A varied bell can be a landmark or a false start

The bell pattern’s order is for arriving followed a listener who has to learn, from the strokes themselves, how often a drummer’s pulses go missing, and found two things about the standard bell pattern, the seven strokes in twelve steps whose gaps run 2 2 1 2 2 2 1. Its order is the quickest of all the orders of those gaps to place oneself by, and that lead lives almost entirely in the first cycle. But in a thin texture — both pulses, the one in threes and the one in fours, heard only half the time — another order, 2 2 2 1 2 1 2, keeps a listener placed better from the second cycle on, by a factor of about five. The bell spreads its two short gaps as far apart as they will go, and they look alike; the other order puts them close together, where they make one figure that happens once a cycle.

That essay ended on what players actually do. Lead drummers vary their parts, and a bell player sometimes shifts a stroke. A varied cycle is a cycle with a distinctive figure in it — a named landmark — and the finding predicted where such a variation would help: not in the first cycle, where the plain pattern already arrives fastest, but later, in thin passages, where its alike short gaps are weakest. The computation it named was the same learner followed through a cycle with one stroke displaced, asking where the displacement buys most and whether a single named variation every few cycles gives the bell back what the other order takes from it.

It does not, and some variations do the opposite of what a variation is for.

Where the earlier essay left the bell #

The gap this essay tries to close is the one in that figure. With both feels heard nine strokes in ten every order of the gaps keeps a listener placed to within a thousandth of a bit a step, and the bell’s lead on arrival is the whole of the difference between them. From half downward the bell falls behind, and at a quarter it has lost its arriving lead as well. The background is three essays deep. Against a pulse the bell pattern is the easiest to place found the order’s arriving advantage; against both feels the bell pattern’s order stops mattering found it vanish when both pulses sound in full; a feel heard half the time gives the bell half its lead brought it back in proportion to what was missing. And the reason the bell is the most even order at all — spreading its onsets as evenly as possible — is the same reason its two short gaps look alike, which is what costs it in a thin texture.

That makes the variation question sharper than it first looks. A variation does not change the bell’s order; it adds, once in a while, a cycle that is not the bell. Whether that helps depends on whether the inserted cycle tells the listener something the plain cycles cannot, and the next sections count it.

A timeline several cycles long #

A variation every few cycles makes the bell’s timeline longer than one cycle. If one stroke is moved every third cycle, the timeline is thirty-six steps long: two plain cycles and one varied. The listener here is the earlier essay’s — it carries a belief about where in the timeline it is, charged six nats for every hypothesis a stroke contradicts, each step’s evidence fading to half over three steps, and it learns from the strokes how often each pulse is lost — and it knows the varied timeline as it knew the plain one. What it does not know is where in the thirty-six steps the music began.

What matters to a dancer or a player is their place in the twelve-step cycle, so that is what is scored: the belief over the long timeline is summed over every position that falls on the same step of the cycle, and the measure is the earlier essay’s surprise — minus the log of the belief on the true step — averaged over the twelve ways the bell sits against the pulses and over thirty seeded draws of which strokes are lost. With no variation at all this is the earlier essay’s computation, and it gives its numbers: with both pulses heard half the time, the plain bell is placed with 0.944 bits of surprise a step in its first cycle and 0.0078 a step afterwards, and 2 2 2 1 2 1 2 with 1.021 and 0.0016.

Ten ways to move one stroke #

A displacement here moves one of the bell’s seven strokes by one step into an empty neighbour. There are ten such moves, and they turn out to fall into three kinds according to what the varied cycle becomes.

Six of the ten make a figure the plain bell never contains: a gap of three steps, with short gaps around it. The best of them, which moves the bell’s second stroke back beside its last short gap to make the gaps 1 3 1 2 2 2 1, brings the staying cost down from 0.0078 to 0.0055, and the next, 3 1 1 2 2 2 1, to 0.0056. Two others help a little, 0.0062 and 0.0066; two do nothing, 0.0077 and 0.0080.

Two turn the varied cycle into the other order of the gaps, 2 1 2 2 2 2 1 and 1 2 1 2 2 2 2, which are rotations of 2 2 2 1 2 1 2. They make staying worse, at 0.0107.

Two turn the varied cycle into the plain bell begun at another step — 2 2 2 1 2 2 1 and 2 2 1 2 2 1 2 are the bell’s own gaps read from a different stroke. They make staying more than three times worse: 0.0252 and 0.0253.

No displacement every third cycle comes near the other order’s 0.0016. The best recovers about a third of the gap between the plain bell and its rival.

A shifted cycle sounds like being lost #

The worst displacements are the clearest result, and the reason is worth drawing out.

The second row puts three strokes on three consecutive steps — the bell’s last short gap followed at once by another — which the plain pattern never does in any cycle, at any alignment. A listener who hears it can recognise the varied cycle as itself, and in recognising it learns exactly where in the cycle it is: it is a named landmark in the sense of two landmarks in a cycle help only if they can be told apart, a point that cannot be mistaken for any other.

The third row is the opposite. Moving one stroke turns the cycle into a perfect copy of the bell pattern begun two steps later. To a listener who knows the pattern and is following it, that cycle is indistinguishable from the experience of having misjudged the cycle’s start: everything it hears is consistent with a different place, and nothing in that cycle says otherwise. So every time the variation comes round, a share of the listener’s belief moves to the wrong place, and the following cycles have to move it back. That is what costs three times the plain bell’s staying surprise. A variation that the timeline’s own rotations already contain is not a landmark but a false start.

The two displacements that make the other order sit between. The varied cycle 1 2 1 2 2 2 2 is not a rotation of the bell, but it shares most of its local figures with the bell read from other strokes, and so it partly misleads and partly informs, and on balance it costs. The pattern that keeps a listener best placed when played every cycle is a poor thing to insert once in three, because what helps is not the order’s distinctiveness in itself but its distinctiveness against what surrounds it.

How often a variation must come #

The benefit falls off with the variation’s rarity, almost in proportion. With the best displacement every other cycle the staying cost is 0.0034, which recovers 70 per cent of the gap between the plain bell and its rival. Every third cycle it is 0.0055, 36 per cent; every fourth, 0.0063, 24 per cent; every sixth, 0.0069, 14 per cent. The harm of a false start hardly falls off at all: 0.0234 every other cycle, 0.0253 every third, 0.0243 every fourth and 0.0220 every sixth — each false start costs nearly as much as the last, and the cycles between are not enough to recover.

So the lead’s question has a quantitative answer. A single named variation every few cycles is not enough to give the bell back what 2 2 2 1 2 1 2 takes; a named variation every other cycle very nearly is. And a lead drummer who varies the bell carelessly — moving whichever stroke comes to hand — has a two-in-ten chance of producing a variation that makes the listener’s job three times harder rather than easier.

What every variation costs on arrival #

The earlier essay found the bell’s order is for arriving. Varying it has a price there too.

The plain bell arrives at 0.944 bits a step. Every one of the ten varied timelines arrives worse: the six that make a new figure at 0.957 to 0.994, the two that make the other order at 1.07, the two false starts at 1.03. The best variation for staying, 1 3 1 2 2 2 1, costs 0.013 bits a step on arrival and saves 0.0023 a step for every step after; it repays its arrival cost in about six cycles.

The arrival cost has a simple source. A listener arriving at a varied timeline has to place itself among thirty-six steps rather than twelve, and although only its place in the cycle is scored, the varied cycle is a place where the possibilities it is weighing make different predictions — so until it has heard the varied cycle once, part of its belief is spent on hypotheses about which cycle of the three it is in. A variation is information about place only once it has been heard, and before that it is a source of doubt.

In thinner and thicker textures #

The comparison so far has both pulses heard half the time. The answer depends on how thin the texture is, as the earlier essay’s did.

The proportion holds. With a quarter of the strokes heard, the plain bell’s staying cost is 0.0523 bits a step, the varied bell’s 0.0345 and the other order’s 0.0060: the variation recovers 38 per cent of the gap, against 36 per cent at half. With three quarters heard all three are within a few ten-thousandths of each other — 0.0005, 0.0004 and 0.0003 — and the question has stopped mattering, because a texture that thick keeps any order placed.

So a variation every third cycle is worth about a third of what the better order is worth wherever the difference is worth having, and the thinner the texture the more that third amounts to in bits. In a quarter-heard texture it is 0.018 bits a step, which over twelve steps a cycle is a fifth of a bit of surprise saved every cycle — small against the first cycle’s eleven bits, and accumulating for as long as the texture stays thin. That is the regime in which the bell pattern is slowest only to a perfect memory placed the listener’s memory as the deciding variable, and the three-step half-life used throughout is what makes a variation every third cycle land just as the last one’s evidence has faded.

What this says about varying a timeline #

Three things follow, and they are claims about what a listener can get from a variation rather than about what players intend by one.

The first is that a variation helps a listener keep its place only when it makes a figure the pattern’s own rotations do not contain. A bell player who moves a stroke beside one of the pattern’s short gaps — making a cluster of strokes closer together than the pattern ever puts them — has made a landmark. One who moves a stroke so that the cycle reads as the pattern begun elsewhere has made a false start, and a false start is worse than no variation at all. The rule is the one the rotation the necklace cannot see stated from the other side: a timeline’s rotations are the same necklace and different places, and a listener’s whole task is telling them apart.

The second is that the help is roughly proportional to how often the variation comes. Once every two cycles recovers most of what a better-placed order would give; once every six recovers very little. That is the arithmetic of a landmark that has to be revisited before the listener’s evidence fades — the three-step half-life the listener’s memory has here — rather than anything about the variation itself.

The third is that the bell’s job as the pattern to arrive by is not compatible with being varied often. Every variation makes arriving harder. A texture that wants to place a newcomer quickly is served by the plain bell; one that wants to keep a listener placed through a thin passage is served by frequent, well-chosen variation — or by the other order. A late dancer needs the landmarks, not the rhythm found landmarks worth most to someone arriving late; here a landmark that has to be learned first is worth most to someone who has already arrived.

What the model assumes #

The listener knows the variation. It knows where in the timeline the varied cycle falls and what it contains, as it knows the plain pattern. A listener hearing a variation for the first time would treat the moved stroke as a contradiction of every hypothesis and pay six nats for it, which is the case the false-start figures already describe. Real listeners presumably fall between: they know that a lead drummer varies, not exactly how.

One stroke moved one step, at a fixed place in the timeline. Real variations add and remove strokes, move several, and come at irregular intervals. The ten single displacements are the smallest variations there are, and the finding that some are landmarks and some false starts would hold for larger ones; their sizes would not.

Strokes are lost independently, the six-nat mismatch cost and the three-step half-life are the earlier essays’, and the averages are over thirty draws at each of twelve alignments — enough to reproduce the earlier essay’s plain-bell figures to a few thousandths, and the smallest differences reported here, a few ten-thousandths of a bit a step, are near that precision.

What the arithmetic cannot say #

What players do and why. A lead drummer’s variations answer to the dance, to the other drums and to the music’s own shape, and the bell’s variations in particular traditions are not modelled here. The arithmetic can say which displacements inform a listener and which mislead one; it cannot say that players choose the first kind, or should.

And sound. A real bell has a timbre, and a moved stroke may be played differently — louder, damped, on another part of the instrument — which would mark it as a variation regardless of where it falls. That would turn every false start here into a landmark, by a cue the model does not have.

Still open: a variation that marks itself #

The false starts are false only because the moved stroke sounds like any other stroke. A bell player who plays the varied cycle on a different part of the bell, or damps it, gives the listener a second cue — the variation’s timbre — that says “this cycle is varied” whatever its figure. The computation is this listener with each stroke carrying a label, varied or plain, heard with some reliability, and it would say whether a reliably marked variation every few cycles closes the gap to the other order, and how reliable the mark has to be before even a displacement that makes the bell’s own rotation stops being a false start and becomes a landmark.