Acoustics & Music Theory
why it sounds like that
Concept

Bell pattern — where it appears

The seven-stroke timeline of twelve pulses, grouped 2 2 1 2 2 2 1, that West African and Afro-Caribbean ensembles use as their fixed reference rhythm. It is the maximally even placement of seven onsets in twelve, and it is one of the patterns tested for how quickly a listener can find a place in it.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

The bell against every division of its cycle. A 12-step pattern of 7 onsets drawn round a circle, with 4 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 1 of the 2, 2 of the 3, 2 of the 4, 3 of the 6. No division has all of its beats struck and none has none of them, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.

The bell is not a polyrhythm

Five earlier essays have set one pattern against another. In the practice the word cross-rhythm was borrowed from, there is one pattern and it is played against the beat — against several beats at once. Counted against the four divisions of a twelve-cycle, the Ewe bell marks half of the two, two-thirds of the three, half of the four and half of the six, and all of none of them.

rhythm · Polyrhythm 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 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 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 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.

TimelineEuclidean rhythmInformationMemory decayRotationCross-rhythmCyclic-rhythmEntrainmentEntropyInferencePolyrhythmRhythmic oddity

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