Rhythm and metre

The bell pattern's order is for arriving

Against a feel that comes and goes, the standard bell pattern is the quickest order of its gaps to place, and the lead grows the less a listener knows about the drummer. Followed past the first cycle, the lead is gone, whether or not the listener ever learns how often strokes are lost. And when both feels thin, a different order keeps a listener placed better than the bell does, because its two short gaps make one figure where the bell's make two that look alike.

14 min read 6 figures What the listener supplies Time is a circle The limits of the listener

Assumes: A feel heard half the time gives the bell half its lead · Against both feels, the bell pattern's order stops mattering

Every listener in the essays on the bell pattern has been followed for one cycle, and every one of them has been told something no listener is told.

The line of argument runs through three essays. Against a pulse the bell pattern is the easiest to place found the standard order of the bell’s gaps the quickest to place against one feel; against both feels, the bell pattern’s order stops mattering found the advantage gone when both feels sound in full, because the two pulses together are a position code of their own. A feel heard half the time gives the bell half its lead then gave its listener a number: the share of a pulse’s strokes that actually reach the ear. With it, a silence where a stroke was due can be weighed correctly — as strong evidence when strokes are rarely lost, as weak evidence when they often are — and the standard bell pattern, gaps 2 2 1 2 2 2 1 in twelve steps, turned out to lead the other orders of its gaps in proportion to how much of one feel was missing. It also found the opposite listener, who takes every silence as proof, more confident and worse placed.

A real listener is neither. Nobody announces how often the drummer rests. The share has to be inferred from the same strokes the listener is using to find their place, and the earlier essay closed on that:

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.

Both halves of that can be computed now, and the answer is the first. Then the question turns round, because keeping one’s place has a best order too, and it is not the bell’s.

A listener who learns the drummer

The new listener carries two beliefs. The first is the familiar one: where in the twelve-step cycle they might be, charged six nats for every hypothesis that contradicts a stroke heard, with every step’s evidence fading to half over three steps. The second is new: how often the pulse in fours actually sounds.

That second belief is kept as a running count — strokes heard where a stroke was due, and silences where one was due — and “where a stroke was due” is itself weighed by the first belief, so a listener unsure of their place is unsure what each silence means. The estimate starts at every stroke is heard, held with the weight of four strokes’ evidence, so the listener begins as the one who takes silence as proof and is talked out of it as the evidence comes in. It never fades. Where one is in a cycle is a matter of the last few steps; how reliable a drummer is, is a fact about the whole evening.

A silence is then charged by the current estimate, exactly as the told listener charges it by the truth. With the estimate frozen, this is the listener who never learns; with it set to the truth, it is the told listener of the earlier essay.

Following a listener for six cycles rather than one rules out exact enumeration — there are up to 2²⁸ patterns of lost strokes — so the expectation is taken over forty seeded draws at each of the twelve alignments. The check is that the told listener’s first cycle, sampled this way and measured the earlier essay’s way, must reproduce its exact figures, and it does: 0.612, 0.591 and 0.513 bits against the exact 0.613, 0.593 and 0.513. The averages are over all twelve alignments of bell against pulses, which are the rotations the necklace cannot see and a listener has to.

The measure throughout is surprise — minus the log of the belief placed on where the listener actually is — rather than how uncertain they feel. The earlier essay showed why: a listener who over-trusts silence can feel certain and be wrong, and only surprise sees it.

The first cycle is nearly everything

With the pulse in threes heard in full and the pulse in fours half the time, the three listeners are placed like this.

Where a listener's uncertainty about the bell's place is spent. The surprise at where the listener actually is — minus the log of the belief placed on the true step — in bits, averaged over each cycle heard and over every alignment of the standard bell pattern against a pulse in threes heard in full and a pulse in fours heard half the time; memory half-life 3 steps. A listener who told how reliable the feel is: 0.485, 0.000, 0.000, 0.000, 0.000, 0.000. A listener who learns it from the strokes: 0.593, 0.000, 0.000, 0.000, 0.000, 0.000. A listener who takes every silence as proof: 0.677, 0.005, 0.004, 0.004, 0.004, 0.004. For every listener the first cycle is nearly everything. Placed once, all three keep their place, and a listener who never learns that strokes go missing is surprised by only 0.005 bits a cycle afterwards. — Fig. 1 Each listener’s surprise at where they are, cycle by cycle, on a logarithmic axis, for the standard bell pattern with the pulse in threes in full and the pulse in fours half the time. All three pay nearly everything in the first cycle. Afterwards the told and learning listeners are placed to under a thousandth of a bit; only the listener who never learns keeps paying, at half a hundredth.

The told listener pays 0.485 bits a step over the first cycle, the learning listener 0.593, the listener who takes silence as proof 0.677. The learner sits between the two, which is where it should be: it begins close to the naive listener and spends the cycle becoming less naive.

After that first cycle the bill almost stops. From the second cycle on, the told and learning listeners are each surprised by less than a thousandth of a bit a step, which in the figure sits on the floor. The one who never learns keeps paying, but only 0.004 or 0.005 bits a step — a hundredth of what it paid on arrival.

The reason is the feel that is heard in full. Once a listener is placed, every stroke of the pulse in threes, together with the bell, confirms the place; the lost strokes of the fours only have to be not contradicted, and a silence can never contradict a place that has just been confirmed. Arriving means choosing among twelve places; staying means keeping one, and the evidence a listener needs for that is far less.

The bell’s lead goes with it

The bell pattern's lead is for arriving. The standard bell pattern's lead over the nearest other order of its own gaps, in bits of surprise a step, cycle by cycle, with the pulse in threes heard in full and the pulse in fours half the time. A listener who told how reliable the feel is: 0.043, 0.000, 0.000, 0.000, 0.000, 0.000. A listener who learns it from the strokes: 0.063, 0.001, 0.000, 0.000, 0.000, 0.000. A listener who takes every silence as proof: 0.144, 0.006, 0.005, 0.005, 0.004, 0.004. The lead is all in the first cycle, for every listener; after it the three orders keep a listener placed equally well. The listener who has to learn the reliability gets more from the bell than one who is told it and less than one who never learns. — Fig. 2 The bell pattern’s lead over the nearest other order of its gaps, cycle by cycle, for the three listeners, with the pulse in fours heard half the time. The whole of it is in the first cycle.

In the first cycle the bell pattern leads the nearer of its two rivals by 0.043 bits a step for the told listener, 0.063 for the learner, and 0.144 for the listener who never learns. After it the lead is below a thousandth for the first two and about 0.005 for the third — and in every case the later cycles together add up to a small fraction of the first.

So the earlier essay’s question has a clear answer. The order of the bell pattern’s gaps is for arriving. It is worth something to every listener in the cycle in which they find their place and almost nothing in the cycles after, as long as one feel is reliable enough to hold them there.

That fits the way the pattern is used. The bell sounds against several beats at once rather than as one layer of a polyrhythm, and it is the part that starts first, the one the other players and dancers enter against, and in a late dancer needs the landmarks the question was always how quickly someone arriving mid-stream could find the cycle. The arithmetic says that is the job the order does.

Learning is slow, and it does not need to be fast

The learner’s estimate of the drummer is worth watching, because it moves far more slowly than its place in the cycle does.

Fig. 3 The learning listener’s estimate of the share of the pulse in fours that is heard, cycle by cycle, for true shares of a quarter, a half and three quarters. Every estimate falls each cycle and is still well above the truth after six.

With half the strokes lost, the estimate starts the first cycle at 84 per cent and is still at 59 after six. With three quarters lost, it is at 39 per cent after six cycles against a truth of 25. The count grows by three strokes of the fours a cycle, and four strokes’ worth of misplaced confidence takes a good many of those to wash out.

Yet from the second cycle on, the learner is placed as well as the listener who knows the truth. Placing oneself does not require knowing how reliable the drummer is; it requires only not treating the first silence as proof, and the learner stops doing that within the first cycle, long before the estimate is anywhere near right. A listener who is wrong about the evening’s reliability by a factor of two can keep their place for as long as the music lasts.

That separates two things the earlier essay could not: a listener’s model of the texture and their use of it. The model converges over minutes. What the model is for is already done in seconds.

Who the order protects

The first-cycle lead depends both on how many strokes are lost and on what the listener knows about it.

Fig. 4 The bell pattern’s first-cycle lead against the share of the pulse in fours heard, for the three listeners. It grows as strokes are lost, for every listener, and it is largest for the listener who knows least.

At nine strokes in ten, the three listeners get between 0.007 and 0.016 bits a step from the bell’s order. At a quarter they get 0.066 if told, 0.111 if learning, and 0.236 if they never learn. The less a listener understands the texture, the more the order is worth to them.

That is the arriving result read as a statement about people. The order of the gaps protects exactly the listener who has not yet worked out what a silence means: the newcomer, the dancer who has just come in from outside, the listener in a crowd who has not yet heard enough to know which drum is intermittent. A crowd costs a late dancer the silences, not the steps found the same listener in an additive metre, misled by silences it trusted too far. By the time they have worked it out, they are placed, and the order has done its work. The earlier essay’s naive listener was the same person caught at the moment of greatest need.

When both feels come and go

Everything so far keeps the pulse in threes reliable, and that is what made staying cheap. Take it away — let both feels be heard half the time, as in a thin texture or a noisy room — and staying costs something again.

Fig. 5 The three orders of the gaps with both pulses heard half the time, cycle by cycle, for a listener told the share. The bell pattern is placed soonest. In every cycle afterwards it is surprised more than the order 2 2 2 1 2 1 2, by a factor of about six.

In the first cycle the bell is still the quickest to place: 0.686 bits a step against 0.740 and 0.784. From the second cycle on it is not the best order to keep: it costs 0.0067 bits a step against 0.0012 for the order 2 2 2 1 2 1 2. The other rival, 2 2 2 2 1 1 2, is worse than both at 0.028.

The amounts are small — thousandths of a bit a step against a first cycle of seven tenths — but they accumulate, and the first cycle does not. The bell’s first-cycle advantage over 2 2 2 1 2 1 2 is 0.054 bits a step, and the rival recovers 0.0055 a step in every cycle after, so over about ten cycles the rival has caught up. A listener who is still listening after ten cycles of a thin texture has been better served by the other order.

Fig. 6 Surprise once placed, averaged over cycles two to six, for each order of the gaps, against the share of both pulses heard, on a logarithmic axis. With the feels heard nine times in ten or more the bell keeps its place as well as any order. From half downward it is behind 2 2 2 1 2 1 2, and at a quarter it has lost its arriving lead as well.

The trade depends on how thin the texture is. With both feels heard nine strokes in ten, every order stays placed to within a thousandth and the bell’s arriving lead is the whole story. At three quarters the bell and 2 2 2 1 2 1 2 are level for staying. At half the rival is five times better at staying, and at a quarter the bell has also lost its arriving lead — 0.926 against 0.915 — and is behind on both counts. At a tenth the rival is better by a tenth of a bit a step on arrival and by a factor of twelve for staying.

Why the other order stays and the bell arrives

The difference between the two orders is where their two short gaps sit, and the reason it matters is a rule found earlier in a different place.

The bell pattern spreads its short gaps as far apart as they can go: 2 2 1 2 2 2 1. Each short gap has long gaps on both sides. That is what makes it the most even order — the one spreading seven onsets as evenly as possible produces — and it is what makes it quick to arrive by: from any starting point a listener meets a short gap soon, and each short gap, taken with the pulses, cuts the possibilities sharply.

But the two short gaps look alike. Each is a quick stroke between runs of long ones, and with a memory of three steps a listener sees roughly one of them at a time. When the feels are thin and a stroke of the pulse goes missing near one of them, the listener cannot easily tell whether it was the first short gap or the second. The order 2 2 2 1 2 1 2 does the opposite. Its two short gaps are close enough to be heard together as a single figure — quick, long, quick — that happens once a cycle and looks like nothing else in it.

Two landmarks in a cycle help only if they can be told apart found exactly this in a colotomic cycle: several recognisable points add up like checkpoints only if the listener can tell which one is which, and alike points are worth a fraction of that. Here the “points” are the pattern’s own short gaps, and the rule decides between two orders of the same seven gaps. The bell’s two short gaps are alike landmarks; the rival’s make one named landmark. Alike landmarks spread evenly are the fastest way to narrow down a place from nothing. A named landmark is the surest way to confirm a place already held when the evidence is poor.

This also sharpens the bell pattern is slowest only to a perfect memory, which found that the ranking of a timeline’s orders depends on the listener’s memory. It depends on what the listener is doing as well. An order’s evenness is worth most to someone arriving; its distinctiveness is worth most to someone staying; and the standard bell pattern is the most even order and not the most distinctive.

What the model assumes

Strokes are lost independently. A drummer who rests drops strokes in runs, and a run of silences is exactly what lets a named landmark be heard across a gap and an alike one be confused. Independent losses are the gentlest case for the bell. Losses in runs would probably favour 2 2 2 1 2 1 2 more, but that has not been computed here.

The learner’s prior is a stated convenience. It starts at every stroke is heard, held with four strokes’ weight. A listener who arrives already expecting an intermittent drum is closer to the told listener; one who holds the belief more stubbornly is closer to the naive one. The learner’s first-cycle numbers move with that choice. The conclusion that the bell’s lead lives in the first cycle does not, since it holds for all three listeners.

The mismatch cost, six nats, and the half-life of three steps are the earlier essays’ and are held. The surprise measure is used throughout rather than entropy, because entropy flatters a listener who is confident and wrong, and every absolute number here would be different under the other measure.

Six cycles are sampled, not enumerated. Forty draws at each alignment reproduce the first cycle’s exact figures to two thousandths, and the smallest staying differences reported — a thousandth of a bit between the bell and 2 2 2 1 2 1 2 at three quarters — are at the edge of that precision, which is why that pair is described as level.

What the picture cannot show

Whether any ensemble plays 2 2 2 1 2 1 2 for this reason. It is a legitimate order of the bell’s seven gaps, and it is the one that keeps a listener placed best in a thin texture, but the standard pattern is the standard for reasons that include history, dance and how it sounds against the other parts. A computation about a listener’s place cannot say those reasons are wrong; it can only say which job each order is better at.

And no listener here hears anything but strokes. A real listener arriving in a room hears the timbres that say which instrument is the bell and which the pulse, the dancers, the song and the harmony. All of that is evidence about place, and all of it lowers every number above. What survives is the ordering, which rests on where the short gaps sit.

Still open: whether a timeline’s variations are named landmarks

Players do not repeat the standard bell pattern identically for an hour; lead drummers vary their parts, and some ensembles let the bell player add or shift a stroke now and then. A varied cycle is a cycle with a distinctive figure in it — a named landmark — and the finding here predicts where such variations would do most for a listener’s place: not in the first cycle, where the plain pattern already arrives fastest, but later, in thin passages, where the plain pattern’s alike short gaps are at their weakest. The computation is the same learner followed through a cycle in which one stroke of the bell is displaced, asked where in the cycle the displacement buys most and whether a single named variation every few cycles is enough to give the bell back what 2 2 2 1 2 1 2 takes from it.