The felt channel carries the inner beats, not the downbeat

A felt drum reaches only where the heard one already does gave a late dancer’s downbeat a second way in. A low drum near a stage is felt in the body as well as heard, and talk cannot mask a vibration, so a drum-marked downbeat felt as well as heard always arrives. It was then worth less than expected: 19.8 points of recognition in a loud crowd, against 23.4 in a quiet room, because the crowd had already taken the melody’s softer strokes between the downbeats, and a marked downbeat is worth most when the stream around it is thinned but not empty. The felt channel held the downbeat at the top of a range whose ceiling the soft strokes set.

That essay ended on the obvious next arrangement. A band whose drummer plays a kick drum on every beat — the four-on-the-floor of much dance music, or, for the additive dances these essays use, a kick on every landmark — puts every landmark into the channel talk cannot mask. Felt that way, the stream would arrive whole in the noisy half of the time. But a stream in which every stroke is felt alike might have no downbeat at all: the felt channel carries a stroke’s arrival, and whether it carries the difference in level that marks the downbeat is another question. The lead’s worry was that once everything is felt, the accent becomes too small a difference in vibration to be felt, and the gain is lost.

The arithmetic says the gain is not lost, and that the accent was never most of it.

Four ways to carry the landmarks #

The model is the earlier essays’. The dancer knows ten additive dances and comes in at a random step; the landmarks — the starts of bars and of long beats — are what recognition runs on, as a late dancer needs the landmarks, not the rhythm found; the crowd covers half the time in stretches of sixteen steps; the dancer hears the crowd too and treats a silence in a noisy stretch as no evidence. Each stroke played in the noise gets through with a probability set by its level against the crowd’s in the stroke’s best band, the calculation in a crowd the downbeat belongs to the drum made, and a stroke felt as well as heard always gets through. The measure here is the share of entries in which the dance is recognised within the perceptual present, rather than a mark’s worth over an unmarked stream, so that arrangements with different inner strokes can be compared directly.

The four curves separate cleanly. With a drum downbeat and the melody between, recognition falls from 58.4 per cent with the crowd level with the band to 37.4 at ten decibels over and 28.9 at thirty. Feeling the downbeat holds it at 37.3 from fifteen decibels up: the earlier essay’s ceiling, expressed as recognition rather than worth. A kick on every landmark, heard, does much better while it can be heard — the quiet room’s 80.9 up to five decibels over the band, 68.1 at ten, 50.2 at fifteen — and collapses to the melody’s level by twenty, when the crowd has taken the softer kicks too.

The fourth curve is the one the lead asked about. With the dial at nought, so that the felt channel carries every stroke and no accent — the downbeat’s extra level is felt as nothing — a kick felt on every landmark gives the quiet room’s 80.9 up to fifteen decibels over the band, where the accent is still heard, and then 75.8 at twenty, 71.6 at twenty-five and 70.1 at thirty. In the loudest crowd modelled, a stream felt alike recognises the dance in seventy entries in a hundred, nearly twice the felt downbeat’s thirty-seven.

The accent is the small half #

Turning the dial prices the accent directly. With the crowd thirty decibels over the band — where the heard drum gets through one time in a hundred, so that the accent depends entirely on the body — recognition rises from 70.1 with no felt accent to 71.5 at a fifth, 74.3 at three-fifths, 76.4 at nine-tenths, and 80.9, the quiet room, when every felt accent is felt. At twenty-five decibels the rise is from 71.6 to the same 80.9.

So the whole of the accent is worth about ten points once every landmark arrives, and the inner strokes arriving are worth about thirty-three over the felt downbeat. The lead’s worry — that a stream felt alike has no downbeat — is right about the downbeat and wrong about what matters. A late dancer finding a dance among ten does most of the work with the pattern of landmarks: which steps carry a stroke and which do not. The accent says which landmark is the start of the bar, and that is worth something, but the rival dances that share a pattern of landmarks with a different downbeat are few. The silences between landmarks, which the crowd turns into non-evidence, are what the felt channel restores.

The curve has one step in it worth noticing. From nine-tenths to certainty it jumps by four and a half points, more than the rest of the dial gives in any tenth. That jump is the listener’s caution. A listener who knows a felt accent is sometimes missed must treat a landmark felt without an accent as weak evidence against a rival that marks it there, not as proof; once the accent is certain, a missing accent is proof again, and every rival that marks the wrong landmark is dismissed at once. A dancer who hears the crowd can trust the quiet found the same price paid for knowing that evidence can be lost; here it is paid on the accent.

Step by step in a loud crowd #

Set side by side at twenty-five decibels over the band, the arrangements make the division plain. A heard drum downbeat with the melody between: 30.4. The downbeat felt: 37.3, the earlier essay’s gain of seven points. A kick on every landmark, heard only: 30.4, no better than the melody, because at that level the crowd takes the kick’s softer strokes as surely as it took the melody’s. Every landmark felt with no accent: 71.6, a gain of thirty-four over the felt downbeat. Every landmark felt with its accent: 80.9, nine more, which is the quiet room.

The order of the gains is the reverse of the order the earlier essay’s question put them in, and it rests on the same masking calculation that in a loud room the accent has to be loud began, where a soft stroke was always the first thing a crowd took. The felt downbeat was the obvious use of a felt channel and is the smallest gain on the chart; the felt inner beats were the afterthought and are the largest. A crowd that hides bars is kinder than one that hides onsets found a dancer preferring the structure of a stream to its count; a stream whose every landmark arrives is all structure, and the accent is one more fact about it.

Which dances the felt stream rescues #

The ten dances a listener knows are drawn from additive cycles of nine to twenty-five steps, and the long ones dominate the count: there are many more ways to group twenty-four steps into twos and threes than nine.

The felt downbeat does well on the short cycles: 80 to 84 per cent of entries on dances of nine to eleven steps, where a single downbeat in the window already narrows the field, since few rival dances share that cycle length. On cycles of fifteen steps and more it finds the dance in only 34 to 45 per cent of entries. The felt stream lifts every length, but the long dances most: seventeen steps from 40 to 82 per cent, twenty from 40 to 75, twenty-four from 34 to 68. A long dance has many rivals of its own length, and a downbeat every twenty-odd steps says little about which; the felt landmarks between say how the cycle is grouped, which is exactly what tells rivals of one length apart.

That makes the felt stream most valuable for the dances that are hardest to recognise in the first place — the long, irregular cycles that a late dancer needs the landmarks, not the rhythm found could be identified only by their grouping. A felt downbeat is a solution for dances that were never in much difficulty.

How close the kick’s inner strokes must be #

The heard kick’s collapse between ten and twenty decibels is a matter of level. Its inner strokes were played eight decibels under the downbeat, as the melody’s were, and the crowd takes them first. A drummer can play them louder.

At ten decibels over the band, inner strokes within six decibels of the downbeat keep recognition at the quiet room’s 80.6 to 80.9; at eight decibels under it falls to 68.1, at twelve to 54.2. At fifteen decibels over the band the inner strokes have to be within two decibels to hold 80.0; four decibels under gives 66.5, eight gives 50.2. At twenty decibels nothing a heard kick can do recovers the quiet room — even inner strokes as loud as the downbeat give 58.5, because the downbeat itself now gets through only four times in five.

So a heard kick on every landmark is a good remedy in a crowd up to fifteen decibels over the band, provided it is played nearly flat, and the accent then has to be a small one. That is a trade the felt channel does not ask for: felt, the kick’s inner strokes arrive whatever their level, and the downbeat’s accent can be as large as the drummer likes, since it only matters where it is heard. In loud crowds the drummer who plays four on the floor with a heavy accent and the drummer who plays it flat are, for a dancer close enough to feel them, giving the same information.

Across the room #

The felt channel is local. The earlier essay found it following the drum’s airborne level in a reverberant room, so that the drum is felt mainly where it is loud.

With the drum felt half the time at four metres, a kick felt on every landmark holds a dancer at 80.9 per cent within a metre and a half of the stage and 80.0 at two metres; it falls to 59.3 at three, 52.2 at four, 41.3 at eight and 38.1 at thirty-two. The heard kick falls faster — 54.3 at two metres, 39.3 at four — and both the downbeat arrangements sit between 34 and 38 everywhere. Beyond eight metres every arrangement is within four points of every other, because the drum is neither felt nor heard over the crowd there.

That is the same verdict the earlier essay reached for the downbeat, and for the same reason. The felt channel is driven by the drum’s own level, which in a reverberant room falls to the reverberant floor within a few metres; inside that radius the felt kick is worth a great deal, and outside it there is nothing to feel. What changes is the size of the prize inside the radius: the felt downbeat added a few points near the stage, while at two metres the felt stream adds twenty-six over the same kick heard and forty-three over the felt downbeat. A dance floor arranged so that the dancers who need orienting are the ones near the drum is using the felt channel for what it is good at.

The accent’s old price, paid in another currency #

The earlier essays priced the accent several times, and the felt stream puts those prices in a new light. The accent buys two units, however loud it is found that in a quiet room a marked downbeat shortens the time to recognise a dance by about two steps whatever its loudness, because the accent is a binary piece of evidence: this landmark starts the bar or it does not. A dancer who comes in late needs the downbeat marked found that for a late entrant that piece of evidence was worth a large share of recognition.

Both findings were made with every landmark arriving. Here the comparison is between a stream in which every landmark arrives and the accent may be lost, and one in which only the accent arrives in the noise. The first is worth twice the second. So the accent’s value in the earlier essays was a value conditional on the landmarks being there to be marked; with the landmarks gone it buys far less, and with the landmarks present but the accent lost the dancer loses only the ten points the dial prices. A band deciding where to spend a felt channel is deciding between the conditional and the condition, and the arithmetic favours the condition.

What the arithmetic says about the lead #

The lead imagined a trade. Everything felt would bring the noisy half of the time back, but at the cost of the accent, and the question was which effect was larger. The figures say the trade exists and is lopsided: bringing the inner beats back is worth three times what the accent is worth, so even a dancer who feels no accent at all is far better off with every landmark felt than with the downbeat felt and the rest lost.

They also explain why the earlier essay found the felt downbeat so disappointing. It was the right channel carrying the wrong thing. The downbeat was already the stroke most likely to get through a crowd by being heard; the inner strokes were the ones the crowd took. A crowd costs a late dancer the silences, not the steps found that what a crowd destroys is the evidence in the gaps between strokes; the felt channel restores that evidence only if the strokes that bound the gaps are the ones being felt.

The practice claim is narrow. It concerns dance music played loud in a room with a crowd talking, with a low drum close enough to the dancers to be felt — a club, a dance hall, a village dance with a big drum — and additive dances of the kind these essays use, where recognising which dance is under way is what a late dancer needs. It is not a claim about why four-on-the-floor became a convention, which has many causes.

The model under the floor #

A felt stroke always arrives; a felt accent arrives with a stated probability. The body is treated as a channel that carries every stroke within its radius and carries the difference between a loud stroke and a soft one only sometimes. The second number is swept, because the threshold for feeling a level difference in vibration is not something the figures can supply.

The listener knows the rates. When the accent is uncertain, the listener treats a landmark felt without one as weak evidence against a rival that accents it there, at the odds the accent’s reliability gives. A listener who did not know the accent could be lost would be misled by every missed accent, and would do worse than the figures say until the accent was certain.

The crowd, the room and the recognition are the earlier essays’: a speech spectrum at a stated level, a single critical distance and a diffuse field, ten known dances entered at random.

What the figures cannot show #

How the body hears a stream. A dancer may lock onto a felt pulse in a way that makes the accent irrelevant, or may need the accent to know where to start a step. The figures treat the felt stream as evidence about which dance is playing, and say nothing about how it is used to move.

Whether a felt kick is the drum the band would play. The kick here sits on every landmark of an additive dance — the starts of the long and short beats of a 2 2 2 3 — which is not the evenly spaced pulse of a four-on-the-floor. Whether a band playing such dances puts a low drum on every landmark is a question about the repertory the arithmetic leaves open.

Still open: a felt pulse that does not follow the landmarks #

Every felt stroke here falls on a landmark, so the felt stream carries exactly the evidence the dancer uses. A drummer who keeps a steady felt pulse under an additive dance — strokes evenly spaced, at the dance’s fastest subdivision, with the landmarks marked only by the melody — gives the body a stream that carries no landmark at all, and arguably makes the crowd’s job easier by filling the silences with strokes that say nothing. The computation is this one with the felt strokes on every step rather than on every landmark, and it would say whether a felt pulse that ignores the dance’s grouping helps a late dancer, as a beat to hold onto, or hurts, as evidence that every rival dance agrees with.