Acoustics & Music Theory
why it sounds like that
Concept

Timeline — where it appears

A short repeating rhythm played unchanged throughout a piece, against which everything else is heard. What a timeline has to do besides repeat is say where in itself it is, and that requirement pulls against the evenness the Euclidean patterns maximise.

Named by 18 essays across one field — each of them below, with the objects they name alongside it.

Euclidean up to the one thing a timeline is for. Five named timelines, each drawn above the Euclidean pattern with the same number of onsets in the same number of steps, with the rotation between them found by search. 3 of 5 are rotations of the Euclidean pattern and 2 are not Euclidean at any rotation — son clave, 3–2 and rumba clave, 3–2, whose gap sequences are 3·3·4·2·4 and 3·4·3·2·4 against the algorithm's 3·3·3·3·4. Where the match holds it holds only up to rotation, and a rotation is not a small difference: the algorithm has no way to produce a starting position, and a starting position is what a timeline is.

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.

rhythm · Euclidean 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.

rhythm · Euclidean 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.

rhythm · Euclidean 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.

rhythm · Euclidean 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.

rhythm · Euclidean 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.

rhythm · Euclidean 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.

rhythm · Euclidean rhythm A timed expectation would erase a slow cycle's cost, and a listener cannot time a slow cycle that well. Bits of position a listener with a 3.5-second memory is still missing over the first cycle of son clave, against how long the cycle takes, for a newcomer with no expectation, a listener timing the cycle with the Weber fraction a duration that long is judged with, and a listener timing it to ten per cent. a newcomer, no expectation: 2 s 0.94, 8 s 0.97, 24 s 1.41, 40 s 2.02, 60 s 2.47; timing as well as listeners do: 2 s 0.68 (w 0.150), 8 s 0.69 (w 0.150), 24 s 0.85 (w 0.150), 40 s 1.90 (w 0.375), 60 s 2.38 (w 0.375); timing the cycle to ten per cent: 2 s 0.47, 8 s 0.48, 24 s 0.55, 40 s 0.73, 60 s 1.02. At ten per cent even a sixty-second cycle is placed about as well as a newcomer places a two-second one. At the precision a listener actually has for durations of half a minute or more, the expectation is worth a tenth of a bit.

An expectation cannot rescue a cycle too slow to time

A listener who knows a piece arrives with an expectation of where in the cycle they are, and the size of that expectation was the number the last essay said nobody had. It can be given one: a listener who has been timing the cycle carries a spread of their Weber fraction times the cycle, which is the same number of steps at any tempo. Timed to ten per cent, a forty-second cycle would be placed better than a newcomer places a two-second one. But forty seconds is judged in the band where the Weber fraction is nearer forty per cent, and there the expectation is worth a tenth of a bit.

rhythm · Cyclic-rhythm Timed from every layer, a slow colotomic cycle is placed almost at once. Bits of position a listener with a 3.5-second memory is still missing over the first cycle, against how long the cycle takes, for a 16-step colotomy with layers every 16, 8, 4 steps: a newcomer, a listener timing from the last gong, a listener timing from the last stroke of every layer, and a listener timing from the gong to ten per cent; with son clave, timed from its own cycle, for comparison. a colotomy, newcomer: 1.6 s 0.71, 2 s 0.71, 3 s 0.72, 4 s 0.73, 6 s 0.75, 8 s 0.78, 12 s 0.85, 16 s 0.93, 24 s 1.14, 32 s 1.37, 40 s 1.60, 48 s 1.83, 60 s 2.14; a colotomy, timing a whole cycle: 1.6 s 0.27, 2 s 0.45, 3 s 0.45, 4 s 0.45, 6 s 0.46, 8 s 0.46, 12 s 0.47, 16 s 0.48, 24 s 0.54, 32 s 1.21, 40 s 1.45, 48 s 1.70, 60 s 2.02; a colotomy, timing from every layer: 1.6 s 0.00, 2 s 0.00, 3 s 0.00, 4 s 0.00, 6 s 0.00, 8 s 0.00, 12 s 0.01, 16 s 0.01, 24 s 0.01, 32 s 0.03, 40 s 0.11, 48 s 0.17, 60 s 0.20; a colotomy, from the gong to ten per cent: 1.6 s 0.34, 2 s 0.34, 3 s 0.34, 4 s 0.34, 6 s 0.34, 8 s 0.35, 12 s 0.35, 16 s 0.35, 24 s 0.37, 32 s 0.42, 40 s 0.49, 48 s 0.60, 60 s 0.80; son clave, timing the cycle: 1.6 s 0.32, 2 s 0.68, 3 s 0.68, 4 s 0.68, 6 s 0.68, 8 s 0.69, 12 s 0.70, 16 s 0.73, 24 s 0.85, 32 s 1.63, 40 s 1.90, 48 s 2.12, 60 s 2.38. At 40 seconds the clock leaves 0.11 bits, against 1.45 timing from the gong and 0.49 for a ten-per-cent timer.

The strokes between the gongs are a clock

An experienced listener arriving in a forty-second gong cycle was expected to be no better placed than a newcomer, because a whole cycle is timed in the band where a Weber fraction is near forty per cent. But nobody times a gong cycle as one interval. A listener who resets at the last gong, the last kempul and the last kenong times three short intervals instead of one long one, and arrives knowing their position to a tenth of a bit — better than a listener timing the whole cycle to ten per cent. A single timeline, whose strokes are all one instrument, offers nothing to reset on.

rhythm · Cyclic-rhythm At every number of strokes, the clock that halves at each level places a slow listener best. Bits of position unknown over the first cycle of a sixteen-step cycle at 40 seconds, for a listener timing from the last stroke of every layer, for every nesting of layers whose periods divide sixteen, grouped by how many strokes a cycle the layers make together. 1 stroke: 16 1.61; 2 strokes: 16 : 8 0.44; 4 strokes: 16 : 8 : 4 0.11, 16 : 4 0.39; 8 strokes: 16 : 8 : 4 : 2 0.10, 16 : 8 : 2 0.16, 16 : 4 : 2 0.37, 16 : 2 0.90. In every group the best nesting is the one in which each layer is twice as fast as the one above it.

The best gong cycle halves at every level

A colotomic cycle is a clock whose layers are its hands, and a clock can be built many ways from the same number of strokes. Searched over every nesting of a sixteen-step cycle, the one that places a slow listener best is, at every budget of strokes, the one in which each layer is exactly twice as fast as the layer above it. A gong with one other layer does best at a ratio of four, and three halving layers with the same four strokes beat it by a factor of three and a half. A layer finer than the one whose interval is already timed to half a step adds almost nothing.

rhythm · Cyclic-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.

rhythm · Euclidean rhythm The three orders of the bell's gaps with the pulse in fours heard 50 per cent of the time. Bits of position unknown over the first cycle, averaged over alignments, for each order of the gaps 1 1 2 2 2 2 2 against a pulse in threes heard in full and a pulse in fours of which 50 per cent of strokes are heard, by a listener who knows the share; memory half-life 3 steps. 2 2 2 1 2 1 2: 0.613; 2 2 2 2 1 1 2: 0.593; 2 2 1 2 2 1 2 (the standard bell pattern): 0.513. The bell pattern leads by 0.080 bits. The expectation over which strokes are lost is exact; drawn instead from 150 random patterns of loss it comes to 0.509 against 0.510 at one alignment.

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.

rhythm · Euclidean rhythm An elaboration's most useful boundary is one the skeleton does not have. Bits of position unknown over the first cycle of a sixteen-step cycle with gong, kempul and kenong, timing from every layer, when an elaborating part adds a figure a whole cycle long whose one recognisable boundary falls at the step drawn, at 40, 80, 160 seconds a cycle. Dashed: the skeleton alone. Dotted: the skeleton with a kethuk every two steps. 40 s: skeleton 0.112, with a kethuk 0.104; boundary at 2 0.029, 4 0.004, 6 0.004, 8 0.006, 10 0.005, 12 0.004, 14 0.028. 80 s: skeleton 0.329, with a kethuk 0.301; boundary at 2 0.125, 4 0.094, 6 0.067, 8 0.087, 10 0.057, 12 0.072, 14 0.089. 160 s: skeleton 0.646, with a kethuk 0.457; boundary at 2 0.360, 4 0.340, 6 0.256, 8 0.323, 10 0.251, 12 0.295, 14 0.292. A boundary on the gong is the gong's own timer and adds nothing. Anywhere in the middle of the cycle it re-times the digit the gong reads worst, and does more than a whole layer of kethuk strokes.

An elaboration is a clock hand where the skeleton has none

A colotomic cycle is a clock whose hands are its punctuating layers, and the finest hand a slow cycle can use runs out as the cycle slows. The elaborating parts that fill the cycle were predicted to supply that finer hand. They can, and a two-step figure recognised at every boundary is a kethuk exactly. But the figure that does most is not a fine one. It is a long one with a single recognisable point in the middle of the cycle, where no skeleton stroke falls: at eighty seconds a cycle it leaves a listener a fifth as uncertain as the skeleton alone, where a kethuk takes off a tenth — and it still does more than a kethuk when a listener catches that point one time in ten.

rhythm · Cyclic-rhythm A second recognisable point is worth a great deal if it can be told from the first. Bits of position unknown over the first cycle of a sixteen-step cycle with gong, kempul and kenong, when the elaborating parts add none, one, two or three recognisable points, each set at its best placement, read by a listener who can tell the points apart and by one who cannot, at 80 and 160 seconds a cycle. 80 s: none 0.329; 1 told apart 0.057 at 10, alike 0.057 at 10; 2 told apart 0.001 at 6/13, alike 0.005 at 6/12; 3 told apart 0.000 at 6/10/14, alike 0.007 at 4/9/14. 160 s: none 0.646; 1 told apart 0.242 at 11, alike 0.242 at 11; 2 told apart 0.094 at 6/13, alike 0.180 at 7/13; 3 told apart 0.038 at 5/11/15, alike 0.117 at 7/10/13. Points told apart add like checkpoints, each taking away most of what is left. Points alike add a little, and at eighty seconds a third alike point leaves the listener worse placed than two.

Two landmarks in a cycle help only if they can be told apart

One recognisable point inside a slow gong cycle, placed where the skeleton has no stroke, did more for a listener than a whole layer of kethuk strokes. A second point does as much again, a third nearly as much again — provided the listener can tell which point they have just heard. If the points sound alike, a second one is worth a fraction of that, seventy of the hundred and five ways of placing it leave the listener worse off than one point would, and two alike points half a cycle apart undo most of what the first one gave. Whether the listener keeps one timer or several turns out not to matter at all.

rhythm · Cyclic-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.

rhythm · Euclidean 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.

rhythm · Euclidean rhythm A newcomer to a slow cycle needs about four hearings to tell two marks apart, and the best pair is not the quickest to learn. A sixteen-step cycle of 160 seconds with two recognisable points that sound alike, and a listener who learns that they are two events by hearing them at different distances from the gong: the bits of position left unknown in each of the first ten cycles, for four pairs of points, with each pair's clock once fully learned dashed. steps 6 and 13 (d′ 1.30 a hearing, nine listeners in ten naming them after 3.7 cycles): 0.219, 0.147, 0.117, 0.103, 0.098, 0.095, 0.094, 0.094, 0.094, 0.094; learned 0.094; steps 7 and 13 (d′ 1.08 a hearing, nine listeners in ten naming them after 4.9 cycles): 0.180, 0.146, 0.126, 0.116, 0.110, 0.106, 0.105, 0.104, 0.103, 0.103; learned 0.102; steps 4 and 12 (d′ 1.69 a hearing, nine listeners in ten naming them after 2.6 cycles): 0.445, 0.252, 0.206, 0.195, 0.192, 0.191, 0.191, 0.191, 0.191, 0.191; learned 0.191; steps 1 and 15 (d′ 2.49 a hearing, nine listeners in ten naming them after 1.7 cycles): 0.301, 0.255, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253; learned 0.253.

A newcomer learns a cycle's landmarks in four hearings

Two recognisable points in a slow gong cycle help a listener only if they can be told apart, and a newcomer at first cannot: two marks that differ only by where they fall in the cycle sound alike until the listener has heard enough cycles to notice. Timed from the gong, the best pair of points is learned by nine listeners in ten after about four cycles — the same four at any slow tempo, so ten minutes at 160 seconds a cycle. The pairs learned fastest sit right after the gong and are poor clocks once learned; for the first four cycles a newcomer is best served not by the best clock but by a pair whose confusion costs less while it lasts.

rhythm · Cyclic-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.

rhythm · Euclidean rhythm

Named alongside it

The objects these essays reach for when they reach for this one.

Memory decayEuclidean rhythmInformationRotationBell patternCycleMaximal evennessWeber fractionClaveDurationEntropySyncopation

All concepts