A dancer who hears the crowd can trust the quiet

A dancer joining an additive-metre dance partway through has to work out which metre is playing, and where in it the band is, from what can be heard in the first three or four seconds. A late dancer needs the landmarks, not the rhythm found that in a quiet room the stream of landmarks — the heads of bars and long beats, with the downbeat marked — does that better than every onset. Then the room filled up.

A crowd costs a late dancer the silences, not the steps let some of the music go unheard, and found the cost was mostly not what was lost. A listener who knows that a played onset can disappear under talk can no longer treat a silence as proof that a rival metre is wrong, since the rival’s predicted onset might have been played and lost. Every silence becomes weak evidence, rivals take several silences each to dismiss, and that caution cost the landmark stream more points of recognition than the losses themselves. A crowd that hides bars is kinder than one that hides onsets gathered the losses into bursts, as a real crowd’s laughter and talk gather them, and found bursts kinder — but left the caution untouched, because the caution belonged to the listener, not the room.

Its closing section named the assumption that makes the caution necessary. Every listener so far treats what it does not hear as a silence that might be a loss. A dancer in a room hears the noise as well. A laugh that covers a bar is not a silence; it is a sound, and a listener who hears it knows that whatever was played under it is gone, and that a silence in the quiet stretch after it is a real silence. That listener can discount exactly the silences the noise produced and take every other at full value. The question was how much of the caution’s cost it buys back.

Nearly all of it.

The listener with a second channel #

Everything about the dances and the listener is 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 — in which the metre must be recognised. The listener keeps the odds of the true metre against every rival entered at every step, and dismisses a rival at nineteen to one. The room’s noise is the same two-state chain: a quarter of the time is lost, in stretches of a stated mean length, and whatever is played under noise is not heard.

The new listener hears the chain. For each step it knows whether the room was noisy or quiet. A step under noise tells it nothing: neither an onset nor a silence is heard there, and it moves on. A step in quiet is heard perfectly, so a silence where a rival predicts an onset is proof against the rival, exactly as it was in a room with nothing lost. A heard onset the rival does not predict still dismisses it at once, as before.

The cautious listener of the earlier essays applies the same rule to every silence: odds multiplied by four for each silence where a rival predicts an onset, since at a quarter lost a silence is four times likelier under the truth than under that rival. It needs three such silences per rival to reach nineteen to one. The listener who hears the noise needs one, provided it falls in quiet.

The cautious listener’s count follows from one line of arithmetic: it needs the smallest whole number of silences n for which (1/p)ⁿ reaches nineteen. At a tenth of the time lost that is two; at a quarter, three; at a half, five. Each further silence a rival needs is a further step of the present spent waiting for it, and in a stream as thin as the landmarks, where a rival’s predicted onsets can be several steps apart, a second or third silence may not come within the present at all. The listener who hears the noise has the count at one whatever the room’s loss rate, because a quiet step is quiet whatever happens elsewhere.

Losing a quarter and knowing where #

The first figure’s two lines differ by more than twenty points at every length of stretch. With losses one step at a time the cautious listener recognises 27 per cent of metres within the present and the listener who hears the noise 51. With stretches of sixteen steps, 32 against 54; with stretches of sixty-four, 35 against 59.

The comparison that says what was bought is with the dashed line. A cautious listener with nothing lost — every onset heard, but allowing for losses at a quarter — recognises 48 per cent. The listener who hears the noise, losing a quarter of the music, does better than that at every stretch length. A dancer who loses a quarter of the music and knows where does better than one who loses none of it and has to allow for losing it. The caution was worth more than the music.

The split shows where the cost goes. For the landmark stream, with stretches of sixteen steps, nothing lost gives 81 per cent; the caution alone takes that to 48; the losses on top take it to 32. Hearing the noise puts it back to 54. So of the forty-nine points the cautious listener loses, about thirty-three were caution and the listener who hears the noise recovers twenty-two of them. What it cannot recover is the twenty-seven points between 81 and 54 that are the music itself, played under a laugh and gone.

The full stream tells a different story, and the difference is the whole of the earlier essay’s argument. Its caution costs little — 79 per cent with nothing lost, 74 with the caution — because every onset gives it plenty of silences at places rivals predict onsets, and three of them come soon. Its cost is the losses, and hearing the noise only lifts it from 48 to 53. The landmark stream is thin: its silences are what tell rivals apart, and the caution made each of them a third of a proof. Give a listener its proofs back and the thin stream recovers.

The stream to follow, when the noise is heard #

The earlier essays turned a noisy room into an argument for following every onset rather than the landmarks. On the draws used here, with nothing lost and no caution, the landmarks lead narrowly, 81 per cent against 79. Lose a tenth of the time, one onset at a time, and the cautious listener’s full stream leads by fifteen; lose a quarter in stretches of sixty-four, and it leads by nineteen.

Every one of those leads belonged to the listener who cannot hear the noise. For the listener who can, a tenth lost one at a time leaves the full stream ahead by only three points, and in stretches of sixteen or sixty-four the landmarks lead by two. At a quarter lost, losses one at a time still favour the full stream by ten, because single lost steps are scattered through the thin stream where each one matters; but in stretches of sixteen the landmarks lead by one, and in stretches of sixty-four by two.

So the earlier advice — follow everything in a noisy room — was advice to a listener who could not tell a laugh from a rest. A dancer who can hears a bursty room much as a quiet one with some bars cut out of it, and in a quiet room the landmarks were the better stream, if only just. A dancer who comes in late needs the downbeat marked is the reason the landmarks win when silences are trustworthy: a marked downbeat and the long beats are where metres differ, and a silence in the landmark stream is a statement that a bar or a long beat does not begin there.

Whether a listener hears a room as a room and a band as a band is the business of auditory scene analysis, and a crowd’s laughter against a band’s instruments is about as clear a case of two streams as there is. The second channel this listener has is not an exotic assumption. It is the ordinary separation of sources that every other figure in these essays quietly denied the dancer.

How far the quiet must be trusted #

The listener above treats quiet as perfect: a silence in quiet is proof. A real quiet room still loses the occasional onset — a soft drum stroke, a note masked by the dancer’s own footfall — and a listener who allowed for that would weaken its proofs back toward evidence.

The answer has a sharp edge, and the edge is set by the listener’s own threshold. A silence where a rival predicts an onset, at a quiet loss rate q, multiplies the odds against the rival by 1/q. The listener dismisses a rival at nineteen to one. So while 1/q is at least nineteen — a quiet room allowed to lose fewer than one onset in nineteen — one silence in quiet still dismisses a rival outright, and recognition does not change at all: 54 per cent for the landmarks at q of nought, one in a hundred, two, four or five in a hundred. Past one in nineteen, every rival needs two silences, and the landmark stream drops at once to 37 per cent, then to 32 at a quarter, which is exactly where the listener who could not hear the noise was.

That is a statement about the listener’s threshold rather than about rooms, and it generalises. A listener who demands odds of K to one before dismissing a rival keeps the whole of what hearing the noise buys as long as it trusts the quiet to one onset in K. A more demanding listener must trust the quiet more. The full stream feels the same edge less — it falls from 53 to 49 — because it never depended on single silences.

With more metres to choose between #

Knowing every metre is slower than knowing none found that a larger repertoire means more rivals, more of them close to the truth, and longer to dismiss them all. Each of those rivals is dismissed by silences in the landmark stream, so the value of trusting a silence grows with the repertoire. Knowing five metres, hearing the noise lifts the landmarks from 44 per cent to 65; knowing ten, from 32 to 54; knowing thirty, from 17 to 36; knowing a hundred, from 4 to 11. The full stream gains four to seven points at every size.

At a hundred metres neither stream recognises much within the present, heard or not, which is the limit how long a limping bar can be and a twenty-five is a nine until its last unit set on the present itself rather than on the room: a long metre does not reveal itself in sixteen steps however clearly those steps are heard.

What a stretch of noise still costs #

The listener who hears the noise has one weakness the cautious listener does not, and it shows in the first figure’s curve. For the cautious listener, longer stretches always helped. For the listener who hears the noise, stretches of two and four steps are slightly worse than losses one at a time — 50 per cent against 51 for the landmarks, and for the full stream 50 against 60 — before longer stretches help again.

The reason is that single lost steps cost a listener who knows about them almost nothing: a gap of one step in the full stream is surrounded by heard steps that do the same work. A stretch of four steps removes a whole region of evidence at once. Only when the stretches get long enough that most presents are entirely clean — the effect the earlier essay found — does burstiness help this listener too. So the earlier result that bursts are kinder than scattered losses was partly a result about caution: for a listener who has no caution to lose, scattered losses are the kinder kind for the full stream, and bursts are kinder only when they are long.

The model the counts rest on #

Additive metres of nine to twenty-five steps from groups of two and three; a listener knowing a repertoire of them, entered at a random step, keeping log-odds of the true metre against every rival at every entry point and dismissing a rival at nineteen to one; a present of sixteen steps. A quarter (or a tenth) of the time is under noise, in stretches drawn from a two-state chain with the stated mean length. Under noise nothing is heard. The cautious listener multiplies the odds by the loss rate’s reciprocal for each silence where a rival predicts an onset; the listener who hears the noise ignores steps under noise and treats a silence in quiet as proof, or, where a quiet loss rate is allowed, as evidence at that rate. Six repetitions of a stride through the repertoire’s metres, each with sixteen independent draws of the room, seeded as in the earlier essays.

What the model leaves out #

Noise that masks partly. A real laugh does not erase a drum stroke; it makes it harder to hear. A listener who heard the noise would then have to judge, step by step, how likely a played onset was to survive it, and the clean division between steps that count and steps that do not would blur into a weighting. The two listeners here are the ends of that range.

Noise that follows the music. As before, the room’s noise is independent of the dance. A crowd that cheers on the downbeats would hide precisely the landmarks, and no listener, hearing the noise or not, could get them back.

Wrong recognitions. A listener who hears the noise is correct about which silences count; the cautious listener is overconfident inside a long stretch of noise, as the earlier essay noted, and may dismiss the true metre’s rivals on repeated evidence that is really one piece. Neither listener’s wrong answers are counted, since the true metre is never tested against itself, so the cautious listener’s figures flatter it and the gap between the two listeners is if anything larger than drawn.

Whose rooms #

The rooms are the same as before: a wedding hall with a band playing a kopanitsa or a karşılama, a folk-dance club with a recording under talk, a festival square. The new claim is about dancers, not rooms. A dancer who listens through a crowd as a crowd — who hears the laugh as a laugh — has the evidence a quiet room gives, cut short where the laugh was, and on this arithmetic should follow the landmarks there as in a quiet room. One who cannot separate the two, or does not try, does better following every onset, as the earlier essays advised. What the arithmetic cannot say is which kind of listener a dancer is.

Still open: noise that is loud enough to hear but not to mask #

Every stretch of noise here hides everything played under it. A room’s noise usually does not: talk under a band masks the soft strokes and leaves the loud ones. A listener who hears the noise then knows that the silences under it are unreliable but that the onsets it does hear under it are real, and that a loud downbeat probably survives where a light upbeat does not. The computation gives each step under noise a chance of hearing what was played, set by how loud it was against the noise — the downbeat accent then counts twice, as a mark and as a better chance of being heard — and would say whether the marked downbeat is worth more in a noisy room than in a quiet one, which is a question about where a band should put its accents when the room is loud.