A crowd costs a late dancer the silences, not the steps

A late dancer needs the landmarks, not the rhythm took a listener who joins an additive-metre dance at an unknown point and gave them a thinner stream than the music: only the onsets that begin a bar or a long beat, every other onset removed. Against every expectation, the thinner stream did better. Knowing ten metres, a listener given the landmarks with the downbeat marked recognised 82 per cent of dances within the sixteen steps a psychological present holds, against 74 for every onset with the downbeat accented; knowing a hundred, 23 against 14. The onsets between landmarks were making rivals look alike, and the downbeat that a dancer who comes in late needs the downbeat marked had shown to be the decisive cue was one of the landmarks anyway.

That listener heard every landmark, and the essay’s closing section said why that matters more for the landmarks than for anything else. A dancer in a crowded room hears the strong parts of the rhythm because they are strong, and still loses some of them — a step under talk, a drum stroke under a trumpet. Since every landmark carries information and the onsets between carry almost none, the prediction was that a lost landmark would cost far more than a lost ordinary onset, and that the landmark stream’s lead would be the first thing to go.

The lead does go, and early. What takes it is not the lost landmarks.

A silence stops being a proof #

The listener in the earlier essay reasoned by contradiction. A rival metre, entered at some step, predicted an onset where there was silence, or silence where there was an onset, and on that step it was ruled out for good. Recognition was the step after the last rival fell.

In a stream where onsets can be lost, half of that reasoning survives and half does not. An onset that is heard is still decisive: if a rival predicts silence there, or predicts an unmarked onset where a marked downbeat sounded, the rival is wrong, because a crowd can hide a sound but cannot invent one. A silence is different. If a rival predicted an onset and none was heard, either the rival is wrong or the onset was played and lost, and a listener who admits that losses happen cannot tell which.

So the listener here keeps odds. Suppose they know that a share $p$ of what is played goes unheard. A silence where the true metre predicts nothing and a rival predicts an onset is $1/p$ times more likely under the truth than under the rival. A silence where the truth’s own onset was lost, and the rival predicted nothing, counts the other way by the same factor. A listener who wants to be sure dismisses a rival once its odds against have reached some figure — nineteen to one in the figures here, a conventional ninety-five per cent — and a heard onset that contradicts it dismisses it outright, as before.

At $p$ = 0 every silence is worth infinite odds and this is the earlier computation exactly; the figures check that it reproduces the earlier essay’s numbers to the last trial. The whole difference is what happens as soon as $p$ is not zero.

One silence, then two #

With the listener asking for nineteen to one, a single silence settles a rival whenever $1/p$ is at least nineteen — whenever fewer than one onset in nineteen is lost. Above that it takes two silences, until $1/p^2$ falls below nineteen, at one onset in about four and a half; above that, three.

That arithmetic is the staircase in the dashed lines. A listener who merely allows for losses, while losing nothing at all, recognises exactly as well as a perfect listener until the loss rate they allow for passes one in nineteen, and then drops in one step. For the landmark stream at ten metres known, the drop is from 82 per cent to 57. For every onset it is from 74 to 69. The second step, at about 0.23, costs the landmarks nine more points and every onset nothing.

Then there is the solid line, where onsets really are lost at the rate allowed for. It follows the dashed line down the step and then keeps falling as more is lost: at one in ten, the landmarks recognise 47 per cent and every onset 58; at one in five, 38 and 47; at four in ten, 15 and 22. The landmark stream’s lead of eight points at perfect hearing becomes a deficit of ten at the first step and stays a deficit at every loss rate past it.

Most of the damage is the caution #

The two lines in the hero figure can be pulled apart at one loss rate, and the split is the result.

At one lost onset in ten, knowing ten metres, the landmark stream falls from 82 per cent to 57 through caution alone — a listener who hears every landmark perfectly but cannot trust a silence — and then to 47 when the losses are real. Of its 35-point loss, 25 is the caution and 10 is the losing. Every onset falls from 74 to 69 through caution and to 58 through losses: 5 points and 11. Knowing a hundred metres the pattern is the same: the landmarks lose 14 points to caution and 3 to losses, every onset 6 and 2.

The prediction said a lost landmark would cost more than a lost ordinary onset, and per onset it does: the kopanitsa’s landmark stream has two onsets a bar where the full stream has five, so at the same loss rate the landmark listener loses fewer onsets and still loses about as many points. But a crowd does not hide a count of onsets. It hides a share of the time — talk swells, a trumpet enters, a partner steps across — and whatever is played in that time is lost from either stream in the same proportion. At equal rates of loss the landmarks cost almost exactly what ordinary onsets cost, ten points against eleven. The real difference between the two streams is something neither lost.

Why a thin stream depends on its silences #

The explanation is in what the landmark stream is made of, which the earlier essay’s own analysis already contains.

A stream of landmarks is two clicks a bar in a kopanitsa and two or three in a long aksak bar, unevenly spaced. What identifies the metre is the position of the gaps: how long after the downbeat the long beat’s head comes, and how long after that the next downbeat does. A rival metre with a different arrangement predicts a click somewhere the true stream has a gap, and in the earlier essay’s arithmetic that gap was a contradiction the moment it passed.

Almost all of the landmark stream’s evidence arrives that way, as a silence where a rival expected a sound. It cannot arrive the other way very often, because a heard landmark contradicts a rival only if the rival predicted silence at that exact step — and a rival that predicts clicks at a different rate will mostly be caught by one of its own clicks falling into a gap. The full stream is the opposite. It has an onset on every short beat, so it is dense with sounds a rival can be caught failing to predict, and a heard onset is a proof whatever the listener believes about losses.

The proportions can be counted over the whole dictionary. Across the 1,820 arrangements from nine to twenty-five units, a metre sounds an onset on 0.41 of its units and a landmark on 0.20: the landmark stream keeps 48 per cent of the onsets and is silent on four units in five. Its short beats are what it threw away, and a short beat is exactly the kind of onset that contradicts few rivals when heard — how unequal a beat is allowed to be found them never more than two against three, so every metre with a run of them makes the same regular pattern — but it is still a sound, and a sound is the one kind of evidence the caution does not touch.

So the landmark stream’s advantage lived in its silences, and a listener who allows for losses has devalued exactly those. The thinness that made the landmarks better was a bet that nothing would be missed, and a listener who hedges that bet loses the advantage before the room has hidden anything.

How sure the listener wants to be #

The step at one in nineteen is not a fact about rhythm. It is where the value of one silence, $1/p$, crosses the odds the listener asks for, so it moves with the odds.

At one lost onset in ten, a silence is worth ten to one. A listener who will dismiss a rival at three, five or nine to one is convinced by one silence, and the landmark stream leads again — 72 per cent to 66. At nineteen, forty-nine or ninety-nine to one, two silences are needed and the landmarks trail, 47 to 58. At a hundred and ninety-nine it takes three and the gap widens, 39 to 57.

So whether a dancer in a crowd is better served by attending to the landmarks is not decided by the crowd alone. It is decided by the crowd and by how much evidence the dancer demands before committing to a step. A dancer willing to guess on thin evidence — to join on the first bar that fits — does best on the landmarks at any loss rate below one in nine or so. A dancer who waits to be sure does best on every onset, because every onset keeps supplying heard sounds, and heard sounds are the only evidence a crowd cannot cast doubt on.

Against the size of the repertoire #

The repertoire comparison the earlier essay made its headline can be redrawn at any loss rate, and the dial moves it.

At one lost onset in ten every onset leads at every repertoire size up to a hundred, where the two tie at 6 per cent. At five metres the gap is largest, 76 against 59, which reverses the perfect-hearing order of 91 against 88. At one in twenty — dialled to 0.05 — the landmarks are back in front at every size, because a single silence still reaches nineteen to one.

The dial makes one more thing visible. Every onset degrades smoothly as the loss rate rises; the landmarks fall off a ledge. A listener relying on landmarks is either doing better than the full stream or clearly worse, and which is decided by a threshold the dancer cannot see — the point at which their own caution stops accepting one silence as enough.

What the numbers rest on #

The metres are every arrangement of twos and threes from nine to twenty-five units, each rotation a metre of its own, as in the earlier essays. Repertoires of five to a hundred are drawn at random, always containing the metre played, with a random entry; each trial is repeated with six independent draws of which onsets are lost, each played onset lost independently at the stated rate. The present is sixteen steps.

The listener knows the loss rate exactly and uses it. A heard onset that a rival does not predict, or a marked downbeat heard where the rival’s onset is unmarked, dismisses the rival. A silence contributes $\ln(1/p)$ to the odds against a rival that predicted an onset there, and the same amount in the rival’s favour when the truth’s own onset was lost where the rival predicted silence. A rival is dismissed when its odds against reach $K$ to one, and a metre is recognised one step after its last rival is dismissed, or never within a hundred and ten steps.

Where the listener is idealised #

One tempo, one present. The sixteen-step present is three and a half seconds at the tempo every metre essay here has used, and a cycle that says where it is and the earlier recognition essays found the long bars the hardest to place inside it — a twenty-five is a nine until its last unit is the extreme case. A long bar’s window was already close to the present with nothing lost, so it has less room to absorb the steps a loss adds, and the averages above mix bars with room to spare and bars with none.

Losses are independent. A crowd does not hide onsets one at a time at random. It hides stretches — a burst of laughter covers a bar — and a run of lost onsets is worse for a stream of two clicks a bar than for five, since it can remove every landmark for a whole bar and leave the full stream with its short beats. Correlated losses would make the landmarks’ position worse than this; nothing here has a correlation length in it.

The listener knows the loss rate. A real dancer does not, and a listener who underestimates it treats silences as stronger than they are and will sometimes dismiss the true metre after a lost landmark — a failure that cannot happen here, because the true metre is never tested against itself. What an overconfident listener gains in speed it would pay for in wrong entries, and the arithmetic of that exchange is a separate count.

Nothing is invented. A crowd can mask a sound and it can also supply one: a heel on the floor, a clap from another group, a drummer’s stray stroke. Those are false onsets, and a single one contradicts the truth outright in this model, which has no way to recover from it.

What the arithmetic cannot say about dance floors #

What share of steps a crowded room actually hides. The whole figure turns on the loss rate, and the rate for a village wedding or a folk-dance club is not in any published source known here. The step at one in nineteen is where the answer would have to be compared, and a room that hides one landmark in twenty is on the good side of it and one that hides one in ten is not.

Whether dancers listen this way at all. Knowing every metre is slower than knowing none found that a repertoire changes what counts as evidence, and a dancer who does not name the metre, only falls into step with it, needs far less than recognition. What the landmarks are worth to a dancer who only needs to find the long beat — beats of unequal length found it defined by its duration — may be decided by something other than any of this.

And the eye. A line dance is watched as much as heard. A dancer who can see the leader’s feet has a landmark stream that a crowd’s noise does not reach, and the step that decides everything here is a step in an auditory quantity only.

Whose dances, and which stream the music supplies #

The dances these metres come from — the kopanitsa’s 2+2+3+2+2 and the long aksak bars of the southern Balkans — are usually accompanied so that the long beat is marked by a bass drum and the dancers’ own step, with the short beats filled in by higher instruments. A late dancer needs the landmarks read that arrangement as a gift to the late dancer, because it is the landmark stream with its downbeat marked.

This essay reads it differently. The arrangement does not give a dancer the landmark stream instead of the full stream; it gives both, at different levels, and a crowd removes the quiet one first. What reaches a dancer in a loud room is the drum and the step — the landmarks — and on this arithmetic that dancer should either commit early on thin evidence or wait for the short beats to reappear. The accent buys two units however loud it is found that a downbeat accent is worth the same at any strength once it is heard; the finding here is its counterpart, that the stream it marks is only as good as the listener’s willingness to believe a silence.

Still open: a crowd that hides bars, not onsets #

The independence assumption is the one a real room breaks first, and it breaks in the direction that matters most here. Noise in a crowded room comes in bursts of a second or more, and a burst the length of a kopanitsa bar removes both of its landmarks at once while leaving the full stream a short beat or two on either side.

The computation would draw losses as stretches with a stated mean length rather than as independent onsets, holding the overall loss rate fixed, and ask how the split between caution and losses changes as the stretches lengthen. The prediction is that the landmark stream suffers disproportionately once a stretch reaches about a bar — because then a single burst can hide every landmark in a cycle — while every onset suffers in proportion to the time lost. If so, the length of a crowd’s bursts relative to the bar is the quantity that decides which stream a dancer should trust, and long aksak bars, which hold more landmarks per burst, would survive a noisy room better than short ones.