A feel heard half the time gives the bell half its lead

The standard bell pattern — twelve steps, gaps 2 2 1 2 2 2 1, the most even arrangement of those gaps and the one spreading seven onsets as evenly as possible produces — was placed against the other two cyclic orders of its own seven gaps in against a pulse the bell pattern is the easiest to place, and the result was lopsided. Heard with a pulse every three steps, a listener with a memory of a quarter of the cycle is still missing 0.560 bits of position over the first cycle with the bell pattern, and 0.734 or 0.738 with either rival: a lead of 0.174 bits. Against a pulse every four steps the lead is 0.107. The bell pattern’s order of gaps looked chosen for how it sits against a beat.

Against both feels, the bell pattern’s order stops mattering then put both pulses under it together, as the ensembles that play it do, and the lead vanished. The two pulses coincide once a cycle and are a position code of their own — the resultant of two even pulses, with each stroke’s owner audible. With both sounding, the three orders are within fifteen thousandths of a bit, and the bell pattern is fractionally second. That essay ended with the question this one answers. An ensemble is never quite one texture or the other. Drums drop out and come back, a dancer marks a feel some of the time (a late dancer needs the landmarks, and gets them only when someone plays them), and a listener in a crowd hears some strokes and loses others. So at what share of the weaker feel does the bell’s order start to matter again?

Two answers would each have meant something definite. If the lead reappeared only when the weaker feel was nearly silent, the bell pattern’s order would be a property of introductions and thin textures. If it reappeared as soon as a feel was intermittent, the full texture of an ensemble would be, to a listener, mostly a one-feel texture, and the order would be shaped for the music as it is usually heard.

It is neither, and the answer is simpler than both. The lead comes back in proportion to the share of strokes lost. Half the strokes heard gives back half the lead — 0.080 of 0.174 when the pulse in fours thins, 0.055 of 0.107 when the pulse in threes does — and the share it gives back stays between 0.47 and 0.55 at every memory from a step and a half to a whole cycle.

A silence that might be a lost stroke #

The listener of the two earlier essays hears, at each step, a symbol: does the bell sound, does the pulse in threes, does the pulse in fours. They hold a belief over the twelve places in the cycle they might be, and at each step every place whose symbol disagrees with what was heard is charged six nats. That is enough to make it very unlikely without ruling it out. Every step’s evidence fades, halving every three steps. The cost is the entropy of the belief — how many bits of position are still unknown — averaged over the first cycle heard and over the twelve ways the bell can sit against the pulses — the rotations that the necklace cannot see and a listener must.

That listener needs one change to hear an unreliable pulse, and the change is about silence. With every stroke sounding, a silence where the hypothesis expected a pulse stroke is as strong as any other evidence. With only a share $\rho$ of strokes heard, the same silence is what a correct hypothesis produces a fraction $1-\rho$ of the time. A listener who knows the share charges it $-\ln(1-\rho)$ nats rather than six — 0.69 at half, 2.3 at nine strokes in ten, and six again only once fewer than one stroke in four hundred is lost. A stroke heard where none was due is still a full mismatch, because strokes are dropped by players and listeners and are not invented.

The first check is that this reduces to the old listener when nothing is lost. At $\rho = 1$ the silence costs six nats again, and the new calculation reproduces the old one exactly, to every printed digit, at every order and alignment.

The expectation is then taken over which strokes are lost. It does not need sampling. In the first cycle a listener hears at most seven pulse strokes, four in threes and three in fours, so there are never more than 128 patterns of loss. Each is weighted by $\rho$ for every stroke kept and $1-\rho$ for every one lost, and all are enumerated. A seeded sample of 150 random loss patterns at one alignment gives 0.509 bits against the enumeration’s 0.510, a second route that agrees.

Every order gets harder, and the bell gets harder slowest #

The dial holds the pulse in threes steady and thins the pulse in fours. At every stop, every order of the gaps is harder to place than it was, since a listener has less to go on. What changes with the share is how fast each order degrades.

With both pulses in full, the three orders sit at 0.425, 0.422 and 0.437 bits. At half the fours they are at 0.513, 0.593 and 0.613. With the fours silent they are at 0.560, 0.734 and 0.738 — exactly the earlier essay’s figures against the pulse in threes alone, which is the second check the calculation has to pass. The bell pattern’s cost rises by 0.135 bits across the whole range; its rivals’ rise by 0.30 and 0.31.

Read the other way, those numbers say what the pulse in fours is worth to each order. Adding the fours in full to the pulse in threes saves a listener 0.135 bits with the bell pattern and 0.30 or 0.31 with either rival. So the fours tell a listener with the bell pattern less than half of what they tell a listener with a rival order, because much of what the fours say about position the bell pattern’s order has already said. The two are redundant, and the rivals’ orders are not redundant with the fours in the same way. The bell does not get better as the fours go; it loses least, because it was already carrying much of what the fours carry. Which step of the cycle that shared information lives on is a question the averages do not answer, and the per-alignment breakdown the earlier essay drew is where to look for it.

Why the return is a straight line #

A lead that grows in proportion to the loss is the least dramatic shape a curve could have. It is also, looked at the right way, the expected shape rather than a coincidence of these numbers.

Start from the case where it is exact. If a feel dropped out for whole cycles at a time rather than stroke by stroke — the drums rest, then come back — and a listener knew which kind of cycle they were in, every cycle would be either a two-feel texture or a one-feel texture. The average cost would be the two textures’ costs weighted by the share. So would the lead: the straight dashed line on the first figure, exactly.

Stroke-by-stroke loss is not that. In a single cycle, some strokes of the fours arrive and some do not, and the listener cannot know in advance which. Yet each stroke that arrives does nearly the whole of its job, because a stroke heard is as good as it ever was: nothing about losing its neighbour makes it less informative. And each stroke that is lost costs nearly what it would cost if the feel were absent, because the knowing listener charges its silence almost nothing at low reliability. So each stroke is close to an independent coin toss between the two textures, and a coin toss per stroke averages to the same straight line as a coin toss per cycle, to first order.

The curves do depart from their lines, slightly and in opposite directions. When the fours thin, the curve runs just under its line: 0.080 at half against 0.087 on the line. When the threes thin, it runs over its line at high reliability — 0.012 at nineteen strokes in twenty against 0.002 on the line — and then joins it. So the first few lost strokes of the threes give back a disproportionate share of the lead, and the first few of the fours slightly less than their share. The model does not say why the two feels differ in this way, and nothing below depends on it. What matters is the size of the departures: they are small beside the lead itself, and neither is a threshold.

The same half at every memory #

The proportion would be a curiosity if it held only at one memory, and the memory is the number these essays have been most careful never to fix. The bell pattern is slowest only to a perfect memory found that a timeline’s ranking among its orders reverses as memory shortens, and every later result has been stated at more than one.

Here the whole lead changes a great deal with memory. With the fours gone, it is 0.248 bits at a half-life of a step and a half and 0.077 at a whole cycle, because a listener who remembers a whole cycle has less need of any particular order. But the share of it returned by half the strokes moves only between 0.47 and 0.55 when the fours thin, and between 0.51 and 0.54 when the threes do. The proportion is a property of the texture, not the listener.

So the question the earlier essay posed has a definite answer, and it is a quantitative one. The order of the bell pattern’s gaps matters in an ensemble to the extent that a feel is missing, strokes and cycles alike. A performance in which the pulse in fours is audible to a listener three strokes in four gives the bell pattern a quarter of its one-feel advantage. One in which it is audible one stroke in four gives three-quarters. There is no stretch of reliability where the order stops mattering all at once, and none where it starts all at once.

The listener who takes silence as proof #

Everything above belongs to a listener who knows how reliable each feel is. The alternative is the listener of the earlier essays applied unchanged: one who charges every silence six nats, as though a lost stroke were impossible. That is the listener who has not noticed that the drummer sometimes rests.

Measuring that listener needs a second number, because the one used so far flatters them. Entropy is how uncertain a listener feels. A listener who treats silence as proof rules places out too fast, and a belief concentrated on too few places has low entropy whether or not the true place is among them. So alongside the entropy the figure draws the surprise: $-\log_2$ of the belief the listener places on where they actually are. For a listener whose model is right the two agree on average; for one whose model is wrong they part.

They part at once. At nine strokes of the fours in ten, the listener who takes silence as proof feels more certain than the one who allows for loss — 0.442 bits of uncertainty against 0.454 — and is further from the truth: 0.457 bits of surprise against 0.431. Confidence and accuracy go in opposite directions. By half the strokes the gap is 0.678 against 0.485, and with the fours silent, a listener who still expects them does worse than one who never heard of them: 0.991 bits of surprise. The bell is played against several beats at once, and this listener misreads exactly the one of them that comes and goes. That is the same trap a crowd costs a late dancer the silences, not the steps found for landmarks in an additive metre. A silence is the most useful evidence a listener has while it is reliable, and the most misleading once it is not.

For this listener the bell pattern’s order matters much more. Measured by surprise, the bell leads its rivals by 0.148 bits at half the fours, against 0.045 for the listener who allows for loss. The bell pattern’s two short gaps are what stop a naive listener from confidently choosing the wrong place when a missing stroke points them there. Measured by the uncertainty they feel, the same naive listener sees the lead return only a third as fast. They cannot see that the bell is protecting them.

The other timeline in twelve #

The standard bell pattern is one of two named twelve-step timelines, and the other is fume-fume, five onsets in twelve with gaps 2 2 3 2 3. Its orders were placed against the same grid in the earlier essay, and it too lost most of its lead against both feels, though not all: it keeps 0.012 bits. With the fours at half it leads by 0.100 against 0.178 with the fours gone, and with the threes at half by 0.055 against 0.107. Measured from its small two-feel lead, that is again about half, 0.53 and 0.45. Two timelines with different numbers of onsets and different gaps return the same share. That is the best available evidence that the proportion comes from the grid rather than from either timeline.

What the model assumes about losing a stroke #

Strokes are lost independently, at a fixed rate. A real ensemble’s losses are neither. A drummer resting drops strokes in runs; a listener’s attention lapses in runs; a loud passage masks a feel’s weaker instrument for bars at a time. The argument above says why that matters less than it seems. Losses in whole cycles give the straight line exactly, and independent losses give nearly the same line, so correlated losses in between should land between the two. But that is an interpolation, and a loss pattern tied to the bell itself — a feel that drops exactly where the bell is loudest — would be a different calculation.

The listener knows the rate. The knowing listener has been given $\rho$. A real listener would have to estimate it from the strokes they hear, in the same first cycle they are trying to place themselves in. Such a listener sits between the two drawn here. They start close to the one who takes silence as proof and move toward the one who allows for loss as the evidence of loss accumulates.

The measure has a bias the surprise exposes. Even the listener who allows for loss is not perfectly calibrated: at full reliability their uncertainty is 0.425 bits and their surprise 0.405. The mismatch cost of six nats makes them slightly underconfident about evidence that is in fact certain. Measured by surprise rather than entropy, the leads in this essay are about half as large: 0.092 bits with the fours gone rather than 0.174. The proportion survives the change of measure — half the strokes return 0.52 of the lead by surprise — but every absolute number in these essays inherits the choice of six nats, and the choice is a stated convenience rather than something measured.

The texture is the one the earlier essays used. A pulse every three steps and a pulse every four, both aligned to the cycle’s first step, with the bell pattern slid against them. Ensembles also sound feels in sixes and twos and on offbeats, and the grid’s position code — which is what makes the bell redundant with both feels present — depends on which feels are sounding.

Still open: whether the bell’s order is for arriving or for staying #

The result closes the question the last essay asked and opens a sharper one. If the bell pattern’s order is worth exactly as much as the feel that is missing, the order is insurance, and insurance is worth most to the listener who needs it most. That listener is the one this essay found worst placed: the one who does not know how often the drummer rests. Measured by what they actually get wrong, that listener gains three times as much from the bell’s order as the one who allows for loss.

Still open, then: whether a listener learning the reliability over the first few cycles — starting naive and becoming knowing as silences accumulate — is served by the bell’s order mostly in the first cycle, when they are still naive, or throughout. The calculation needs the listener’s estimate of $\rho$ carried forward alongside their belief about position, which is a second quantity inferred from the same strokes. If the bell’s advantage concentrates in the first cycle, the order is for arriving; if it persists, it is for keeping one’s place in a texture that never settles.