A crowd that hides bars is kinder than one that hides onsets

A crowd costs a late dancer the silences, not the steps gave a listener joining an additive-metre dance at an unknown point a room in which some of what is played goes unheard. The listener knows the rate and so can no longer treat a silence as proof that a rival metre is wrong, and that caution turned out to be most of the cost. At one onset in ten lost, allowing for the loss cost the landmark stream — the bar heads and long beats, with the downbeat marked — twenty-five points of recognition before anything was lost, and the losses themselves a further ten. The landmark stream’s lead over the full stream of onsets, which a late dancer needs the landmarks, not the rhythm had found with nothing lost, flipped at one onset in nineteen.

Its closing section named the assumption a real room breaks first. The losses were independent, one onset at a time, as though the crowd’s noise were a fine rain. A crowd’s noise comes in bursts: a laugh, a shout, a chorus of talk, each a second or more long. The prediction followed from the landmarks’ thinness. A burst the length of a bar can remove every landmark in a cycle while leaving the full stream some beats on either side, so the landmark stream should suffer disproportionately once a burst reached about a bar. Long bars, which hold more landmarks per burst, should survive a noisy room better than short ones.

Both halves are wrong. Holding the share of time lost fixed, longer stretches of loss help recognition in every stream, help the full stream more than the landmarks, and help long bars no more than short ones. A crowd that hides whole bars costs a late dancer less than one that hides the same amount of music a step at a time.

Losing stretches at a fixed rate #

The dances and the listener are the earlier essays’. Metres of nine to twenty-five steps built from groups of two and three, a repertoire of ten of them known to the listener, an entry at a random step, and a present of sixteen steps — three and a half seconds at the tempo used throughout — within which the metre has to be recognised. The listener keeps the odds of the true metre against every rival at every entry point and dismisses a rival when the odds reach nineteen to one. A heard onset that a rival does not predict dismisses it at once; a silence where the rival predicts an onset multiplies the odds by one over the loss rate, since a silence where something was played is now only evidence, not proof.

The one change is how the losses are drawn. Instead of each step being lost independently with probability $p$, the room alternates between stretches of noise and stretches of quiet as a two-state chain over steps. A lost step stays lost with probability $1 - 1/b$, so stretches of noise have mean length $b$ steps. A heard step turns lost with probability $p / ((1-p) b)$, which keeps the share of time lost at exactly $p$ whatever $b$ is. As $b$ grows, the same amount of noise is gathered into fewer, longer stretches. A kopanitsa bar is eleven steps long, so bursts of eight to sixteen steps are about a bar and bursts of sixty-four are several.

The listener’s own rule is unchanged. It still allows for losses at the true rate and still treats each silence as independent evidence, which is now the wrong model of the room; that is taken up below.

More stretches of nothing #

The first figure’s curves all rise to the right. At a tenth of the time lost, the landmark stream goes from 46 per cent recognised within the present with losses one by one to 50 per cent with stretches averaging 64 steps; the full stream from 61 to 66. At a quarter lost the rises are larger: from 27 to 35 for the landmarks and from 40 to 53 for the full stream. Nothing gets worse, and nothing gets better until the stretches reach about four steps.

The reason is almost entirely one quantity.

With a quarter of the time lost one step at a time, a sixteen-step present is almost never clean: the chance that all sixteen steps are heard is $0.75^{16}$, about one in a hundred. With the same quarter lost in stretches averaging sixteen steps, 55 per cent of presents are clean, and with stretches of sixty-four, 69 per cent. The closed form for the chain, $(1-p)\,(1 - p/((1-p)b))^{n-1}$ for a present of $n$ steps, agrees with twenty thousand drawn presents to within a point everywhere.

So gathering the noise into bursts does not change how much of the music is lost. It changes how the loss is shared out among presents: most presents lose nothing and a few lose a great deal. A listener joining in one of the clean presents is exactly the earlier essay’s cautious listener with nothing lost. One joining in a noisy present may lose the dance entirely, but for recognising within the present, losing everything is not much worse than losing a quarter, since a quarter lost one step at a time already defeats most entries.

Caution stays; the losses shrink #

The earlier essay’s split, between what the listener’s caution costs and what the losses themselves cost, is the clearest way to see what the stretches change.

The caution’s cost does not change, because it is a property of the listener’s rule, not of the room: the listener allows for a quarter of the time lost whether the loss comes in stretches or not. What changes is the losses’ own cost. For the landmark stream it falls from 21 points with losses one by one to 14 with stretches of thirty-two steps; for the full stream from 34 to 23.

The full stream’s numbers explain why it gains more. With a quarter of the time lost, its caution costs nothing at all: at that rate the listener needs three silences to dismiss a rival, and the full stream supplies plenty of silences at places where rivals predict onsets, so the caution does not bind. All of the full stream’s cost is losses, and the losses are what stretches reduce. The landmark stream pays most of its cost as caution, which stretches cannot touch.

Where a stretch as long as a bar sits #

The prediction put the danger at a stretch about a bar long, and the curves say nothing special happens there. A kopanitsa bar is eleven steps and the metres here run from nine to twenty-five, so bars sit between the fourth and sixth marks on the horizontal axis of the first figure, and every curve rises smoothly through them. There is no dip where a burst first becomes long enough to swallow a cycle’s landmarks.

The reason is that swallowing a cycle’s landmarks is only one of the things a stretch that long does, and the others outweigh it. A bar-length stretch at a quarter of the time lost also means the other three quarters of the time come in long clean stretches, and a present that falls in one of them is heard perfectly. At stretches of sixteen steps more than half of all presents are clean. The listener entering in one of those gets the dance exactly as the earlier essay’s cautious listener with nothing lost would, and that single fact outweighs the damage the long stretches do to the presents they hit.

What does appear at the longest stretches is a small new failure: the metre is sometimes never recognised at all within the passage. With stretches averaging sixty-four steps and a quarter of the time lost, about seven entries in a hundred never recognise the metre, where losses one by one never produce that outcome. A stretch of noise that long can cover most of the passage a listener hears, and no rule can recover what was never heard. It is a failure of a different kind from the ones the caution produces — a dancer who joins during a toast and never hears the band — and it grows only when the stretches are several bars long.

A cycle that says where it is and beats of unequal length both treated the dancer’s problem as one of information arriving at a steady rate. The stretches make it a problem of information arriving in bursts: long periods in which everything is heard, separated by gaps in which nothing is. On this arithmetic a dancer is better served by that than by a steady trickle with the same total, because a present with everything in it is worth more than several presents each missing a quarter.

The stream to trust in a noisy room #

That reverses the earlier essay’s advice only in degree, but the degree is large. With nothing lost, the landmark stream leads, 82 per cent against 74. With a tenth of the time lost one onset at a time, the full stream already leads, 61 against 46, because the landmarks’ caution is so expensive. With the same tenth lost in stretches of sixty-four steps the full stream leads by 16 points, 66 against 50. At a quarter lost, its lead grows from 13 points with losses one by one to 18 with stretches of sixty-four.

So the prediction had the stream right and the direction wrong. The stream the length of a crowd’s bursts decides is the full stream, not the landmarks, and it decides it in the full stream’s favour. A dancer in a room whose noise comes in bursts does better listening to everything than to the heads of bars, by more than in a room whose noise is evenly spread.

Long bars are no safer #

The second half of the prediction was that long bars, holding more landmarks per burst, would survive a noisy room better than short ones.

They do not. In every setting, recognition falls as the bar gets longer. For the landmark stream in stretches of sixteen steps it runs from 53 per cent for bars of nine to twelve steps to 30 for bars of twenty-one to twenty-five; for the full stream in stretches, from 58 to 47. The stretches help every length a little, and they help the longest bars no more than the others: 4 points for the landmarks’ longest group against 3 for their shortest.

The reason is the one a twenty-five is a nine until its last unit found with nothing lost at all. A long bar takes nearly the whole sixteen-step present to reveal itself, so it has no spare time for a loss, however the loss is distributed. The prediction reasoned about how many landmarks a burst could hide in a long bar, but a long bar’s difficulty was never about its landmarks. It was about its length against the present — the limit how long a limping bar can be set on the metre itself — and a burst does not change that.

At every size of repertoire #

The pattern holds as the repertoire grows. Knowing five metres, stretches of sixteen steps lift the landmark stream from 41 per cent to 44 and the full stream from 56 to 60. Knowing thirty, from 12 to 17 and from 17 to 27. Knowing a hundred, recognition within the present has collapsed either way: 3 to 4 per cent for the landmarks and 3 to 6 for the full stream. Whatever the repertoire, gathering the noise helps, and helps the full stream more.

A listener with the wrong model of the room #

One thing about this listener is now wrong, and it is worth saying what it gets wrong and in which direction. The listener treats every silence as independent evidence at the room’s loss rate. In a room whose noise comes in stretches, a silence next to another silence is much weaker evidence than that: if the step before was lost, this one is very likely lost too, and the silence says almost nothing about whether an onset was played. A listener who knew the stretches would discount runs of silence and count isolated silences at nearly full weight.

The direction of the error is knowable. Inside a stretch of noise the listener treats each silent step as fresh evidence against every rival that predicts an onset there, and a long stretch of silence can push a rival’s odds past nineteen to one on the strength of what is, in fact, one piece of evidence repeated. That tends to make this listener faster than a correct one, and sometimes wrong. The first figure is therefore an upper bound on what a listener who modelled the stretches would achieve in recognition speed, though not necessarily in accuracy; the wrong entries such a listener makes are not counted here, since the true metre is never tested against itself.

A second listener is the natural counterpart. It would keep, alongside its odds, a belief about whether the room is currently noisy — a hidden state inferred from the pattern of heard and unheard steps — and count each silence according to that belief. It is a forward pass over a two-state chain per rival, and it is the calculation the section below points to.

Where the stretches come from #

The chain is the simplest model of bursty noise. It gives stretches of noise lengths drawn from a geometric distribution with the stated mean. Real noise in a room has heavier tails, with a few very long stretches such as a toast, a song from another table or an announcement, and many short ones. A heavier tail puts more of the loss into fewer presents, so it should push every curve here further in the same direction.

The noise does not follow the music. A crowd’s noise here is independent of the dance, whereas in a real room it may rise at the loud parts and fall at the quiet ones, or cluster at the phrase ends where people talk. Noise that followed the downbeats would be the prediction’s worst case after all, and nothing here tests it.

One present, one tempo. As before, the present is sixteen steps at one tempo. A slower dance puts fewer steps in the present and makes every stretch relatively longer, which should help the full stream more still.

Whose rooms #

The rooms this is about are the ones a dancer who comes in late needs the downbeat marked had in mind: a wedding hall in the Balkans or Turkey with a band playing a kopanitsa or a karşılama, a folk-dance club with a recording over talk, a festival square. The loss rate of any of these rooms is not published, and neither is its burst length. What the arithmetic says is that the second number matters as much as the first, in the direction that makes rooms kinder than a rate alone suggests, and that the room’s burstiness should change which part of the music a dancer listens to. Knowing every metre is slower than knowing none found that a repertoire changes what counts as evidence. A room’s burstiness does too.

Still open: a listener who hears the noise as well as the music #

Every listener in these essays infers the metre from what it hears and treats what it does not hear as a fixed rate of loss. A dancer in a room also hears the noise: the laugh that covers a bar is not silence, it is a sound, and a listener who hears it knows that bar is unreliable. That listener has a second channel, a noise stream saying when to trust silences, and can discount exactly the silences a burst produced. The computation adds that channel to the listener’s odds — a silence during audible noise worth nothing, a silence in quiet worth its full weight — and would say how much of the caution’s cost, the largest cost in every figure here, a listener can buy back by listening to the room as well as the band.