measure destroys the result entirely.

part 4 · rhythm

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.

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.

part 5 · rhythm

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.

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.

part 6 · rhythm

With a fading memory the slowest order of the gaps 1 1 2 2 2 2 2 is not the one a perfect memory finds. For each of the 3 cyclic orders of the gaps 1, 1, 2, 2, 2, 2, 2 in 12 steps, the bits of position still unknown once listening has settled, against how many steps a listener's memory of a step takes to halve, with a mismatch costing 6. 2 2 2 1 2 1 2: 12 → 4e-9, 6 → 4e-5, 4 → 0.007, 3 → 0.070, 2 → 0.540, 1.5 → 1.130, 1 → 1.786; 2 2 2 2 1 1 2: 12 → 2e-5, 6 → 0.035, 4 → 0.335, 3 → 0.795, 2 → 1.405, 1.5 → 1.694, 1 → 2.005; 2 2 1 2 2 1 2 (the standard bell pattern): 12 → 2e-5, 6 → 0.027, 4 → 0.231, 3 → 0.535, 2 → 1.028, 1.5 → 1.368, 1 → 1.819. With perfect memory the slowest to locate is the standard bell pattern; at a half-life of 6 steps the highest floor is the order 2 2 2 2 1 1 2, and at 1 it is the order 2 2 2 2 1 1 2.

The bell pattern is slowest only to a perfect memory

Among the orders of its own gaps, a named timeline is usually both the most even and the slowest to locate — for a listener who never forgets. Give the listener a memory that halves and the result comes apart. Of six timelines slowest among their orders with perfect memory, only the fume-fume stays slowest for every forgetting listener, and the standard bell pattern, which is the fume-fume with onsets and rests exchanged and settles at exactly the same floors, is second of its three orders for every memory of half its cycle or less. The census ranking survives better, and in fourteen of twenty-one censuses it was the arithmetic of a pattern that repeats.

part 7 · rhythm

Against a pulse, the bell pattern is the quickest of its orders to place. The bits of position a listener is still missing, averaged over the first cycle heard, for each cyclic order of the gaps 1 1 2 2 2 2 2 in 12 steps, heard alone, against a pulse every three steps and against a pulse every four, at perfect memory, half-life 3 steps, half-life 1.5 steps. 2 2 2 1 2 1 2: alone 1.08, 1.23, 1.73; against a pulse every 3 steps 0.69, 0.75, 0.96; against a pulse every 4 steps 0.63, 0.67, 0.85. 2 2 2 2 1 1 2: alone 1.22, 1.63, 2.02; against a pulse every 3 steps 0.60, 0.73, 0.92; against a pulse every 4 steps 0.83, 1.07, 1.36. 2 2 1 2 2 1 2 (the standard bell pattern): alone 1.25, 1.45, 1.82; against a pulse every 3 steps 0.54, 0.56, 0.67; against a pulse every 4 steps 0.55, 0.57, 0.68. Alone, the bell pattern is not the quickest order to place at any memory. Against either pulse it is the quickest at every memory.

Against a pulse the bell pattern is the easiest to place

Heard alone, the standard bell pattern is not the quickest order of its own gaps to place in its cycle, for a listener with any memory. Heard against a pulse every three steps or every four — which is how anyone hears it — it is the quickest, at every memory and at every alignment of pulse and bell, and by a wide margin: at a memory of a quarter of the cycle, 0.56 bits unplaced over the first cycle against 0.73 for either rival against a pulse in threes. Six of eight named timelines do the same. A timeline's order of gaps looks chosen for how it sits against the beat, not for how it sounds alone.

part 8 · rhythm

With both pulses sounding, the bell pattern's lead over the other orders of its gaps disappears. The bits of position a listener with a memory half-life of 3 steps is still missing over the first cycle, averaged over every way each order can sit against the pulses, for each order of the gaps 1 1 2 2 2 2 2 in 12 steps, heard alone, against a pulse every three steps, against a pulse every four and against both; the dashed line is the two pulses alone, with no bell, at 0.53. 2 2 2 1 2 1 2: alone 1.234, every 3 steps 0.738, every 4 steps 0.672, both pulses 0.437. 2 2 2 2 1 1 2: alone 1.630, every 3 steps 0.734, every 4 steps 1.037, both pulses 0.422. 2 2 1 2 2 1 2 (the standard bell pattern): alone 1.452, every 3 steps 0.560, every 4 steps 0.564, both pulses 0.425. The standard bell pattern leads its nearest rival by 0.174 bits against the pulse in threes and 0.107 against the pulse in fours; against both, the three orders are within 0.015 bits and its lead is -0.003.

Against both feels, the bell pattern's order stops mattering

Against a pulse in threes, or a pulse in fours, the standard bell pattern is placed a sixth or a tenth of a bit sooner than either other order of its own gaps. A twelve-step cycle in the ensembles that play it is heard against both feels at once, and the two pulses together are a position code of their own: they coincide once a cycle and place a listener within three steps. Heard against both, the three orders are within a seventieth of a bit of one another and the bell's lead is gone — the advantage its order was shaped for lives only where one feel is dropped.

part 9 · rhythm

A feel heard half the time gives the bell half its lead

Against one feel the standard bell pattern is the quickest order of its gaps to place; against both feels at once its lead is gone. An ensemble's texture is neither: each feel is heard some of the time. Given a listener who knows how often a pulse's strokes are lost, the lead comes back in proportion to the loss — half the strokes missing restores half of it, at every memory from a step and a half to a whole cycle. There is no threshold at either end, and a listener who treats every silence as proof comes out more certain and worse placed.

part 10 · rhythm

The bell pattern's lead is for arriving. The standard bell pattern's lead over the nearest other order of its own gaps, in bits of surprise a step, cycle by cycle, with the pulse in threes heard in full and the pulse in fours half the time. A listener who told how reliable the feel is: 0.043, 0.000, 0.000, 0.000, 0.000, 0.000. A listener who learns it from the strokes: 0.063, 0.001, 0.000, 0.000, 0.000, 0.000. A listener who takes every silence as proof: 0.144, 0.006, 0.005, 0.005, 0.004, 0.004. The lead is all in the first cycle, for every listener; after it the three orders keep a listener placed equally well. The listener who has to learn the reliability gets more from the bell than one who is told it and less than one who never learns.

The bell pattern's order is for arriving

Against a feel that comes and goes, the standard bell pattern is the quickest order of its gaps to place, and the lead grows the less a listener knows about the drummer. Followed past the first cycle, the lead is gone, whether or not the listener ever learns how often strokes are lost. And when both feels thin, a different order keeps a listener placed better than the bell does, because its two short gaps make one figure where the bell's make two that look alike.

part 11 · rhythm

A displaced stroke helps a listener keep its place only if it makes a figure the bell never makes. The standard bell pattern, 2 2 1 2 2 2 1, with both pulses heard half the time, a listener who learns how often strokes are lost: the surprise a step once placed, averaged over the second to sixth cycles, when every third cycle one stroke of the bell is moved one step into an empty step. Each bar is one of the 10 possible displacements, labelled by the varied cycle's gaps, on a logarithmic axis; the lines are the plain bell (0.0078) and the other order of the gaps, 2 2 2 1 2 1 2 (0.0016). 1 3 1 2 2 2 1 (a figure the bell never makes): 0.0055; 3 1 1 2 2 2 1 (a figure the bell never makes): 0.0056; 2 2 1 2 1 3 1 (a figure the bell never makes): 0.0062; 2 2 1 3 1 2 1 (a figure the bell never makes): 0.0066; 2 2 1 1 3 2 1 (a figure the bell never makes): 0.0077; 2 2 1 2 3 1 1 (a figure the bell never makes): 0.0080; 2 1 2 2 2 2 1 (the other order): 0.0107; 1 2 1 2 2 2 2 (the other order): 0.0107; 2 2 2 1 2 2 1 (the bell, shifted): 0.0252; 2 2 1 2 2 1 2 (the bell, shifted): 0.0253.

A varied bell can be a landmark or a false start

In a thin texture the standard bell pattern keeps a listener placed five times worse than another order of its gaps, because its two short gaps look alike. A lead drummer's variation — one stroke moved, once every few cycles — was predicted to give the bell back what it loses. The best single displacement every other cycle recovers seventy per cent of the gap; every third cycle, a third; every sixth, a seventh. Every variation makes the bell slower to arrive by. And the displacements that turn the varied cycle into the bell begun somewhere else make keeping one's place three times worse, because they sound exactly like having lost it.

part 12 · rhythm

A mark does little until it is nearly always heard, and then it does everything, whichever stroke was moved. The standard bell pattern with a varied cycle every 3, both pulses heard half the time, a listener who learns how often strokes are lost: the surprise a step once placed, averaged over the second to sixth cycles, against how reliably each stroke's mark is heard. The plain bell reads 0.0078 and the other order of the gaps 0.0016. the bell shifted: 50% 0.0252, 60% 0.0238, 70% 0.0204, 80% 0.0154, 90% 0.0092, 95% 0.0062, 100% 0.0027; a new figure: 50% 0.0055, 60% 0.0055, 70% 0.0052, 80% 0.0048, 90% 0.0041, 95% 0.0036, 100% 0.0025; nothing moved: 50% 0.0076, 60% 0.0076, 70% 0.0075, 80% 0.0070, 90% 0.0057, 95% 0.0047, 100% 0.0026.

A bell can mark a cycle without moving a stroke

A lead drummer's varied cycle helps a listener or misleads one, according to which stroke it moves, because a moved stroke can make the bell look as if it had started somewhere else. Give the varied cycle a different sound as well and the distinction goes. Heard nine times in ten and a misleading variation is no longer one; heard always, it no longer matters which stroke moved, or whether any did. A mark heard seven times in ten does almost nothing, and the arithmetic of that threshold is the same as a coin's.

part 13 · rhythm