Acoustics & Music Theory
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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.
15 min read 5 figures Time is a circleWhat the listener suppliesThe design was forced

Assumes: The bell pattern is slowest only to a perfect memory · A cycle that says where it is

The bell pattern is slowest only to a perfect memory took the standard West African twelve-step bell pattern — seven strokes with gaps of two, two, one, two, two, two and one — and put its gaps in every other cyclic order they allow. There are three. With a perfect memory, the bell pattern is the slowest of the three for a listener to place, meaning to know where in the cycle they are; with a memory that fades, another order becomes slower and the bell pattern is no longer the extreme.

That essay heard the bell pattern alone, and its closing section named the problem with that. Nobody hears a timeline alone. In an ensemble the bell sounds against a pulse the drums and the dancers’ feet mark, and a cycle that says where it is had already found that regular layers change what a listener can learn about position. The question left was whether the bell pattern’s standing among the orders of its own gaps survives the pulse, and whether a pulse in threes and a pulse in fours rank the orders the same way.

It does not survive. It reverses.

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.
Fig. 1 The bits of position still unknown, averaged over the first cycle, for the three orders of the gaps 1 1 2 2 2 2 2, heard alone, against a pulse every three steps and against a pulse every four, at a perfect memory and at half-lives of three and one and a half steps. Alone the bell pattern is never the quickest to place. Against either pulse, at every memory, it is: 0.54, 0.56 and 0.67 bits against three, and 0.55, 0.57 and 0.68 against four.

What placing a timeline costs

A listener who arrives in the middle of a twelve-step cycle has twelve possible positions, 3.58 bits of uncertainty. Each step heard either contains a stroke or does not, and every position inconsistent with what has been heard is ruled out. The ideal observer of the earlier essays rules a position out for good; a listener with a fading memory weights each heard step by how long ago it was, halving every stated number of steps, so a position contradicted long ago can drift back into contention. The cost used here is the uncertainty averaged over the first cycle a listener hears — the price of arriving.

A pulse changes the evidence. At each step the listener now hears two things: whether the bell strikes, and whether the pulse does. A pulse every three steps divides the twelve-step cycle into four beats; a pulse every four divides it into three. The pulse alone says nothing about which beat a listener is on, since every beat is the same; what it adds is where each bell stroke falls relative to the beats.

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.
Fig. 2 The earlier essay’s figure: for each order of the bell pattern’s gaps heard alone, the uncertainty a forgetting listener settles at, against the memory’s half-life. With a perfect memory every order is eventually placed; as memory shortens the orders settle at different floors, and the bell pattern is not the order that settles highest.

The reversal

Heard alone, over the first cycle, the order 2 2 2 1 2 1 2 is the quickest to place at every memory — 1.08 bits unplaced with a perfect memory, 1.23 at a half-life of three steps, 1.73 at one and a half. The bell pattern is slower at every memory — 1.25, 1.45 and 1.82 — and the order 2 2 2 2 1 1 2 slowest at the shorter memories.

Against a pulse every three steps the bell pattern costs 0.54, 0.56 and 0.67 bits, and the other two orders cost 0.69, 0.75 and 0.96, and 0.60, 0.73 and 0.92. Against a pulse every four steps the bell pattern costs 0.55, 0.57 and 0.68, and the others 0.63, 0.67 and 0.85, and 0.83, 1.07 and 1.36. At a half-life of three steps — a memory of a quarter of the cycle — the bell pattern against either pulse leaves 0.56 or 0.57 bits unplaced, and neither rival leaves fewer than 0.67.

So the pulse does not merely help every order a little. It helps the bell pattern most, by enough to move it from the middle or bottom of its family to the top, and it does so for both pulses and at every memory drawn.

The two pulses do disagree about the other two orders. In threes, 2 2 2 2 1 1 2 is the second quickest and 2 2 2 1 2 1 2 the slowest; in fours the order is the other way round, and 2 2 2 2 1 1 2 is by far the slowest, at 1.07 bits for the quarter-cycle memory. That answers the earlier essay’s second question: a pulse in threes and a pulse in fours rank the orders differently — except at the top, where both put the bell pattern.

Why the bell pattern sits well against both

The bell pattern’s standing does not come from any one alignment with the beat. Turning the timeline against the pulse by each of its twelve steps shows how little the alignment matters.

Where the pulse falls hardly matters; which order of the gaps does. The bits of position still unknown over the first cycle, for each order of the gaps 1 1 2 2 2 2 2, with the timeline turned by each of its 12 steps against a pulse every three and every four steps, at a memory half-life of 3 steps. against a pulse every 3 steps: 2 2 2 1 2 1 2 from 0.73 to 0.75; 2 2 2 2 1 1 2 from 0.73 to 0.74; 2 2 1 2 2 1 2 (the standard bell pattern) from 0.56 to 0.56. against a pulse every 4 steps: 2 2 2 1 2 1 2 from 0.65 to 0.70; 2 2 2 2 1 1 2 from 1.02 to 1.07; 2 2 1 2 2 1 2 (the standard bell pattern) from 0.56 to 0.57. Turning a timeline against the pulse moves its cost by a few hundredths; changing the order of its gaps moves it by several tenths.
Fig. 3 The bits unplaced over the first cycle for each order of the gaps, with the timeline turned by each of its twelve steps against the pulse, at a half-life of three steps. Against threes the bell pattern costs 0.56 at every turn and the others between 0.73 and 0.75. Against fours the bell pattern costs 0.56 to 0.57 and the others 0.65 to 0.70 and 1.02 to 1.07.

Against a pulse in threes the bell pattern costs 0.56 bits at every one of its twelve alignments, and the other orders between 0.73 and 0.75. Against fours the bell pattern ranges from 0.56 to 0.57, and the others from 0.65 to 0.70 and from 1.02 to 1.07. The bell pattern’s worst alignment is quicker to place than either rival’s best, against either pulse. Where the pulse falls moves a cost by a few hundredths of a bit; which order the gaps are in moves it by tenths.

Against a pulse in fours the reason can be read straight off the beats. Cut into three beats of four steps, the bell pattern’s strokes fall as 0101, 0110 and 1011 — three different arrangements — while each of the other two orders repeats one: 0101, 0101 and 1011, and 0101, 0101 and 0111. A listener who has heard one beat of the bell pattern against a four-pulse knows which beat it was; a listener hearing 0101 from either rival does not. Against a pulse in threes the explanation is subtler. Cut into four beats of three steps, every order carries three different arrangements and one repeat, so a count of distinct beats does not separate them; what does is the sequence the beats come in, which the cost measures and a count cannot.

How fast the position arrives

Averaging over the first cycle hides how quickly the pulse does its work, which is visible step by step.

A pulse gives away most of the bell pattern's position within half a cycleThe bits of position a listener with a memory half-life of 3 steps is still missing, step by step through two cycles of the standard bell pattern (2 2 1 2 2 1 2), alone, against a pulse every 3 steps, against a pulse every 4 steps. alone: 2.63 after 1, 1.77 after 3, 1.02 after 6, 0.58 after 12, 0.54 after 24; against a pulse every 3 steps: 1.77 after 1, 0.40 after 3, 0.01 after 6, 0.00 after 12, 0.00 after 24; against a pulse every 4 steps: 1.93 after 1, 0.31 after 3, 0.01 after 6, 0.00 after 12, 0.00 after 24. At the start every listener is missing 3.58 bits, the 12 possible positions.aloneagainst a pulse every 3 stepsagainst a pulse every 4 steps051015200.01.02.03.0steps heardbits of position unknown
Fig. 4 The bits of position unknown, step by step through two cycles of the bell pattern, for a listener with a half-life of three steps, alone and against each pulse. Alone the listener still misses 1.02 bits after six steps and settles near 0.54. Against a pulse in threes, 0.40 bits after three steps and none after six; in fours, 0.31 after three. The dial moves the memory half-life from one and a half steps to twelve.

Alone, the bell pattern leaves a listener with 2.63 bits missing after one step, 1.77 after three, 1.02 after six — half a cycle — 0.58 after a whole cycle, and it settles at 0.54 bits, never fully placed, because a memory of three steps cannot hold enough of the pattern to rule out every rotation at once. Against a pulse every three steps the listener misses 1.77 bits after one step, 0.40 after three — one beat — and 0.01 after six. Against a pulse every four the listener misses 0.31 after three steps and 0.01 after six. With a pulse the forgetting listener is fully placed within half a cycle; without one, never.

Six of eight

A result about one timeline could be an accident of its gaps, so the comparison was run for every named timeline in the collection whose gaps can be put in more than one order, each at a memory of a quarter of its cycle.

Most named timelines are the quickest of their orders to place once a pulse is heard. For each named timeline whose gaps can be put in more than one order, how many of those orders a listener with a memory of a quarter-cycle places faster than the timeline itself, and how many slower, heard alone and against each pulse that divides its cycle. son clave (6 orders): alone 2 faster, 3 slower; against a pulse every 4 steps 0 faster, 5 slower. rumba clave (6 orders): alone 4 faster, 1 slower; against a pulse every 4 steps 2 faster, 3 slower. the standard bell pattern (3 orders): alone 1 faster, 1 slower; against a pulse every 3 steps 0 faster, 2 slower; against a pulse every 4 steps 0 faster, 2 slower. cinquillo (2 orders): alone 1 faster, 0 slower; against a pulse every 4 steps 0 faster, 1 slower. fume-fume (2 orders): alone 1 faster, 0 slower; against a pulse every 3 steps 0 faster, 1 slower; against a pulse every 4 steps 0 faster, 1 slower. shiko (2 orders): alone 1 faster, 0 slower; against a pulse every 4 steps 0 faster, 1 slower. soukous (12 orders): alone 9 faster, 2 slower; against a pulse every 4 steps 0 faster, 11 slower. gahu (6 orders): alone 0 faster, 5 slower; against a pulse every 4 steps 3 faster, 2 slower. Against a pulse, 6 of the 8 are placed faster than every other order of their own gaps.
Fig. 5 For eight named timelines, how many other orders of the same gaps a listener places faster, and how many slower, heard alone and against each pulse that divides the cycle. Alone, the soukous timeline is beaten by nine of its eleven rivals and the son clave by two of five. Against a pulse every four steps, both are placed faster than every rival. Six of the eight are the quickest of their orders against a pulse; the rumba clave and the gahu pattern are not.

Against a pulse, six of the eight are placed faster than every other order of their own gaps: the son clave, the bell pattern, the cinquillo, fume-fume, shiko and the soukous timeline. The soukous case is the most striking. Heard alone, nine of the eleven other orders of its gaps are placed faster; against a pulse in fours, none is, and all eleven are slower. The son clave goes from two faster and three slower alone to none faster and five slower.

The two exceptions are informative. The rumba clave, which is the son clave with its third stroke moved one step later, has two orders of its gaps that are placed faster against a pulse in fours. The gahu pattern, which alone is the slowest of its six orders, against a pulse is beaten by three of them. Both are timelines with a stroke deliberately displaced from where a closely related pattern puts it, which may be exactly what costs them against a pulse — a displacement is heard as tension against the beat, and a tension is a departure from the arrangement that sits most informatively on it.

Settled, not only arriving

The arrival cost is what a listener pays in the first cycle. What a forgetting listener settles at — the uncertainty left once listening has gone on as long as it likes — is the quantity the earlier essay ranked the orders by, and the pulse reverses that ranking too.

Alone, at a memory half-life of three steps, the three orders settle at 0.07, 0.79 and 0.54 bits: the order 2 2 2 1 2 1 2 almost fully placed, the bell pattern half a bit short, 2 2 2 2 1 1 2 nearly a bit short. At a half-life of one and a half steps, 1.13, 1.69 and 1.37. A listener with a short memory never fully places any of them.

Against a pulse in threes, at a half-life of three steps, the orders settle at 0.01, 0.14 and 0.00; at one and a half steps, 0.40, 0.49 and 0.11. Against a pulse in fours, at three steps, 0.00, 0.32 and 0.00; at one and a half, 0.25, 0.92 and 0.08. The bell pattern is fully placed at the longer memory against either pulse, and at the shorter one it leaves a tenth of a bit where the rivals leave between a quarter of a bit and nearly a whole one.

So the earlier essay’s result — that the bell pattern is slowest to place only for a perfect memory, and loses that distinction when memory fades — was a result about a listener hearing the bell by itself. For a listener hearing it against the beat, it is the order of its gaps that is placed soonest and settles most completely, at every memory the earlier essay drew.

What this does to the evenness trade

The Euclidean patterns have been about a trade. As evenly as possible found Bjorklund’s algorithm producing patterns spread as evenly as their counts allow, and what the clave buys with its unevenness found the son clave giving up some of that evenness to be placed sooner: heard alone, the most even pattern of five in sixteen needs almost the whole cycle heard before its position is unambiguous, and the clave needs nine steps. Evenness and placeability pulled in opposite directions, and a timeline sat somewhere on the frontier between them.

The pulse changes the terms of that trade. Against a pulse, placing a timeline no longer depends only on the pattern’s own gaps but on how they fall against the beats, and an even pattern’s strokes fall against the beats in a variety of ways that a bunched pattern’s do not. The bell pattern is a rotation of a Euclidean pattern — the most even arrangement of seven strokes in twelve — and against a pulse it is also the quickest to place of its gaps’ orders. For a listener who hears the beat, the evenness and the placeability are no longer a trade; the even order is the placeable one.

That does not make the clave’s unevenness a mistake. The longest silence is not a third axis and the other censuses keep evenness, not locating found that the named timelines keep the evenness their censuses allow, and the son clave is one of the six timelines here that are the quickest order of their gaps against a pulse. What changes is the explanation: an ensemble’s timeline need not be uneven to be placed, because the ensemble supplies the beat that an even pattern needs to be placed against.

Which computation produced the numbers

Each timeline is a cycle of steps with strokes; its orders are every cyclic arrangement of its gaps, up to rotation. The pulse is a stroke every three or four steps, placed on the first step of the pattern as given and turned against it in the alignment figure. At each step the listener’s evidence is the pair of symbols — bell or not, pulse or not — and the posterior over the cycle’s positions is updated by a mismatch penalty of six for each step whose evidence contradicts a position, weighted by 2 to the minus the steps since, divided by the half-life. The uncertainty is the entropy of that posterior in bits, averaged over every starting position; the arrival cost averages it over the first cycle. Timelines and their gaps are the collection’s own table.

What the listener model leaves out

A pulse is heard perfectly and bars are not. The pulse here marks beats but not which beat is first. A listener who also hears a bar’s downbeat, from a bass drum or a dancer’s weighted step, would place any timeline faster, and the ranking among orders could change again.

The pulse is taken as given. A listener here is told the beat and has to find the timeline against it; in practice the beat itself is inferred from the ensemble, and rhythm is a circle found that where a bar line is drawn on a cycle is a choice. A listener still finding the beat is solving a harder, joint problem than the one computed, and the orders could rank differently while the beat is uncertain.

Only the strokes’ presence is heard. Real bells have two pitches, and drums have strokes that differ in pitch and timbre, all of which carry position. The model gives each order of gaps the same bare evidence so that only the order differs.

One pulse at a time. African and Afro-Cuban ensembles sound several regular layers at once, often in threes and fours together, which is where a cycle that says where it is found the combination carrying position. The two pulses here are compared, not combined.

What placing a timeline does not say

Whether timelines are designed to be placed. The finding is that the named timelines are, in most cases, the most placeable order of their gaps against a pulse. What the clave buys with its unevenness and the frontier and the ruler found other properties the named timelines maximise, and a timeline that became standard for any of those reasons could have been the most placeable by coincidence of the same small-number arithmetic.

Whether listeners place themselves by timeline. A dancer in an ensemble orients by many things. What the computation says is how much position the timeline and a pulse together make available, not that anybody uses it.

Whose timelines

The bell pattern and the claves are timelines in exactly the sense that matters here: repeating patterns that ensembles in West and Central Africa and the Afro-Caribbean diaspora use to keep a common reference while other parts vary. They are played against a beat that the other parts, the dancers and often a hand-clap make explicit, in threes, in fours, or both. On this arithmetic, the orders of their gaps that tradition kept are the ones a listener can place against that beat soonest — which is a property worth having in an instrument whose job is to tell an ensemble where it is.

Still open: the pulse in threes and fours together

The two pulses ranked the other orders in opposite ways, and a twelve-step cycle in the ensembles that play these timelines is heard in both at once: a four-beat and a three-beat feel over the same bell. With both pulses sounding, a listener’s evidence at each step is a triple — bell, pulse in threes, pulse in fours — and the two pulses together already carry some position, since they coincide only once a cycle.

That makes the combined case the sharpest test of the finding. If the bell pattern is still the quickest order to place with both pulses sounding, its gaps are positioned to be read against the whole metrical grid. If the pulses together carry so much position that every order is placed within a beat, the bell pattern’s advantage lives only in textures where one of the two feels is dropped, and the question becomes which feel an ensemble drops and when.

Part 8 of 8

One essay in the series on euclidean rhythm. The essays either side of this one:

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 Part 7 — 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.

Where this goes next

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. A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players. A cycle that says where it is Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke. 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.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Bell patternEuclidean rhythmInformationMemory decayRotationTimeline