Rhythm and metre

Repetition buys least where it is needed most

Every locating figure so far is a listener arriving — the uncertainty averaged over the first cycle heard. Cyclic music comes round dozens of times, and the same model already carries the answer for a listener who has settled: a floor of uncertainty that nothing had read. At two seconds a cycle the repetitions close the whole gap. At sixty they close eight per cent for a single timeline and twelve for a layered code. A slow cycle is worse on the first hearing and gains less from the second, and the two disadvantages compound.

14 min read 5 figures Time is a circle The limits of the listener The second time is different

Assumes: The cycle that outruns the memory · A cycle that says where it is

The essay that put a clock into the comparison put a clock into a comparison that had none, and found that a listener’s memory span decides which cyclic design locates them: at a clave’s two seconds the memory covers twenty-eight steps and forgets nothing, and at a gong cycle’s forty it covers 1.4 and forgets almost everything.

Every figure in it is a listener arriving — the position uncertainty averaged over the first cycle heard, which is what somebody walking into the room has. A cycle cannot cadence, so there is no first time and no last time to arrive at, and a listener’s entry point is genuinely arbitrary. Cyclic music is not heard once. A gong cycle comes round dozens of times in a piece and a clave hundreds — the repetition rhythm is a circle takes as the starting point — and a listener who has been hearing one for two minutes is not the listener that essay modelled.

The model already had the answer and nobody had read it. The decaying position code’s rows settle: after enough steps the uncertainty stops falling and sits at a floor, which is what is left when every piece of evidence old enough to be forgotten has been. The arrival figure is an average over the first cycle; the floor is where a listener who has stopped arriving lives.

What a second hearing is worth

A cycle already known, against a cycle just arrived at. How many bits of uncertainty about position a listener has, against how long the cycle takes, for a single timeline and for a layered colotomy — each drawn twice, once as a listener arriving and once as a listener who has been hearing it long enough to settle. At 1.6 seconds a cycle the timeline goes 0.94 bits arriving and 0.00 settled, and the colotomy 0.71 and 0.00; At 16 seconds a cycle the timeline goes 1.10 bits arriving and 0.08 settled, and the colotomy 0.93 and 0.23; At 60 seconds a cycle the timeline goes 2.47 bits arriving and 2.27 settled, and the colotomy 2.14 and 1.89. The gap between each pair is what the repetitions are worth, and it narrows as the cycle slows. The two designs are drawn at their own step counts rather than at equal strokes, so the levels here are not the earlier ones and the gaps are. — Fig. 1 Bits of uncertainty about position, against how long the cycle takes, for a single timeline and for a layered colotomy — each drawn twice, arriving and settled. The gap between each solid line and its dashed partner is what the repetitions buy.

At a cycle of two seconds the single timeline’s listener arrives at 0.94 bits and settles at zero. The whole of the uncertainty goes, because a two-second cycle fits inside a three-and-a-half-second memory and nothing about it has to be forgotten. The gap is the largest it ever gets, and it is the case where it matters least — a listener who is already almost certain gains the little that is left.

At sixty seconds a cycle the same listener arrives at 2.47 bits and settles at 2.27. The repetitions buy two tenths of a bit. The cycle’s steps are nearly four seconds apart, a memory that halves every three and a half seconds has forgotten the previous stroke before the next arrives, and hearing the thing forty times does not change that. Every hearing is the same first hearing.

That is the finding and it is the opposite of what repetition is usually credited with. A design that is hard to locate in on one pass does not recover on the next, because the reason it is hard is that the evidence is spread over more time than the listener can hold — and holding is not something practice supplies.

Bits are the wrong unit for the comparison, because a fast cycle has less to gain and a slow one has more. The quantity that compares them is the share of the available gap that the repetitions close: how far a settled listener gets from where they arrived, against how far a listener with a perfect memory would get.

Fig. 2 What the repetitions buy, as a share of the gap a perfect memory would close, against cycle duration. Both designs start at all of it and both fall away.

Both designs close the whole of the gap up to about eight seconds a cycle, which is where the memory span still covers a useful share of the pattern. From there they fall: at sixteen seconds the timeline closes 93 per cent and the colotomy 75; at thirty-two, 31 and 36; at sixty, 8 and 12.

The crossing in the middle of that is the essay’s second result and it is worth separating from the first. Past about thirty seconds a cycle, the layered code closes the larger share. The single timeline starts ahead — its evidence is denser per step — and falls faster, because a line’s information is in the relation between successive strokes and successive strokes are exactly what a decaying memory loses.

A colotomy’s information is not in relations. It is in which instruments are sounding now: a stroke of the gong is a stroke of the gong whether or not a listener remembers the last one. So a colotomy’s evidence is local and a timeline’s is not, and local evidence survives forgetting.

A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players. — Fig. 3 The layered design the comparison uses: several isochronous parts at nested periods, so that the position is in which instruments coincide rather than in what has been heard before. the third essay is where that code was built.

That is a claim about design rather than about memory, and it fits the third essay’s own account: a colotomic cycle says where it is with nothing asymmetric in it, and its position is legible from a single stroke. Something available from a single stroke does not need a memory, so nothing a memory does to it can take it away.

Why this does not overturn the fourth essay

the fourth essay compared the two designs at equal strokes and reported a crossing at twenty-three seconds. Nothing here is that comparison and nothing here contradicts it.

The two designs are drawn at their own step counts, so the levels on these figures are not commensurable with that essay’s and are not meant to be: a three-layer colotomy carries more per step than a binary line by construction, and that is a fact about the encoding rather than a finding. What is commensurable is the gap — how much each design gains from being heard again — because that is a comparison of a design with itself.

Where a cycle of 16 at 16, 8, 4 outruns the listener's memory. The residual uncertainty a listener is left with once the evidence has stopped accumulating, against how long one turn of a 16-step cycle takes. The listener's memory of a step halves after 3.5 seconds throughout; what changes is how many steps that is. At a cycle of 1.6 seconds it is 35 steps and every design reaches certainty, which is the regime a clave is played in. At 60 seconds it is 0.93 steps and none of them does: layers at 16, 8, 4 settles at 1.89 bits, son clave settles at 2.27 bits, the best single line of 7 settles at 1.85 bits. That is the range a gong cycle occupies, and it is the design that wins there. — Fig. 4 The fourth essay’s own figure: the two designs at equal strokes, against cycle duration, for a listener arriving. Its crossing is the one the essays here have recorded, and everything on this page is about the vertical distance between an arriving listener and a settled one rather than about which line is lower.

So the two essays answer different questions about one model. That one puts the two designs on a shelf and asks which is lower; this one asks how much each moves when the piece has been going for a while, and finds that they move least where the shelf is highest.

Local evidence and remembered evidence

The crossing above is worth pulling out as a distinction, because it applies past this pair of designs and it is the reason they behave differently under forgetting.

Some evidence about position is local. A gamelan’s gong stroke says “this is the top of the cycle” on its own, without reference to anything a listener heard earlier. So does the coincidence of two layers, and so does a downbeat marked by a change of instrument. A listener with no memory at all still receives it.

Some evidence is relational. A clave’s third stroke says where a listener is only against the first two: rhythm is a circle and the bar line is a choice, and what locates a listener on that circle is the pattern of gaps rather than any one of them. Relational evidence requires the earlier terms to still be there.

A decaying memory is precisely a filter on relational evidence and not on local evidence. That is why the two curves separate as the cycle slows: at two seconds a cycle both kinds survive, and at sixty the relational kind has gone and the local kind is untouched. The timeline’s advantage is entirely relational and the colotomy’s is entirely local, which is why one falls off a cliff and the other does not.

It also says what a design could do about it. A single timeline played on two instruments — one for some strokes and one for the others — turns part of its relational evidence into local evidence, at no cost in strokes. Whether the traditions that use long cycles do exactly that is a question about instrumentation rather than about rhythm, and the answer looks like yes.

What a slow cycle has instead

If repetition does not rescue a forty-second cycle, something does, because gamelan is not music in which nobody knows where they are.

the fourth essay named the candidate and this one sharpens it. Everything measured here is the punctuating skeleton — the gongs, the kenong, the kempul — and the parts that play over it are dense, rule-governed and derived from the cycle position. The skeleton’s floor rises steeply exactly where the elaboration gets denser, which is what makes the elaboration a second position code with a job rather than an ornament. A cycle that says where it is built the first of those codes and named the second.

This essay adds one thing to that argument. If the skeleton’s floor were merely high on first hearing, a listener who stayed would learn their way out of it, and the elaboration would be doing something else. The floor is what a listener reaches after learning, so the elaboration is not covering a transient. That reading also fits what the essay that found a cycle has no first time and no last found about what cyclic music uses instead of closure: density, computed from the onsets and nothing else. It is covering a permanent deficit, and the prediction is correspondingly sharper: the density of the elaborating parts should track the settled floor rather than the arrival figure, and those two differ by a factor of two at the slow end.

Locating layers at 16, 8, 4 in a cycle of 16, with a memory that fades. How many bits of position are still unknown, against how many consecutive steps have been heard, for listeners whose memory of a step halves after 8, 4, 2 steps, against one who never forgets. Every curve starts at 4.00 bits, which is the whole cycle. The perfect-recall curve reaches zero and stays there, which is what a locating length is. The others do not: they settle at a floor of 0.005 at a half-life of 8, 0.155 at a half-life of 4, 0.712 at a half-life of 2, because evidence old enough to have been forgotten has stopped ruling anything out. Certainty is a property of the memory rather than of the pattern. — Fig. 5 The uncertainty falling toward its floor for one pattern at four rates of forgetting, which is the object every figure above reads two numbers from. With no forgetting it reaches zero; with any forgetting at all it stops somewhere.

Where the floor comes from

The floor is the quantity this essay is built on and it is worth saying plainly what it is, because “the uncertainty a listener settles at” sounds like a statement about a person and is a statement about arithmetic.

A listener’s posterior over where they are in the cycle is built from the steps they have heard, each discounted by its age. A step heard one half-life ago counts half; two half-lives, a quarter. Run that forward and the sum converges: the recent steps carry nearly all of it and the old ones carry a vanishing share, so after enough steps the posterior stops changing shape. The floor is the entropy of that converged posterior, and it is a property of the pattern and the half-life together.

The important consequence is the one the figures rest on. The floor does not depend on how long the listener has been there, so hearing the cycle a hundred more times moves it by nothing at all. A listener at the floor after four cycles and a listener at the floor after four hundred are the same listener, and the only difference repetition makes is getting there.

That makes the gap between arrival and floor a complete account of what repetition buys within this model. It is not a first instalment of a longer learning curve; it is the whole curve, and it is over quickly. At two seconds a cycle a listener is at the floor within the first cycle; at sixty they are at it within about two, and the floor is high.

Which is why the deficit is permanent rather than transient, and why the elaboration argument above has the shape it does. A model in which the floor fell with exposure would predict that a long cycle needs help at the start of a piece and not later. This one predicts it needs help throughout, which is what the parts that play over a colotomy actually do.

Which computation produced the numbers

The position code is one recursion: a listener holds a posterior over rotations of the cycle, each step’s evidence discounted by its age at a stated half-life, a mismatch costing a fixed penalty rather than being fatal, and the answer is the entropy of the posterior in bits. At an infinite half-life it reproduces the ideal observer’s count row for row, which is the check the model carries.

The arrival figure is that entropy averaged over the first cycle’s worth of steps. The floor is the value it converges to, which is not zero for any finite memory. The gap between them is the only new quantity here.

The memory is fixed in seconds — three and a half — and the steps are not, so the half-life in steps is the memory divided by the step duration, and the step duration is the cycle’s length over its step count. That is the fourth essay’s conversion and it is carried unchanged, which is why the horizontal axis is the same one.

The single timeline is the son clave, sixteen steps with five onsets. The colotomy is three isochronous layers at periods of sixteen, eight and four, and its symbols are which layers strike — the third essay’s own construction.

Where the model stops

Learning is not the same as settling. The floor here is what a decaying memory reaches on a stream it has no stored model of. A listener who has heard a gamelan piece a hundred times has a stored model, and a stored model is not a decayed memory — it is a prior, and a prior changes the posterior in a way this arithmetic has no term for. What is computed is the limit of listening, not the limit of knowing.

That is the most important limit on the page and it cuts against the finding. If a listener can learn the pattern as an object, the slow cycle’s disadvantage is a first-hearing problem after all, and this essay has measured the wrong thing. What the arithmetic can say is that the two are different mechanisms and that only one of them is in the model.

And the memory half-life is one number. It is not; a memory for a stream of events has several time constants, and the shortest of them is the one this model uses. Three and a half seconds is the middle of the psychological present as the essays here carry it, and the figures would look materially different at two or at eight — the crossing between the two designs moves with it, and the level at which both collapse moves with it too. What does not move is the direction: a fixed memory in seconds against a cycle measured in steps produces a falling curve at any constant.

The penalty for a mismatch is a convention. A step that disagrees with a candidate rotation costs six units rather than ruling it out, which is what makes the posterior soft and the entropy finite. Raising it sharpens every curve toward the ideal observer’s and lowers every floor; the fourth essay chose the value and this essay keeps it, so the two sets of numbers are on one scale.

What the picture cannot show

It cannot show a listener tapping. The cycle that outruns the memory converted a memory span into a number of steps and this essay reads the same conversion twice; neither has a motor period in it.

It cannot show the tempo changing. A gamelan piece moves through tempo levels, so the same cycle is forty seconds in one section and twelve in another, and a listener carrying a code learned at twelve into a section at forty is a case this model has no state for.

Nor a piece with an ending. A cycle cannot cadence is the essay that found cyclic music has neither a first time nor a last, which is what makes the settled figure the right one to read rather than a limit nobody reaches.

Nor the entrainment. A listener does not merely receive strokes; they tap, sway or count, and a motor period is a memory of a kind that decays very differently from an evidence trace. Everything here treats a listener as a receiver rather than as somebody keeping time.

And it cannot show what being lost feels like. Two bits of uncertainty over sixteen positions is a listener who has narrowed it to four, which is not the same as no idea and is not the same as knowing. Nothing here maps a number of bits onto an experience, and the mapping is the part a reader most wants.

Still open: whether a prior can be given a number

The one limit above that is worth attacking rather than recording is the difference between settling and learning, because it decides whether this essay’s finding is about cyclic music or about strangers to it.

A prior over rotations is exactly the kind of object the model already uses — the posterior is a distribution over rotations, and a listener who knows the piece starts from a sharpened one rather than a flat one. Giving the model a prior costs nothing in implementation: begin the recursion with a distribution concentrated by however much a previous hearing left, and read the floor again.

What it costs is a number nobody has. How much does one hearing of a forty-second cycle sharpen a listener’s expectation of where the next gong falls? That is a measurement, it is a small one, and it is the kind a listening experiment can make directly — present a cycle a stated number of times, interrupt it, and ask where in the cycle the interruption came. Run over a range of cycle durations it would say whether the deficit this essay computed is real for an experienced listener or is an artefact of modelling every listener as a newcomer with a good ear and no history.