Rhythm and metre

The cycle that outruns the memory

A timeline and a colotomy were compared at equal strokes and the comparison had no clock in it. A memory span is a number of seconds and a cycle is a number of steps, so the two only meet through a tempo — and at a clave's two seconds a listener's memory covers twenty-eight steps and forgets nothing, while at a gong cycle's forty it covers 1.4 and forgets almost everything. The single line is the better locator up to twenty-three seconds a cycle and the layered code is better after it, which is very close to where each is actually used.

16 min read 7 figures The limits of the listener Time is a circle What the listener supplies

Assumes: A cycle that says where it is · The listener who forgets

Rhythm is a circle and the bar line is a choice, which is why where am I is a question a listener genuinely has to answer at all. A cycle can say where it is without any of its parts being able to set two designs against each other at the same cost in strokes. A single asymmetric timeline puts the position in the shape of one part; a colotomy puts it in which of several perfectly regular parts sound together. Both were measured the same way, and the answer was a trade: the layered code leaves 2.81 bits unknown after a single step against the best single line’s 3.01, and the single line reaches certainty in five consecutive steps against the layered code’s eight.

That comparison has no clock in it anywhere. Both designs are sixteen steps long, both are scored in steps, and nothing says how long a step lasts.

It matters because the measure’s listener has perfect recall. The residual entropy falls to zero and stays there, and a locating length exists at all, only because a rotation ruled out at step three stays ruled out at step eight — which is a claim about a memory rather than about a pattern. The key-finding ladder put a decay on exactly that assumption and found that forgetting costs a great deal of confidence and very little accuracy. Here it costs something else.

What forgetting does to a locating length

Put the two idealisations on knobs. Each step’s evidence is discounted by its age, halving every so many steps; and a mismatch between what was heard and what a rotation predicts costs a fixed amount rather than being fatal. The posterior over the cycle’s rotations is what the discounted evidence supports, and the answer is its entropy in bits.

At perfect recall and fatal mismatches this is the measure the ladder already has, row for row and to the last decimal. Anywhere else it is not, and the difference is not a matter of degree.

Locating a single line 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 0.000 at a half-life of 8, 0.031 at a half-life of 4, 0.917 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. 1 The son clave, located by four listeners. All four start at four bits, which is the sixteen rotations they are choosing between. The one who never forgets reaches zero at nine steps and stays there, which is the pattern’s locating length. The others do not reach zero at all: they settle at a floor — 0.000 at a half-life of eight steps, 0.031 at four, 0.917 at two — because evidence old enough to have been forgotten has stopped ruling anything out. A listener with a short enough memory never becomes certain of where in the clave they are, however long they listen.

Certainty is a property of the memory rather than of the pattern. A locating length is the number of steps a perfect memory needs; a forgetting listener has instead a floor, which is the uncertainty they are left with once the evidence has stopped accumulating because the oldest of it is leaving as fast as the newest arrives.

That is the quantity to compare two designs on, and it is one number rather than two. The immediacy the layered code was credited with and the certainty the single line was credited with are both special cases of it — the first is the floor for a listener who forgets everything, the second the floor for a listener who forgets nothing.

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. 2 The same four listeners on the layered code — a stroke every sixteen steps, one every eight, one every four. The perfect-recall curve reaches zero at eight steps rather than the clave’s nine, which is the certainty the design was said to be slower at. The floors are 0.000, 0.004 and 0.712, so at every memory span drawn here the layered code is *worse* than the clave rather than better. That is the opposite of what the earlier comparison suggests, and it is measured in steps — which turns out to be the wrong unit.

A memory is in seconds and a cycle is in steps

The comparison above is at a fixed number of steps of memory, and no listener has a memory measured in steps. The window over which a listener holds a stretch of sound as one present thing is a number of seconds, and so is the fading of a musical memory a return has to survive. Both are a few seconds; three and a half is a reasonable figure and is the one this collection uses.

A memory in seconds becomes a memory in steps through the tempo, and there is no tempo in the position measure. Supply one and the two designs stop being comparable in the abstract, because they are not played at the same speed and they are not close.

A son clave cycle lasts about two seconds. A step is 125 milliseconds, and a memory half-life of three and a half seconds is twenty-eight steps — nearly two whole cycles. A slow gong cycle lasts forty seconds or more. A step is two and a half seconds, and the same memory is 1.4 steps: by the time the next punctuating stroke arrives, the one before it is more than half forgotten.

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 bossa-nova pattern settles at 2.38 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. 3 The floor against how long one turn of the cycle takes, for the layered code, the son clave, the perfectly even pattern the bossa uses, and the best single line of the same seven strokes. The listener’s memory halves after three and a half seconds throughout; only the tempo changes. Everything is zero out to about six seconds, which is the whole range a clave is played in — so for that repertoire the perfect-recall measure is not an approximation, it is exact. Past ten seconds the designs separate, and past twenty they separate a great deal.

A clave never faces this problem. Out to a cycle of eight seconds every design here reaches certainty and the floors are all zero or nearly so, which means the entire published account of what a timeline is for — the frontier between evenness and locatability, the son clave’s nine steps against the even pattern’s fifteen — is exactly right in its own repertoire and needed no correction. That is worth saying because it is the more common outcome of putting a parameter back into a model, and it did not happen to the other design.

The crossover is at twenty-three seconds

A colotomy does face it, and past a certain cycle length it faces it better than a single line does.

Below twenty-three seconds a cycle the son clave has the lower floor. Above it the layered code does. At twelve seconds the clave is at 0.008 bits and the layered code at 0.090; at twenty-three they are level at 0.51; at thirty the clave is at 1.07 and the layered code at 0.79; at forty the clave is at 1.64 and the layered code at 1.19.

So the trade the earlier comparison found does not invert and it does not survive either. It relocates. Each design is the better locator over a range of cycle durations, and each range is close to the one its own tradition works in — claves at two to three seconds, gong cycles at tens of seconds. The two designs are not competing answers to one question after all; they are answers to the same question asked at two speeds, and the speed is what the earlier comparison held fixed without saying so.

The reason is worth stating because it is not the reason the trade was originally given. The layered code was said to put its information in an instant, available from one stroke and needing no memory. That is true of a listener arriving at an arbitrary moment, and it is not what a listener who has been present is doing. What a forgetting listener has is a recent window, and the question is how much a recent window discriminates. A single line’s discriminating feature — the son clave’s one gap of two steps — has to be waited for, and at a slow enough tempo the wait is longer than the memory. The layered code’s signatures repeat every four steps, so something discriminating is always recent. It is not immediacy that saves it; it is frequency.

And the coprime finding reverses

The sharpest thing the earlier comparison found was that the traditional nesting is not the best arrangement of three layers. Periods of sixteen, eleven and seven use one fewer stroke than sixteen, eight and four, tell a listener as much from a single step, and reach certainty in six rather than eight. The nesting was priced at two steps of certainty and one stroke, spent on making the layers form a hierarchy the elaborating instruments can subdivide.

With the memory priced too, that bill is smaller than it looked and eventually it is not a bill at all.

Where a cycle of 16 at 16, 11, 7 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, 11, 7 settles at 1.96 bits, son clave settles at 2.27 bits, the best single line of 6 settles at 1.96 bits. That is the range a gong cycle occupies, and it is the design that wins there. — Fig. 4 The coprime arrangement on the same axis. It is the better locator over most of the range — at a cycle of twenty-four seconds it sits at 0.16 bits against the nested set’s 0.55, which is the advantage the earlier comparison found, now measured in the presence of a memory. What it does not do is keep that advantage all the way down.

At forty-four seconds a cycle the nested arrangement overtakes the coprime one, and past that it stays ahead: 1.89 bits against 1.96 at sixty seconds. The mechanism is the one above, applied to the other design. Coprime periods generate more distinct signatures, and more distinct signatures mean each one occurs more rarely; a listener whose memory is under two steps long cannot accumulate evidence across the gaps between rare events, and what they need instead is a signature that recurs. Nesting produces exactly four signatures over sixteen steps, one of them every four steps, which is the arrangement that keeps something discriminating inside a very short window.

A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 11, 7. 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 6 strokes a cycle, 0.38 to the step, spread over 3 players. — Fig. 5 The coprime arrangement drawn out: three isochronous layers at sixteen, eleven and seven, whose strokes almost never coincide. It generates more distinct signatures than the nested set and spreads them unevenly round the cycle, which is what makes it the quicker locator for a listener who remembers several steps and the slower one for a listener who barely remembers the last.

So the design a tradition uses is the better locator in the tempo range that tradition uses, on a measure built to show that it was not. That is a weaker claim than a story about optimal design and it is stronger than the alternative, which was that the nesting is a locating loss paid for out of the hierarchy. The nesting is not a loss at the tempos it is used at.

The best line anybody could have chosen, and the one anybody plays

There is a gap in the comparison above that flatters the single line and it is worth closing, because it changes the size of the result rather than its direction.

The line the layered code has been measured against is the best of every rotation class of seven onsets in sixteen steps — 715 of them, ranked by the same measure and the winner taken. Nobody plays it. It is the pattern that would be chosen by somebody who had computed the census, and at a sixty-second cycle it sits at 1.85 bits against the nested code’s 1.89, which is a tie rather than a defeat.

The patterns anybody does play are a good deal worse than that at any slow tempo. At sixty seconds the son clave is at 2.27 bits and the perfectly even pattern at 2.38, against the nested code’s 1.89 and the coprime one’s 1.96. So the honest statement of the result is in two halves: against the best single line that exists the layered design draws, and against every single line a tradition has actually produced it wins comfortably.

That distinction matters because the census is not a menu anybody was choosing from. A timeline is one arbitrary pattern that has to be learned and taught, and what a tradition arrives at is a pattern that does several jobs at once — it has to be playable, memorable, danceable and distinguishable from its own rotations, and a clustered pattern that would locate a listener instantly is useless as a timeline for reasons that have nothing to do with information. The frontier of usable single lines is much narrower than the census, and every point on it is above the optimum the comparison used.

What the single line is doing instead

None of this makes an asymmetric timeline a worse idea. It makes it an idea with a tempo attached.

How much of a cycle has to be heard before its position is known. A listener who has heard w consecutive steps of a repeating pattern knows the pattern and not the phase, and the number of rotations still consistent with what has been heard is the uncertainty left. Each curve falls from log₂n bits at nothing heard to zero at the length that first tells every rotation apart. The son clave reaches zero at 9 of its sixteen steps and the bossa-nova pattern, which is a rotation of the Euclidean one, at 15. The dashed lines are the Euclidean patterns with the same onset counts, and the even pattern is the slowest to locate in every case. — Fig. 6 The perfect-recall curves for three of the timelines used throughout, with the Euclidean pattern of the same size dashed under each. Every one of these is a clave-speed object: at a cycle of two or three seconds a listener’s memory covers the whole curve and more, so this picture is the whole of what is happening and the floors from the previous figures are all zero. The nine-against-fifteen difference between the son clave and the even pattern is real and it is a difference in how fast certainty arrives, not in whether it arrives.

The thing an asymmetric timeline is good at is being learned. It is one arbitrary pattern of a stated length, and a player who knows it can locate themselves in a fraction of a cycle. That is a completely different asset from the one the layered code has, which is a rule rather than an item — the periods, and that they nest — and it generalises to cycle lengths and instruments a listener has never met.

Six rhythms, all from the same construction. Euclidean rhythms generated by Bjorklund's algorithm, which spreads a number of onsets as evenly as a number of steps allows. The patterns were not transcribed from recordings; they are what the algorithm produces, and the names beside them are where each one is already in use. — Fig. 7 The son clave and the bossa-nova pattern, five onsets in sixteen apiece. The bossa’s gaps are 3, 3, 4, 3, 3 and the son’s are 3, 3, 4, 2, 4: one gap of two, occurring nowhere else, which is the whole of what makes the son locatable. At a two-second cycle that gap is 250 milliseconds away from the onset before it and a listener holds it easily. At a forty-second cycle it would be five seconds away, and by then the listener has forgotten the onset it is a gap from — which is why the floors in the third figure separate exactly where they do.

Which computation produced the numbers

The measure is a strict generalisation of the one the ladder already had, and the generalisation is two parameters.

A listener has heard some number of consecutive steps. Each one is compared against what every rotation of the cycle predicts at that moment, and a disagreement counts against that rotation — weighted by how long ago the step was, halving every so many steps, and costing a fixed amount rather than being fatal. What is left is a distribution over the sixteen rotations, and the reported figure is its entropy averaged over where the listener happened to come in.

Set the half-life to infinity and the mismatch cost high and the distribution is uniform over the rotations still consistent with everything heard, which is the ladder’s own measure exactly. That equality is checked rather than asserted, at every window length, and it is what makes every number here a distance from a published curve rather than a second opinion.

The mismatch cost is held at six throughout, which makes one disagreement worth about a factor of four hundred against a hypothesis — a listener who mishears a step very rarely. Loosening it to two, which is a listener who mishears often, raises every floor and leaves the ordering of the designs at a given tempo unchanged; it is the half-life rather than the mishearing that does the work here.

A colotomy’s cost in strokes is the sum of its layers’ hits, so sixteen-eight-four is seven strokes and sixteen-eleven-seven is six. The single line it is compared against is the best of every rotation class of that many onsets, chosen with hindsight, which is what makes the comparison a comparison rather than a demonstration.

Where the model stops

The half-life is one number and a memory is not exponential. A discount that treats every step alike is the simplest thing with the right shape. A real memory is better for the beginning of a stretch than for its middle, and better for an event that mattered than for one that did not — and in a cyclic music the strokes that matter are exactly the ones the code is made of, which would help the layered design and by an amount nothing here computes. The hierarchy above the bar is the same objection from the other side: a listener who has a metre is not weighting the last few seconds equally either.

Three and a half seconds is borrowed rather than measured. It comes from this collection’s own account of the perceptual present, which is a range from two seconds to eight. At two seconds the crossovers move earlier and at eight they move later, by roughly the ratio; the existence of a crossover does not depend on the number and its position does.

Every stroke is still the same stroke. A gong ageng and a kenong differ enormously in timbre, register and decay, and a listener telling them apart is doing something much easier than telling anonymous clicks apart. The model gives the layered design perfect discrimination between its channels and gives it no help from anything else.

And a listener has not just arrived. The floor is what remains once the evidence has stopped accumulating, which is the right quantity for somebody who has been listening for a while and the wrong one for somebody who has just walked in. Both matter and they are different questions; the earlier comparison asked the second and this one asks the first.

What the picture cannot show

It cannot show the elaboration, and in a slow cycle that is most of what is sounding. The punctuating layers are a skeleton, the elaborating instruments play over it at a density that rises as the cycle slows, and the position code measured here is the skeleton’s alone.

Nor can it show entrainment. A listener who has heard three cycles has a metre rather than a decoding problem, and everything measured here is about locating rather than about knowing. A metre is precisely a memory that is not an exponential discount, which is the caveat above with a name.

And it cannot say which way the causation runs. A tradition whose cycles are slow may use a layered code because a single line would not survive the tempo, or its cycles may be slow because a layered code lets them be. The arithmetic prices the constraint and says nothing about which side of it came first.

Whose music, and when

The nested-period structure is Javanese and Balinese first of all, where it is explicit, named and taught, and the cycle lengths are powers of two by convention. Gong cycles run from a few seconds at the fastest tempos to well over a minute at the slowest, and the same piece traverses that range as the tempo level changes — which puts one structure on both sides of the twenty-three-second crossover in a single performance.

The son clave is Cuban and its cycle is about two seconds; the bossa-nova pattern is Brazilian and comparable; the standard bell pattern is West African and faster still. None of them is ever played anywhere near the range where the floors here become large.

That the two families of design sit on opposite sides of a crossover computed from a published memory span is the finding, and it is a coincidence of two numbers until somebody has measured a third — how long a listener actually holds a gong stroke.

Where this ladder goes next

Four rungs. Rhythm is a circle and the bar line is a choice; a cycle cannot cadence; a cycle can say where it is without any of its parts being able to; and how well either design says it depends on a clock that was not in the measure, with the single line ahead below twenty-three seconds a cycle and the layered code ahead above it.

What is owed is still the elaboration, and it now has a reason rather than a gap. Everything above is the skeleton, and the skeleton’s floor rises steeply exactly where a gong cycle slows — 0.09 bits at twelve seconds, 1.19 at forty. If a listener is nonetheless located, something else is doing it, and the obvious candidate is the part that gets denser as the cycle gets slower. That makes the elaboration a second position code with a specific job, and it turns a vague debt into a testable one: the density of the elaborating parts should rise with cycle duration at about the rate the skeleton’s floor does. It still needs transcriptions this collection does not have, but it now needs far fewer of them — a handful of the same piece at different tempo levels, rather than a repertoire.