Rhythm and metre

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.

15 min read 5 figures Time is a circle What the listener supplies The design was forced

Assumes: Against a pulse the bell pattern is the easiest to place · The bell pattern is slowest only to a perfect memory

Against a pulse the bell pattern is the easiest to place put the standard bell pattern — the twelve-step timeline with gaps 2 2 1 2 2 2 1 — beside the other two orders of the same seven gaps and asked which a listener places soonest. Heard alone, it is not the quickest at any memory. Heard against a pulse every three steps, or against a pulse every four, it is the quickest at every memory and every alignment, by a wide margin. The conclusion was that a timeline’s order of gaps looks chosen for how it sits against the beat rather than for how it sounds alone.

That essay’s own open question was sharper than its answer. The two pulses ranked the other two orders in opposite ways, and in the ensembles that play this timeline a twelve-step cycle is not heard against one pulse. It is heard against two: a four-beat feel of dotted crotchets and a three-beat feel over it, the dancers and the drums marking different ones, often at once. With both 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 position, since they coincide only once a cycle.

If the bell is still the quickest order with both pulses sounding, its gaps are positioned to be read against the whole metrical grid. It is not. Against both pulses the three orders are placed within a seventieth of a bit of one another, the bell pattern is fractionally the second of them, and at every memory its lead has gone.

Four textures, three orders

The listener is the forgetting listener of the two essays before: a memory whose weight on each heard step halves every three steps — a quarter of the cycle — and a mismatch cost of six. The cost is the entropy of their posterior over the twelve positions, averaged over the first cycle heard. And because the ranking against a single pulse held at every alignment but the ranking against two need not, each order’s cost is averaged over all twelve ways it can sit against the pulses.

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. — Fig. 1 Bits of position still unknown over the first cycle, averaged over every alignment, for each order of the gaps 1 1 2 2 2 2 2 in twelve steps, heard alone, against a pulse every three steps, a pulse every four, and both; dashed, the two pulses alone at 0.53. The standard bell pattern: 1.452 alone, 0.560, 0.564 and 0.425. Its rivals: 1.234, 0.738, 0.672 and 0.437; and 1.630, 0.734, 1.037 and 0.422.

The first three columns are the earlier essay’s result, with the alignments averaged. Alone, the bell pattern is the middle of the three orders, at 1.452 bits, with 2 2 2 1 2 1 2 quicker at 1.234. Against the pulse in threes it is placed in 0.560 against its rivals’ 0.734 and 0.738 — a lead of 0.174 bits. Against the pulse in fours, 0.564 against 0.672 and 1.037, a lead of 0.107.

The fourth column is different in kind. Against both pulses the bell pattern reads 0.425, and its rivals read 0.437 and 0.422. The spread across the three orders is 0.015 bits, and the order 2 2 2 2 1 1 2 — the least even arrangement, with its two short gaps together, where the bell pattern is the most even one, the arrangement as evenly as possible produces — is placed three thousandths of a bit sooner than the bell. Evenness, which against a single pulse went with being placed soonest, is worth nothing either way against two. A lead of a sixth of a bit against one feel has become a tie against two.

The grid is a timeline of its own

The dashed line in the last column is the reason. It is the two pulses alone, with no bell at all, and it reads 0.53 bits — lower than any of the three orders heard alone, and not far above any of them heard with both pulses.

Two pulses every three and every four steps are not two regular patterns. Heard together they make a single cycle of four kinds of step: the one step where both sound, the three where only the pulse in threes sounds, the two where only the pulse in fours does, and the six where neither does. That is a position code with no symmetry, and it is exactly the resultant rhythm the third pattern nobody played found emerging from two players playing even pulses — with the difference that here the two pulses are distinguishable, so the listener hears not one resultant but which of its strokes belong to which feel.

Fig. 2 Bits of position unknown, step by step through two cycles, hearing the standard bell pattern alone, the two pulses alone, and all three together; the dial sets the memory’s half-life. At a half-life of three steps, after three steps: 1.77 bits for the bell alone, 0.13 for the pulses alone, 0.01 for all three. After six: 1.02, 0.01 and 0.00.

The curves say how much of the work the grid does. After three steps heard, a listener with the bell alone is still missing 1.77 bits. A listener with only the two pulses is missing 0.13. A listener with all three is missing 0.01. By six steps the grid alone has placed the listener; the bell alone never gets below half a bit at this memory. Dragging the memory shorter shows the same thing more starkly: at a half-life of a step and a half, after half a cycle the bell alone is at 1.43 bits and the pulses alone at 0.10. The first cycle’s average for the bell and grid together is 0.425 bits, and nearly all of it is spent in the first two steps, before the pulses have said anything.

Against both feels, position is carried by the feels, and the bell adds to it a little in the first step or two — which is where the three orders differ, and they differ by thousandths.

Why one feel needs the bell and two do not

The difference between one pulse and two is a difference of symmetry, and it can be counted exactly.

A pulse every three steps in a twelve-step cycle looks the same from four starting points: heard alone, a listener could be at any of four positions forever, which is two bits that no amount of listening removes. A pulse every four steps looks the same from three starting points, 1.58 bits. Against either pulse on its own, then, the listener’s position is decided by the bell, because only the bell breaks the pulse’s symmetry — and how quickly it breaks it depends on how its gaps fall against the pulse, which is exactly what the order of the gaps controls. That is why the order mattered so much against one feel: the order was the only thing in the texture that could tell the four or three candidate positions apart.

Two pulses in twelve steps have no symmetry at all. Three and four share no factor, so the only rotation that maps the pair onto itself is the whole cycle, and a listener who hears both for long enough is placed with no bell at all: the floor is zero, and the first cycle’s average is 0.53 bits. Against both feels the bell is no longer the thing that breaks the symmetry, because there is none to break. It only hurries along a placement the grid would have made anyway, and any of the three orders hurries it about equally.

Put in numbers, the grid does four fifths of the work and the bell a fifth. The bell alone is placed at 1.45 bits over the first cycle and the grid alone at 0.53; together they reach 0.43. Adding the grid to the bell removes a bit; adding the bell to the grid removes a tenth of one. That tenth is where the three orders compete, and they tie in it.

This is also why a comma is a polyrhythm that never closes and the other essays on polyrhythm could treat three against four as an object in its own right: a three-against-four cycle is the smallest pair of pulses whose combination has no symmetry, and so it is itself a timeline in the sense these essays use — a pattern that says where in the cycle it is.

The lead against one feel and against two

The lead could be a quirk of the memory chosen. It is not.

Against one pulse the bell pattern leads its rivals at every memory, and against both it ties them. How many bits sooner the standard bell pattern is placed than the nearest other order of its gaps, over the first cycle and averaged over alignments, against the listener's memory half-life in steps, against a pulse every three steps, a pulse every four and both. every 3 steps: 1 0.161, 1.5 0.248, 2 0.232, 3 0.174, 4 0.134, 6 0.100, 12 0.077, perfect 0.064; every 4 steps: 1 0.187, 1.5 0.192, 2 0.147, 3 0.107, 4 0.098, 6 0.093, 12 0.090, perfect 0.089; both pulses: 1 -0.002, 1.5 -0.002, 2 -0.003, 3 -0.003, 4 -0.004, 6 -0.004, 12 -0.004, perfect -0.004. Below zero, a rival order is placed sooner. — Fig. 3 How many bits sooner the standard bell pattern is placed than the nearest other order of its gaps, averaged over alignments, against the memory’s half-life, against a pulse every three steps, a pulse every four, and both. Against the pulse in threes: from 0.161 to 0.248 at short memories and 0.064 with perfect memory. Against the pulse in fours: 0.089 to 0.192. Against both: between −0.002 and −0.004 at every memory.

Against the pulse in threes the lead peaks at 0.248 bits for a memory of a step and a half and falls to 0.064 for a listener who never forgets; against the pulse in fours it runs from 0.192 to 0.089. It is always there, and it is largest for the forgetful listeners who most need it. Against both pulses it is between −0.002 and −0.004 at every memory, from a half-life of one step to none at all. The bell pattern is never the quickest of its orders against the whole grid, and it is never behind by enough to matter.

Where the bell as written sits

The averaging over alignments hides one thing a player would want to know: how the bell pattern does in the one alignment it is actually played in, with its first stroke on the downbeat where the two feels meet.

Against both pulses, the bell pattern as written sits near its worst alignment. The bits of position a listener with a memory half-life of 3 steps is still missing over the first cycle, for each order of the gaps 1 1 2 2 2 2 2 against a pulse every three steps and a pulse every four together, at every one of the 12 ways the order can sit against the two pulses; the dashed line is the pulses alone, 0.53. 2 2 2 1 2 1 2: 0.436, 0.416, 0.437, 0.468, 0.436, 0.416, 0.437, 0.448, 0.434, 0.433, 0.435, 0.448; 2 2 2 2 1 1 2: 0.437, 0.420, 0.409, 0.415, 0.442, 0.415, 0.409, 0.420, 0.437, 0.415, 0.422, 0.415; 2 2 1 2 2 1 2 (the standard bell pattern): 0.406, 0.420, 0.468, 0.420, 0.406, 0.415, 0.435, 0.429, 0.419, 0.429, 0.435, 0.415. The standard bell pattern as written, with its first stroke where the two pulses meet, reads 0.435; its best alignment reads 0.406 and its worst 0.468, and it is the quickest of the three orders at 3 of the 12 alignments. — Fig. 4 Bits of position unknown over the first cycle for each order against both pulses, at all twelve alignments. The standard bell pattern ranges from 0.406 at its best alignment to 0.468 at its worst; as written, with its first stroke where the two pulses meet, it reads 0.435. It is the quickest of the three orders at 3 of the 12 alignments.

Against both pulses, alignment matters as much as order does. The bell pattern runs from 0.406 bits at its best alignment to 0.468 at its worst, a range four times the spread between the three orders, and it is the quickest of the three at only three of the twelve alignments. As written it reads 0.435, among the worst three of its twelve, three hundredths of a bit from its best.

That is a small effect, and it is the cleanest statement of what the bell is not doing against both feels: its written alignment is not chosen for locating. Its first stroke falls on the one step of the cycle that the grid already identifies, where both feels coincide — and a stroke on the common downbeat is an obvious thing to want for reasons the measure here cannot see, in a dance above all. What the figure shows is only that, once both feels sound, neither the bell’s order nor its alignment is where a listener’s sense of position comes from.

The other twelve-step timeline

There is one other widely named twelve-step timeline, the fume-fume, whose five onsets have two orders of their gaps.

Against both pulses, neither twelve-step timeline keeps an eighth of the lead one pulse gives it. For each named twelve-step timeline, how many other orders of its own gaps are placed faster than it, and by how much it leads the nearest, averaged over alignments, heard alone, against a pulse every three steps, a pulse every four and both, at memory half-lives of a quarter and an eighth of the cycle. the standard bell pattern (3 orders): alone 1 faster, lead -0.218 / 1 faster, lead -0.091; every 3 steps 0 faster, lead 0.174 / 0 faster, lead 0.248; every 4 steps 0 faster, lead 0.107 / 0 faster, lead 0.192; both pulses 1 faster, lead -0.003 / 1 faster, lead -0.002. fume-fume (2 orders): alone 1 faster, lead -0.218 / 1 faster, lead -0.091; every 3 steps 0 faster, lead 0.178 / 0 faster, lead 0.280; every 4 steps 0 faster, lead 0.107 / 0 faster, lead 0.192; both pulses 0 faster, lead 0.012 / 0 faster, lead 0.017. — Fig. 5 For the two named twelve-step timelines, how many other orders of their gaps are placed faster, and the lead over the nearest, averaged over alignments, alone, against each pulse, and against both, at memories of a quarter and an eighth of the cycle. The standard bell pattern leads by 0.174 and 0.107 against one pulse and by −0.003 against both. The fume-fume leads by 0.178 and 0.107 against one pulse and by 0.012 against both.

It behaves the same way. Against the pulse in threes it leads its one rival by 0.178 bits at a quarter-cycle memory, and against the pulse in fours by 0.107. Against both it still leads, by 0.012 — about a ninth of the smaller of the two leads either pulse gave it. So neither of the two named timelines that live in a cycle with both feels keeps more than an eighth of its advantage when both feels sound, and one of them keeps none.

The alignment within one feel

One more comparison separates the two textures sharply. Against a single pulse, the earlier essay found the bell pattern quickest at every one of the alignments a pulse allows — every way of setting it against the pulse in threes, and every way against the pulse in fours. Against both pulses together it is quickest at three alignments of twelve. A single pulse has only three or four distinct alignments, because of its symmetry, and the bell’s order wins all of them; the grid has twelve, and at nine of them one of the bell’s rivals is placed as soon or sooner. What looked like a property of the bell’s order against the beat was a property of the order against a symmetric beat.

What the order of gaps is for

Put together, the four readings of the bell pattern give the timeline’s order a narrow job description. Alone, the bell pattern is not the quickest order to place, as the bell pattern is slowest only to a perfect memory found. Against one feel it is the quickest by a clear margin. Against both it is no quicker than its rivals. Its order is shaped for a texture with one feel in it.

That is not an embarrassment for the ensemble; it is a prediction about the ensemble. The bell is not a polyrhythm described the bell as one pattern played against the beat — against several beats at once — and the arithmetic here says the several beats together make the bell redundant as a locator. A listener in the full texture does not need the bell to know where they are; a listener in a thinner texture does. So the bell’s shape would matter most exactly in the textures where one feel has dropped out: an introduction in which the bell enters over one drum, a passage in which the dancers mark the four and the drums fall silent, a listener attending to one feel and not the other. And in the full texture, whatever the bell’s order is doing, the position-finding measure used here cannot see it.

The listener behind the numbers

The listener hears each step as a symbol: whether the bell sounds, whether the pulse in threes sounds and whether the pulse in fours sounds — three independent features, all heard correctly. The posterior over the twelve possible positions is updated at every step, each mismatch between a heard symbol and a candidate position costing six units of log-evidence, and each past step’s weight halving every so many steps. The reported cost is the entropy of that posterior, averaged over the first cycle and over every starting position. The three orders are every cyclic arrangement of the gaps 1 1 2 2 2 2 2 up to rotation — the necklaces of the rotation the necklace cannot see — and each is set against the two pulses at all twelve relative rotations, since a pulse in threes and a pulse in fours together have no symmetry within twelve steps.

What the grid model cannot hear

That the two feels are equally loud and equally clear. The pulses here are two perfect symbols. In an ensemble one feel is usually the stronger, carried by the dancers’ feet or a bass drum, and the other is implied; a listener who hears the weaker feel only now and then is somewhere between one pulse and two, and the bell’s advantage would be somewhere between its two values.

That a listener attends to both. Hearing two feels at once is not the same as being told two things. A listener who follows one feel and hears the other as texture is a listener with one pulse, and for that listener the earlier essay’s result stands.

That position is what a timeline is for. Everything in these essays on timelines measures how quickly a timeline tells a listener where they are. It says nothing about what a timeline’s shape does for dancing, for the parts that play against it, or for its identity as the pattern of one tradition and not another.

And that memory is the only limit. The listener forgets but never mishears. A texture of three layers is harder to hear correctly than one of two, and errors of perception would weigh against the richer texture in a way this model does not.

The music this concerns

The seven-stroke standard bell pattern and the five-stroke fume-fume are timelines of West African and Afro-Caribbean music in twelve steps, played against a four-beat and a three-beat feel together; two clocks at once and which of the two is the beat examined that three-against-four relation as a relation between two pulses. The claim here is about the information those textures give a listener trying to find their place, under a stated model of memory, and about nothing else in them.

Still open: the feel that drops out

The result makes the bell pattern’s shape a property of textures with one feel, and that turns a question about rhythm into a question about the ensemble: when is one of the feels absent? The computation that would say something about it gives the two pulses unequal reliability — each sounding, or being heard, only a stated share of the time — and asks at what share of the weaker feel the bell’s lead reappears. If it reappears only when the weaker feel is nearly silent, then the bell’s order is for introductions and thin textures; if it reappears as soon as the weaker feel is intermittent, then the full texture of an ensemble is, to a listener, mostly a one-feel texture with the other feel coming and going, and the order is shaped for the music as it is actually heard.