In a loud room the accent has to be loud

A dancer arriving late at a Balkan wedding has to work out which of the dances they know the band is playing, from whatever bar they walk in on, and quickly. A dancer who comes in late needs the downbeat marked and a late dancer needs the landmarks, not the rhythm found what the dancer needs to hear. Not every onset of an additive metre, but its landmarks — the starts of its bars and of its long beats — and the first of them marked. With ten metres known, that stream lets a listener entering at a random step recognise the dance within the sixteen steps of a present, three and a half seconds at the tempo used throughout, in 81 per cent of entries.

Then the crowd. The three essays before this one let the room’s noise take part of the music away, and found the cost was mostly not the music lost. It was the caution a listener needs once a silence might be a loss. A dancer who hears the crowd can trust the quiet gave the listener a second channel: it hears the noise as well as the music, ignores whatever falls under noise, and treats every silence in quiet as proof. That listener lost a quarter of the time to noise and still recognised the dance in 54 per cent of entries.

That listener’s advantage came from a division of the evening into two kinds of time. In quiet, everything is heard and every silence is proof. Under noise, nothing is heard and nothing is learned. The crowd that costs a late dancer the silences had made every silence doubtful; hearing the noise made the doubt local, confined to the stretches the noise actually covered. What it could not do was learn anything inside those stretches.

Its closing section named the assumption that noise hides everything played under it. Talk under a band does not. It masks the soft strokes and leaves the loud ones, and the loud ones are exactly the ones a band accents.

Why loudness enters through masking #

A sound hides another by occupying the part of the ear that would have registered it. One sound hides another set out the rule these essays use for it: a weaker tone near a louder one in frequency is lost when its level falls below the threshold the louder one raises. Talk raises that threshold across the middle of the spectrum by an amount that rises and falls with the talk. A stroke played under it is heard if it rises above that threshold at that moment, and a louder stroke rises above it more often. So the probability that a stroke survives a stretch of noise is, in the end, a statement about its level against the noise’s, averaged over how the noise fluctuates.

That is the only way level enters below. The model does not compute masking from spectra; it takes the two probabilities as given, one for an accented downbeat and one for any other stroke, and sweeps them. Any real room puts a particular pair on that grid, and the figures are drawn so that the pair can be read off.

A stroke that might get through #

The dances, the repertoire, the entry and the present are the earlier essays’. Additive metres of nine to twenty-five steps built from groups of two and three; ten of them known; an entry at a random step; sixteen steps in which to recognise the right one. The listener keeps odds of the true metre against every rival at every offset and dismisses a rival at nineteen to one. The noise is the same two-state chain: a stated share of the time is under noise, in stretches of a stated mean length, sixteen steps unless otherwise said.

What changes is what the noise does to a stroke played under it. A marked downbeat — played louder — is heard through it with one probability; any other stroke, played at the ordinary level, with another. The listener knows both probabilities and still hears the noise. So under noise:

• a stroke it hears is as good as one in quiet, since noise can hide a stroke but not invent one. A rival that predicted nothing there, or predicted a stroke with the wrong marking, is dismissed at once.
• a silence it hears is evidence, weighted by how likely each metre’s predicted stroke was to be masked. If the true metre predicts nothing there and a rival predicts a marked downbeat that cuts through every time, the silence is proof. If the rival predicts a soft stroke that half the time gets through, the silence doubles the odds against it.

With both probabilities at nought this is the earlier essay’s listener exactly, which ignored everything under noise, and the figures reproduce its numbers to the decimal.

How much a silence under noise says #

The second rule is the heart of it, and it can be counted on its own.

Each silence under noise multiplies the odds against a rival by one over the chance its predicted stroke was masked. If the stroke always gets through, one silence is enough. At nine chances in ten, two; three in four, three; half, five; a quarter, eleven; a tenth, twenty-eight — more than the present has steps. If the stroke never gets through, no number of silences says anything, which was the earlier essay’s rule for everything under noise.

So the loudness of a stroke now decides something it decided nothing about before. In a quiet room the accent buys two units however loud it is: a downbeat marked at any strength the listener can detect does all the work an accent can do, because what helps is knowing which onset is the downbeat, not how loud it is. In a noisy room, the loudness of the downbeat is the chance that it gets through. And that chance decides two things, both in the accent’s favour: whether a true downbeat played under noise is heard, and whether a rival that predicts a downbeat where none is heard can be thrown out on that silence alone.

More than in quiet #

The hero figure puts the accent’s worth — recognition from the marked landmarks less recognition from the same landmarks unmarked — against how much of the soft playing the noise lets through, with every marked downbeat getting through.

In a quiet room the accent is worth 23.4 points: 81 per cent recognition against 57. Under noise that hides everything, it is worth less, and the more noise the less: 21.4 points with a tenth of the time noisy, 18.1 with a quarter, 11.3 with half. Both streams lose under noise, and the marked stream, which had more to lose, loses more.

Under noise that every marked downbeat cuts through, the picture inverts as soon as some soft playing gets through too. With a quarter of the time noisy, the accent is worth 22.9 points with no soft strokes heard, 25.9 with half, 26.7 with three-quarters. With half the time noisy, 19.8, 27.0 and 28.9 — five and a half points more than in a quiet room. Only when the soft playing gets through completely does the room cease to exist, and the accent’s worth returns to its quiet value.

The reason the accent gains is that the noise hurts the two streams differently. The unmarked landmarks are all soft, so under noise each of them is heard at the soft rate and each silence is weak evidence. The marked landmarks have a downbeat in every bar that is heard whenever it is played and whose absence is proof whenever it is predicted. Noise that lets soft strokes through partly degrades the unmarked stream and leaves the marked stream’s most useful strokes intact. In a quiet room both streams are perfect; in this room only one of them is.

The accent has to cut through #

The other half of the picture is what happens when the accent itself does not always get through. With a quarter of the time noisy and half the soft strokes heard, an accent that cuts through a quarter of the time is worth 19.7 points; half the time, 21.4; three times in four, 23.2, still a fraction under its quiet value; every time, 25.9. With no soft strokes heard the line is lower all the way along, from 18.1 to 22.9.

So in a noisy room an accent’s worth runs from less than its quiet value to more, and what decides which is how reliably it is heard. The quiet-room result that loudness buys nothing is right about a quiet room. In a loud one, an accent that is only a little louder than the rest is worth less than in quiet, and one that cuts through every time is worth more. That is a precise version of a familiar instruction to a band in a noisy room, to hit the one. The instruction is not merely to play louder all round, which lets more of both streams through and narrows the gap between them, but to make the downbeat the one stroke the noise cannot take. The accent’s worth comes from the difference between the two survival rates, and a band that raises both together spends its effort on the unmarked strokes as much as on the marked one.

Every stream, three rooms #

Drawn for all four streams the earlier essays compared, the effect is confined to the streams with something marked. The marked landmarks go from 54 per cent under masking noise to 59 when the downbeats cut through and 66 when half the soft strokes do too. Every onset with the downbeat accented — a fuller stream with the same accent — goes from 53 to 58 to 68, slightly more, because it has more soft strokes for the noise to let through. The unmarked landmarks stay at 36 per cent until soft strokes get through, and then reach only 40; every onset unmarked, from 14 to 16.

The comparison between streams that the landmark essays made is therefore not settled by the room. In quiet the marked landmarks and the full accented stream are within two points of each other, 81 and 79. Under noise that passes loud strokes and half the soft ones they are within two points the other way, 66 and 68. A dancer in a loud room does almost as well from a band that plays every onset with the downbeat accented as from one that plays only the landmarks — and either is worth half again what an unaccented band gives.

There is an echo here of what two landmarks in a cycle help only if they can be told apart found in a gong cycle: a landmark is worth what makes it distinguishable from the other points of the cycle, not what makes it present. A marked downbeat in a quiet room is distinguishable by its marking alone. In a loud room it has to be distinguishable from the noise as well, and the same accent does both jobs only if it is loud enough for the second.

Short bursts, and the size of the repertoire #

The stretches of noise matter as they did before, in the same direction. A crowd that hides bars is kinder than one that hides onsets found long bursts kinder to the cautious listener. For the accent’s worth to a listener who hears the noise, short ones are kinder. Under noise the accent cuts through, its worth is 33.2 points when the noise comes one step at a time, 28.3 in stretches of eight, 25.9 in stretches of sixteen and 23.3 in stretches of sixty-four, back to its quiet value. Even under noise that masks everything, scattered noise leaves the accent worth 24.2 points against 17.6 in long stretches. Scattered noise leaves a downbeat in quiet in most bars; a long stretch can take a whole bar’s downbeats at once, and then only how reliably the accent cuts through can rescue it.

The repertoire limits the result. A listener who knows five metres gains 13.7 points from the accent in quiet and 21.5 in a room that passes loud strokes and half the soft ones. One who knows ten, 23.4 and 25.9. One who knows thirty gains 30.0 in quiet but only 24.7 in that room, and one who knows a hundred, 11.0 and 8.1. With many metres to tell apart, two of them often share their downbeats over the stretch the listener hears and differ only in their soft strokes, and no accent can help with that; noise that blurs the soft strokes then costs more than the accent can buy back. The larger the repertoire, the more the dancer depends on hearing everything, and the less a loud room’s accent can stand in for it.

The model the counts rest on #

Additive metres of nine to twenty-five steps from groups of two and three; repertoires drawn as in the earlier essays, ten metres unless stated, the entry at a random step, sixteen steps of present. The landmarks are the starts of bars and of long beats; the marked stream accents the bar’s first. The noise is a two-state chain over steps with a stated share of time and mean stretch length, and each stroke played under it is heard with the stated probability — one for a marked downbeat, another for any other stroke — decided by a seeded draw independent of the noise, so that every setting is the same rooms and the same entries. The listener hears the noise, takes every heard stroke as fact, takes a silence in quiet as proof, and takes a silence under noise as evidence at the ratio of the two metres’ chances of a stroke being masked there. A rival is dismissed at nineteen to one.

What the model leaves out #

Loudness as anything but a probability. An accent here is heard or not, and its loudness enters only as the chance of being heard. A real accent heard under noise may be heard as quieter than it is, and a listener might mistake a heard ordinary stroke for an accent; neither confusion is in the model, and both would weaken the marked stream.

Noise that the band responds to. A band in a loud room plays louder, and a band’s dynamics change from bar to bar. The survival probabilities are held fixed across the evening; a band that played its downbeats harder as the room got louder would keep the accent’s worth higher than the figures show.

Whose rooms #

The repertoire is the additive dances of the Balkans and Anatolia, and the rooms are weddings, cafés and village squares where those dances are danced with talk and children and traffic going on. The arithmetic is about a listener inferring a metre from what reaches them, and so it applies to any music whose metre a newcomer must find while others talk. It is not a claim about how any dancer listens.

Still open: a downbeat that is marked by more than loudness #

Every accent here is a matter of level, and a level is exactly what noise interferes with. A band has other ways to mark a downbeat that noise interferes with differently: a drum stroke whose timbre is unlike anything else in the music, a bass note an octave below the rest, a change of instrument on the one. Noise masks by frequency band as well as by level, and talk is concentrated in the middle of the spectrum, so a downbeat marked by a low drum or a high bell may cut through talk where an equally loud accent on the melody does not. The computation is this one with a survival probability that depends on which band the mark is carried in, against the spectrum of a crowd, and it would say which way of marking the one survives a loud room best. That is the arranger’s version of the question this essay answered for the player.