A bell can mark a cycle without moving a stroke

The standard twelve-step bell pattern has a weakness for a listener in a thin texture. Its two short gaps look alike, so someone who has lost their place can settle on the wrong one, and a feel heard half the time gives the bell half its lead found it about five times worse for keeping one’s place than another order of the same gaps. A varied bell can be a landmark or a false start asked whether a lead drummer’s variation, one stroke moved every few cycles, could give that back.

It could, in part, and it could also do the opposite. Six of the ten possible moves make a figure the bell never otherwise contains, and those help: the best brings the cost of staying placed from 0.0078 bits a step down to 0.0055. Two turn the varied cycle into the other order of the gaps and make matters a little worse. And two of them turn the varied cycle into the bell itself, begun at another step, which makes things more than three times worse, at 0.0252. A listener who hears the plain bell in the wrong place is then being told so, every third cycle, by the music.

That essay’s last section pointed out what its listener was missing. A moved stroke is false only because it sounds like any other stroke. A bell player who plays the varied cycle on another part of the bell, or damps it, gives a second cue, one that says “this cycle is varied” whatever its figure. So add that cue to the listener, with some reliability, and ask two things: whether a marked variation closes the gap to the other order, and how reliable the mark has to be before a displacement that makes the bell’s own rotation stops being a false start and becomes a landmark.

The calculation answers both, and then asks a third question of its own that makes the original one look narrower. If the mark does all this, what is the moved stroke for?

A mark as evidence #

The listener is the earlier essay’s. It holds a belief about where in a long timeline it is. Each stroke it hears or fails to hear is evidence against the positions that disagree, a hypothesis contradicted by the bell being charged six nats and one contradicted by a pulse being charged according to how often that pulse has been missed, and it forgets, with each step’s evidence falling to half over three steps. Its place in the twelve-step cycle is scored by summing its belief over every position on the same step. Both pulses are heard half the time.

The mark goes in as one extra observation. Each stroke of the bell is heard with a label, varied or plain, which is right with probability $m$. A hypothesis that expected that label gains $\ln 2m$ nats and one that did not loses $-\ln 2(1-m)$. At $m = 1/2$ both are zero, so a mark that is right half the time does nothing at all, and the calculation returns the earlier essay’s figures to every decimal place on every displacement. That is the control, and it holds because the mark has its own random draws and leaves the strokes exactly as they were.

The mark is applied to every stroke of the varied cycle rather than just the moved one, since that is what playing a cycle on another part of the bell means. This is a stated choice, and a mark on only one stroke would be much weaker evidence.

A mark does little and then everything #

The curve is a hockey stick. For the shifted cycle, the worst of the ten, the cost goes 0.0252 with no information, 0.0238 at sixty per cent reliability, 0.0204 at seventy, 0.0154 at eighty, 0.0092 at ninety, 0.0062 at ninety-five and 0.0027 at a hundred. The first twenty points of reliability take off a fifth of it, and by the end it is a ninth of what it was. For the best displacement, which has less far to fall, the same range goes 0.0055, 0.0055, 0.0052, 0.0048, 0.0041, 0.0036, 0.0025.

The reason is exact and belongs to the coin rather than to the bell. On average a stroke’s mark gives the true position an advantage of $m\ln 2m + (1-m)\ln 2(1-m)$ nats. At seventy per cent that is 0.08 nats, at eighty 0.19, at ninety 0.37, at ninety-five 0.50, and at a hundred 0.69. It grows from a slope of zero at a half, which is the property of any such information measure: a slightly informative label is very nearly no label. With evidence fading to half in three steps, the listener holds three or four strokes’ worth of it at a time, and 0.08 nats that many times over will not overturn six nats’ charge from a contradicting stroke of the bell. A 95-per-cent mark can start to.

So the first answer is available. The false start stops being false at a reliability of about ninety-two per cent. That is where the shifted cycle crosses the plain bell, at every third cycle: below it, moving the stroke that way does harm even with the mark, and above it, the same variation is a net gain. A figure which would mislead a listener in every other respect is turned into a landmark by being marked, and it needs to be marked nearly every time.

That has a practical shape. A bell player’s two sounds, an open stroke and a damped one or the bell’s mouth and its side, are probably very reliably told apart in a quiet room by someone listening for them, and much less so inside a dense ensemble. The same variation can therefore be a landmark for one listener and a false start for the next, according to where the dance is and how loud the drums are. Nothing here estimates what $m$ is in any actual ensemble. It says only that the answer changes sign at ninety-two per cent.

The displacements stop differing #

The previous essay’s most striking finding was how much depended on which stroke moved. A mark removes that.

With no mark the ten displacements differ by a factor of 4.6. With a perfect one they differ by 1.4. All of the earlier essay’s sorting, into new figures that help, rival orders that hurt a little and rotations that hurt a lot, comes from the listener having only the stroke positions to go on. Once the cycle is labelled, the positions matter much less. The ordering does not quite vanish, and the two rotations are still the highest of the ten at 0.0027 and 0.0031, but they have come in from five times the best to about one and a quarter.

That suggests the question the essay was not asked. If it hardly matters which stroke moved, it may not matter whether one did.

With nothing moved at all #

The third curve in the first figure is a cycle in which no stroke has moved. The bell plays its pattern every time, and every third cycle simply sounds different: same strokes, other timbre.

With a mark that always gets through, this costs 0.0026 bits a step. That is what any displacement costs with the same mark: 0.0023 to 0.0031. The moved stroke, the subject of the whole previous essay, is adding almost nothing once the cycle has a reliable sound of its own.

It does still do something when the mark is not reliable. At ninety per cent, the unmoved marked cycle costs 0.0057 and the best moved-and-marked one 0.0041. At eighty per cent it is 0.0070 against 0.0048, and at seventy 0.0075 against 0.0052. That is where the figure earns its keep, as a second and independent cue, since a new figure in the varied cycle is evidence in every stroke’s position whether or not its sound was caught. The two cues add, as cues should, and the better the one is, the less the other contributes.

This is a surprisingly close echo of something already found on a very different instrument. A goal tone that returns should swell only once found that two points in a gamelan’s slow cycle which sound alike cost a listener a good deal, and that told apart by some other mark, they cost almost nothing, whatever their positions. Two landmarks in a cycle help only if they can be told apart had the principle before either. And a newcomer learns a cycle’s landmarks in four hearings ends by asking how different two marks must sound before position stops mattering. A fast twelve-step bell and a slow 160-second gong cycle have no timescale in common. But a listener locating itself in any cycle is doing the same inference, and the same answer keeps coming back: position is slow evidence and sound is fast, when sound can be heard.

How often it has to come #

The other half of the original question was whether a marked variation closes the gap to the other order of the gaps, which reads 0.0016. Every third cycle, it does not, even with a perfect mark.

Every other cycle, a perfect mark beats the other order of the gaps by a factor of five, at 0.0003 against 0.0016. Every third cycle it reaches 0.0026 and does not catch it, and every sixth it only gets back about a third of the plain bell’s cost. So the answer to that half of the lead has a condition on it: the gap closes, but only at a rate which makes the marked cycle a pattern in itself.

The dependence on reliability at that rate is steep as well. The unmoved mark, every other cycle, reaches the other order’s 0.0016 at a reliability of about 97 per cent. With the best displacement added, it gets there at about 87, because the two cues are helping each other. A bell which alternates its two sounds cycle by cycle is, of course, a bell with a twenty-four-step timeline, and the arithmetic says that is an enormous gain. Whether a listener takes it that way, or hears two different cycles, or hears a change of texture, is outside what the belief calculation can say.

Against both feels the bell pattern’s order stops mattering was the first sign that a bell’s order is not the only resource a listener has, and a related claim from elsewhere gives this some context: a cycle cannot cadence found that music on a repeating cycle has none of the usual harmonic means of marking an ending. A sound that marks a place in the cycle takes on that job with its own resources, and every tradition’s cycle has found some way to do it.

Where the benefit is and is not #

The figures so far are all averages from the second cycle onward, once the listener has placed itself. That leaves out arrival, and it conceals how the cost develops with time.

The settled cost, in the later cycles, is what the mark changes most. By the fourth cycle the shifted bell settles at 0.0230 with no information and 0.0022 with a perfect mark, a factor of ten. These are small numbers. The whole cost once a listener is placed is a small fraction of a bit each step, and it is a rate rather than a size: what it represents is an occasional serious loss of place, and that is what a false start was.

What it barely changes is the first cycle, in which almost everything is decided, and that is for an obvious reason.

On arrival a mark saves at most about a tenth of the first cycle’s confusion, since no stroke in that first cycle carries the marked sound. What it does save it saves by ruling out hypotheses in which the listener is already inside a varied cycle, and that is a small part of a large uncertainty. So a mark helps a listener who has found their place and is keeping it, and it does nearly nothing for one who has just walked in. The first of those is what these figures measure, and a dancer already in the circle is a better match to it than a newcomer. It is a different benefit from what a landmark might be thought to give, and the reason for it is clear from the first cycle’s numbers: the plain bell’s first-cycle cost of 0.94 is actually lower than the other order’s 1.02, which is why the bell pattern’s order is for arriving and the other order only pays off afterwards.

What the listener and the mark assume #

The model. A hypothesis-weighing listener with a fixed charge for contradiction, a fading memory of a three-step half-life, two pulses each heard half the time, and a mark on every stroke of the varied cycle, independently right with probability $m$. All of those are the earlier essays’ numbers except the mark, and the mark’s independence is the most generous assumption here, since a noisy ensemble that hides one stroke’s timbre will tend to hide its neighbours’. Correlated errors would make the mark weaker than these curves say.

Averages over twelve alignments and thirty draws. The curves are seeded, with six cycles of listening. Small differences of a few parts in a thousand at the top of each curve are within what another seed could move, which is why they are reported to two figures.

The label is a given. The listener has been told that there are two sounds and which one marks the varied cycle. Learning that, which is exactly what a newcomer’s task is, is left out here.

What a belief calculation cannot hear #

It cannot hear groove. A variation has a musical job: a lead drummer’s move is a signal and a provocation, and whether dancers shift their feet or the other players answer it has nothing to do with a listener locating itself in a twelve-step cycle. All these figures price exactly one effect a variation has, and it may not be the main one.

It cannot hear whether a sound is a mark. A change of timbre on a bell could be taken as a mark, as an error, or as a player’s whim. A listener has to decide which, and a mark only works once it has been learned as a mark, a problem these figures assume has been solved. It takes time, and in some musical cultures the meaning is learned over years.

And nothing here is a measurement of any ensemble. No recording has been counted for how often a bell’s sound changes or how often it is heard. The reliabilities are a sweep rather than estimates.

Whose bell #

The pattern is the seven-stroke, twelve-step standard bell of West and Central African drumming and of its descendants in Afro-Cuban and Afro-Brazilian music, and the idea of a lead drummer varying against a fixed bell is a stylised version of one practice in some of those traditions. In many of them the bell itself is held constant and it is the master drum that varies, so a varied bell in the strict sense is rare, and the argument applies more directly to any recurring part that is treated as a reference. The listener here knows the pattern and is trying to keep its place. That is not how anyone in those traditions would describe what they are doing.

Still open: a bell with two sounds in every cycle #

Everything above puts its mark on a whole cycle now and then. A double bell, like the Ewe gankogui, plays two pitches inside every cycle, so each stroke’s position is labelled all the time by which of the two it falls on, and a timeline of two sounds is a very different object from a timeline with one sound and a variation. The computation is this listener with a mark on each stroke from a fixed assignment of the seven strokes to two pitches, and it would say which assignments keep a listener best placed, whether the worst of the bell’s short gaps can be told apart by giving them different pitches, and whether a two-note bell makes the other order of the gaps unnecessary without any variation at all.