The bell pattern is slowest only to a perfect memory

The other censuses keep evenness, not locating ended on a table of orders. Hold a timeline’s gaps fixed, arrange them every way they can go round the cycle, and a named timeline is usually the order whose onsets sit nearest to perfectly even and the order that takes longest to tell a listener where in the cycle they are. The standard bell pattern’s five gaps of two steps and two of one go round a cycle of twelve three ways; the bell pattern is the most even of the three and needs eleven steps to locate, against ten and eight for the others. Of the eight named timelines whose gaps have more than one order, five were both the most even and the slowest.

Every one of those locating numbers belongs to a listener who never forgets. A rotation that a step rules out stays ruled out, however long ago the step was, and that is why a locating length exists at all. The cycle that outruns the memory built the measure without that assumption and found that certainty is then a property of the memory rather than of the pattern: a forgetting listener’s uncertainty does not reach zero but settles at a floor. What that measure was never asked is whether the floors put the orders of a timeline’s gaps in the sequence the perfect memory did.

They do not, and the bell pattern is where it shows first.

A memory that halves #

The measure has two settings, and the perfect-memory count puts both at an extreme. Each step heard is evidence against every rotation of the cycle that predicts something else at that moment, and the evidence is discounted by its age, halving every so many steps — the half-life. A mismatch costs a fixed amount rather than ruling a rotation out; the cost used throughout is six, which makes one disagreement worth a factor of about four hundred against a rotation, the cost of a listener who seldom mishears. What remains is a distribution over the rotations, and the number reported is its entropy in bits, averaged over every place in the cycle the listener could have come in.

With an infinite half-life and a fatal mismatch this is the perfect-memory count, step for step. With a finite half-life the uncertainty falls for a while and then stops falling, because the oldest evidence is leaving as fast as new evidence arrives. A key-finding listener given the same kind of decay lost far more confidence than accuracy. Here what forgetting costs is the ranking.

Two numbers come out of the measure, and they belong to different listeners. The floor is where the uncertainty settles — what a listener who has been inside the music for a while is left with. The arrival cost is the uncertainty averaged over the first cycle heard, which is what matters to a listener who has just come in. The perfect-memory locating cost the earlier essays ranked by is the arrival cost of a listener with no decay at all.

The bell pattern loses its place #

With perfect memory the three orders of the bell pattern’s gaps are ranked by how long they take to locate: eight steps for 2 2 2 1 2 1 2, ten for 2 2 2 2 1 1 2, and eleven for the bell pattern itself, 2 2 1 2 2 1 2. For every listener whose memory of a step halves within half a cycle, the order 2 2 2 2 1 1 2 settles highest and the bell pattern is second. The margin is small at a half-life of six steps and large at three, where the bell pattern is left with 0.535 bits and the order with its two short gaps side by side with 0.795. The quickest order stays the quickest at every memory drawn.

The solid curves and the dashed ones disagree about the same pair. With perfect memory the bell pattern’s curve lies above that of 2 2 2 2 1 1 2 from the sixth step on and reaches zero a step later. With a memory that halves every three steps, the order with its short gaps together flattens out higher and stays there for as long as anybody listens.

The obvious explanation is the wrong one. A short memory might be expected to penalise long runs of equal gaps, since a listener holding only the last few steps inside a stretch of strict alternation cannot tell one place in it from another. The order 2 2 2 2 1 1 2 does have the longest run, five twos in a row once the cycle wraps round. But the quickest order, 2 2 2 1 2 1 2, has four twos in a row against the bell pattern’s three, and settles far lower than both. Whatever a forgetting listener ranks by, it is not the length of the longest run of equal gaps.

The fume-fume is the bell pattern turned inside out #

The fume-fume, five onsets in twelve with gaps 2 2 3 2 3, is a different timeline, and it is also the bell pattern with every onset made a rest and every rest an onset. To a listener trying to find their place that exchange changes nothing. Each step either agrees with what a rotation predicts or does not, and exchanging onsets and rests on both sides of every comparison leaves every agreement where it was. So a pattern and its complement settle at exactly the same floor at every memory span.

And yet the fume-fume stays the slowest order of its gaps for every listener, forgetting or not, and the bell pattern does not. The two are ranked against different rivals. The bell pattern’s gaps go round the cycle three ways and the fume-fume’s only two, because the complement of the third order has a gap of four and is not an arrangement of the fume-fume’s gaps at all. The order that overtakes the bell pattern has no counterpart among the fume-fume’s orders, so nothing overtakes the fume-fume.

Put both back into their whole censuses, where every pattern of their size competes, and the difference disappears. Seven onsets in twelve and five in twelve are complements of each other pattern for pattern, and in both the evenly spaced pattern is second slowest for every forgetting listener drawn, with the complement of 2 2 2 2 1 1 2 above the fume-fume exactly where 2 2 2 2 1 1 2 is above the bell pattern. A named timeline’s standing among the orders of its own gaps turns out to depend on which orders its gaps happen to admit.

Eight timelines and five listeners #

The table sets the eight named timelines whose gaps have more than one order against five listeners. Six of the eight are the slowest of their orders, or tied for it, with perfect memory: the rumba clave, the standard bell pattern, the cinquillo, the fume-fume, the shiko and the soukous. Only the fume-fume is the slowest in every column.

The others lose the place at different memories. The bell pattern is second of three for every forgetting listener, arriving or settled. The rumba clave is second or third of six. The soukous, tied slowest of its twelve orders with perfect memory, is third for a listener arriving with a quarter-cycle memory, fifth once settled with a half or a quarter, and seventh with an eighth. The cinquillo holds on until its memory halves in an eighth of its cycle, which for a cycle of eight steps is a single step, and is then the faster of its two orders. The shiko holds on for an arriving listener and for a settled one with half a cycle of memory, and is the faster of its two orders once the memory is down to a quarter.

The two timelines that were quickest move the other way, or do not move. The son clave, tied fastest of its six orders with perfect memory, is fourth slowest for an arriving listener and second slowest for a settled one whose memory halves in an eighth of the cycle. The gahu, tied with it before, is the fastest in every column. The exception the locating story was told about, the son clave, is an exception only to a perfect memory.

So the rule the earlier table suggested — that a named timeline is usually the most even order of its gaps and also the slowest to locate — holds for a listener who never forgets. For a settled listener whose memory halves in a quarter of the cycle it holds for two of the five timelines that met it, the fume-fume and the cinquillo, and with an eighth it holds for the fume-fume alone.

A curve that is higher everywhere can settle lower #

The shiko’s reversal is small, and it is worth drawing because it rules out the tempting explanation that a forgetting listener’s ranking is the perfect-memory curve read at some short window of hearing.

With perfect memory the shiko is never less uncertain than its sister order, whatever number of steps has been heard, and it needs two more steps to locate. There is no window at which reading the solid curves would put the shiko lower. With a memory that halves every two steps it settles lower all the same, by 0.043 bits. The margin is small, and its direction is one that no reading of the perfect-memory curves can produce.

That is the precise sense in which the two listeners disagree. The perfect memory ranks patterns by how long the whole of the evidence takes to become decisive. The forgetting listener ranks them by how much a recent stretch of evidence, weighted towards its last few steps, leaves open at every place in the cycle at once — and the two quantities are related, but a pattern can be ahead on one at every window and behind on the other.

The census ranking survives, and was mostly repetition #

The census-level result held up far better, and it has a simpler reason than the earlier essay gave it.

The evenly spaced pattern was the slowest of its census to locate in nineteen of twenty-one censuses. In fourteen of them the number of onsets and the number of steps share a factor, and the even pattern then repeats within its own cycle — two onsets in eight are the same half-cycle twice, eight in sixteen the same two steps eight times. A pattern that repeats never locates at all. Its rotations by a whole repeat are the same sequence, so a listener whose memory covers the whole cycle is left with exactly the logarithm of the number of repeats: one bit for two in eight, three for eight in sixteen. In those fourteen, being slowest is a statement about divisibility: the evenly spaced pattern of k onsets in n steps, when both share a factor d, is d copies of the pattern of k/d onsets in n/d steps.

The seven censuses where the counts share no factor are the ones that test anything: three and five in eight, five and seven in twelve, and three, five and seven in sixteen. With perfect memory the even pattern is the slowest in five of them and second in two. With a memory halving in a half or a quarter of the cycle it is slowest in four and second in three — the bell pattern’s census and the fume-fume’s lose the place, and three in sixteen gains it. With an eighth it is slowest only in five in sixteen, second in five censuses, and seventh of thirty-five in three in sixteen.

A memory that short does one more thing, and it is the possibility the earlier essay held open. In five of the censuses that repeat — two onsets in eight, twelve and sixteen, six in eight and ten in twelve — the even pattern becomes the fastest of its census at an eighth-cycle memory. Once recall is short enough, the pattern that could never be located at all is the one that leaves a listener least uncertain, because a very short memory leaves every other pattern of the census less sure still. That is where evenness and locating stop pulling apart, and it happens only in censuses so sparse or so dense that no timeline anyone plays is among them.

At the speed a timeline is played #

Every memory above is a fraction of the cycle, and no listener’s memory is. The window over which a listener holds a stretch of sound as one present thing is a number of seconds, and three and a half is the figure the essays on memory use for it. A memory that halves in three and a half seconds becomes a half-life in steps only once the cycle has a duration.

The standard bell pattern goes round in less time than a clave’s two seconds. At that speed a memory of three and a half seconds spans more than the whole cycle, every floor is below a millionth of a bit, and the three orders cannot be told apart by where a listener settles. What separates them is how long they take to locate on arrival, which is the quantity the perfect memory ranks, and on arrival the bell pattern is the slowest of its three orders at every cycle from one and a half seconds to three. At the speed it is played, the bell pattern is still the slowest order of its gaps.

The reordering begins where the cycle is long enough for the memory to cover only part of it. On arrival the bell pattern has already lost its place at a cycle of four seconds. At the floor, another order settles more than a thousandth of a bit above it from a cycle of six seconds, where the half-life is seven steps, and by twelve seconds the order with its short gaps together is left with half a bit and the bell pattern with a third. A bell pattern slowed to that speed is no longer a timeline anybody dances to, so the result is not that listeners find the bell pattern easy to locate. It is that the earlier ranking among the orders of its gaps depended on an assumption the timeline’s own tempo happens to satisfy.

The arithmetic #

Each census is every placement of k onsets in n steps grouped into necklaces, as in the essays before this one, and a timeline’s orders are the necklaces with its multiset of gaps. The floor is the entropy of the distribution over rotations once further listening changes it by less than a billionth of a bit, averaged over the n places a listener can come in; the arrival cost is the same entropy averaged over the first n steps heard. A table column’s half-life is that share of the cycle’s own length in steps, so a quarter-cycle memory is three steps in twelve and four in sixteen. A rank counts the patterns whose floor is higher by more than a billionth of a bit, so floors that differ only in their last digits are ties.

Two checks run with the figures. With an infinite half-life and a mismatch cost of two hundred the fading measure reproduces the perfect-memory curve at every step for every order drawn, so every floor above is a distance from a published curve rather than a different measure. And a pattern and its complement settle at the same floor to twelve decimal places, which is the reason the fume-fume and the bell pattern can be compared at all. For the censuses that repeat, the floor at a memory as long as the cycle is checked against the logarithm of the number of repeats.

What the model assumes #

That memory is exponential and the same for every step. A real memory keeps what mattered and loses what did not, and in a timeline the strokes that matter are the onsets. A memory that held onsets better than rests would break the symmetry between a pattern and its complement, and the bell pattern and the fume-fume would stop settling at the same floors.

That a mismatch costs six. The cycle that outruns the memory found that loosening the cost raises every floor and leaves the ordering of its designs alone. The orderings here that turn on thousandths of a bit — the cinquillo’s and the shiko’s — are the ones most exposed to that setting, and the bell pattern’s, which turns on a quarter of a bit at a three-step half-life, is the least.

That the listener has no metre. A listener who has heard a few cycles has a beat, and a beat is a memory that is not a decaying discount of the last few steps. Everything here is a listener decoding position from a single line with nothing else to hold on to.

What these ranks cannot establish #

That the bell pattern is heard with a short memory. At the speed the pattern is played it is not, and the perfect-memory ranking applies. What the ranks establish is how fragile that ranking is: the census result survives forgetting, the order result does not, and the difference is whether the comparison is with every pattern of a size or with the few orders one set of gaps admits.

That traditions chose their timelines for any of this. The named timelines are ten, and the rotation the necklace cannot see found that where they match an even pattern they match it only up to a starting point. A table of eight timelines against five listeners describes those eight; it is not a rate.

Still open: the bell pattern against the pulse #

Every listener here hears the timeline alone, and nobody does. In an ensemble the bell pattern sounds against a pulse the drums and the dancers mark, and a cycle that says where it is measured what regular layers do to locating when the position is carried by which parts sound together. The computation that follows puts the two side by side: the three orders of the bell pattern’s gaps, each sounded with a pulse every three steps, ranked by the floors and arrival costs of a forgetting listener. It would say whether 2 2 2 2 1 1 2 still overtakes the bell pattern once the listener can also hear where the beats fall, or whether the pulse is what keeps the most even order the slowest — and, since a rhythm is a circle and the bar line a choice, whether a pulse in threes and a pulse in fours rank the orders the same way.