A felt drum reaches only where the heard one already does

In a crowd the downbeat belongs to the drum took a late dancer at a wedding — arriving partway through a dance in an additive metre and trying to recognise which of ten dances it is — into a room full of talk, and asked which way of marking the downbeat survives it. Talk has the spectrum of speech, loud between 250 and 400 hertz and weak below and above, so a mark’s survival depends on where its energy lies. A louder melody note, in the register of speech, was lost with the crowd ten decibels over the band. A low drum, with its energy at 63 to 125 hertz where talk is weak, survived until the crowd was twenty-three decibels over. Marked by the drum, the downbeat was worth 19.8 points to the dancer under a crowd fifteen decibels over the band, the melody’s accent 11.6.

That essay ended with a second channel it had not counted. A low drum near a stage is not only heard; the chest and the floor vibrate with it, and talk, which has almost no energy at sixty hertz, cannot mask a vibration at all. A drum heard and felt would survive any crowd, and the question was how far from the band that advantage reaches — how far out a dancer stands before the drum’s lead over every other mark stops being one no other mark can approach.

The answer has two parts, and neither is the one expected. A mark that always arrives is worth less than it sounds, because the crowd has already taken what made marks valuable. And the felt channel cannot be separated from the room.

Where the crowd left the four marks #

Everything is the earlier essay’s model. The dancer knows ten additive dances and comes in at a random step; the landmarks of the dance — the starts of its bars and long beats — are what recognition runs on, as a late dancer needs the landmarks, not the rhythm found; a crowd covers half the time in stretches of sixteen steps, and the dancer, who hears the crowd too, treats a silence in a noisy stretch as no evidence. A mark’s worth is the points it adds to recognising the dance within the present. The melody’s unmarked strokes are eight decibels softer than its accent and sit in the speech band, so they are the first things the talk takes.

A mark that always arrives #

The felt channel’s whole promise is a mark that arrives every time, so the first question is what such a mark is worth.

A downbeat that always arrives adds 28.9 points with the crowd level with the band, 22.1 at five decibels over, and 19.8 at ten — and stays at 19.8 however loud the crowd becomes. Up to fifteen decibels over the band the heard drum is the same mark, since it already gets through every time. Only at twenty decibels does the heard drum begin to fall behind: 17.3 against 19.8, then 12.9 at twenty-five and 11.4 at thirty. So the felt channel is worth 2.5 points at twenty decibels over the band, 6.9 at twenty-five and 8.4 at thirty, and nothing at all in a crowd any quieter.

The flat line is a ceiling, and it is lower than the quiet room’s 23.4. A mark that cannot be stopped by the crowd is still worth less in a crowd than in silence, because the crowd’s damage was never only to the mark.

There is a floor too. A mark that never gets through the crowd still adds 11.3 points, from the downbeats that fall in the quiet half of the time; the melody’s accent sits on that floor from twenty decibels up. Every mark lies between 11.3 and 19.8 once the crowd is loud, and the felt channel’s whole contribution is to hold the drum at the top of that range.

What the ceiling is made of #

The ceiling is set by the soft strokes. With every downbeat arriving and none of the soft strokes, the mark adds 19.8; with a fifth of the soft strokes arriving, 23.1; half, 27.0; four-fifths, 30.9 — more than in a quiet room. With every soft stroke arriving as well, the noisy half of the time is no longer noisy and the mark adds the quiet room’s 23.4.

The peak at four-fifths is the earlier essays’ finding seen from the far end: a marked downbeat is worth most when the stream around it is thinned but not empty, because the mark says where the bar starts and the surviving soft strokes say how it divides, and a listener given both can rule out rivals neither would rule out alone. A crowd that hides bars is kinder than one that hides onsets found the same preference for structure over count. The crowd takes the soft strokes first — 72 per cent arrive with the crowd level with the band, 14 per cent at five decibels over, none at ten — and once they are gone, the best any downbeat mark can do is the 19.8 that a mark alone provides.

So the felt drum has a lower ceiling than the lead imagined. A mark no crowd can stop is a mark worth 19.8 points, not one worth what it was in quiet, and the difference — the eleven points between 19.8 and the 30.9 a thinned stream could give — lives in the soft strokes, which no channel for the downbeat can bring back. A band that wanted those back would have to protect the inner beats as well, by moving them into the drum or the bass, which is the remedy the earlier essay found for the downbeat itself.

The room the drum is felt in #

The lead asked how far from the stage the advantage reaches. A drum’s sound reaches a dancer two ways: directly, falling six decibels for every doubling of distance, and by reflection, as a reverberant field that is nearly the same everywhere in the room. The two are equal at the room’s critical distance, which in a room of a few hundred people with hard walls is a few metres; four is taken here. The band is set to the earlier essays’ 70 decibels at eight metres, and the crowd is local talk, as loud at the back of the room as at the front.

The felt channel is driven by the drum’s own airborne level at the dancer, since it is the low-frequency sound pressure that sets the chest and body vibrating. That means it follows the same law as the hearing. Its threshold for this purpose is not known, so it is stated instead as a distance: the radius at which the drum is felt half the time.

The band’s level falls 11.3 decibels between one metre and eight, and less than one decibel between eight and thirty-two: past the critical distance the room is one level everywhere. The felt channel inherits that shape. A drum felt half the time at four metres is felt at one metre every time, at eight metres a quarter of the time, and at thirty-two metres still 16 per cent of the time, because the drum’s level has stopped falling. A drum felt half the time at sixteen metres is felt at thirty-two metres 47 per cent of the time. Distance from the stage matters only inside the first few metres; beyond them, either the reverberant drum is strong enough to be felt or it is not, all the way to the back wall.

The first eighty milliseconds are a different room found the same division for clarity: inside the critical distance a listener hears the direct sound and outside it the room. For a felt drum the division is sharper still, because what is felt is level alone, and the room’s level is flat.

Where the felt channel pays #

Under a crowd twenty-five decibels over the band, every mark is worth most at the front of the room, where the band is loudest against the talk: at one metre the drum adds 19.8, the bell 19.6, the bass 19.4 — all near the ceiling — and only the melody’s accent, at 12.0, is still buried. By eight metres the drum is down to 12.9 and the rest to the floor, 11.3 to 11.5, and they stay there to the back of the room.

With a drum felt half the time at four metres, the felt channel lifts the drum by almost nothing at one metre, where it was heard anyway; by 1.7 points at two metres; by 2.0 at four; by 1.3 at eight; by 0.9 at thirty-two. Turn the dial to a drum felt half the time at sixteen metres and the advantage grows across the whole room, but it is still largest near the stage. Turn it to two metres and at thirty-two metres the felt drum is worth exactly what the heard one is.

This is the surprise, and it comes from one fact. The felt channel and the heard channel are driven by the same quantity, the drum’s level at the dancer, and both fall the same way across the room. A drum loud enough to be felt at some distance is, at that distance, also loud enough to be heard over much of the crowd; where the drum has fallen too far to be heard, it has usually fallen too far to be felt. Talk cannot mask the vibration, but the room dilutes it exactly as it dilutes the sound, and the felt channel’s advantage therefore sits in the narrow band of distances and crowd levels where the drum is strong enough to be felt but the crowd is loud enough to bury its sound.

That band is easy to locate. At two metres from the stage the felt channel adds nothing until the crowd is twenty decibels over the band, then 1.7 points at twenty-five, 4.6 at thirty and 6.6 at thirty-five — a crowd far louder than the band itself. At eight metres it adds 0.4, 1.3, 1.8 and 1.8. At thirty-two metres, 0.4, 1.0, 1.1 and 1.2. The felt drum’s lead over every other mark is real only for a dancer near the stage in a crowd that is drowning the band, and there it is worth a few points.

What this says about the lead #

The earlier essay’s lead imagined the felt channel as one that “always gets through”, making the drum’s survival close to certain wherever the dancer stands. Neither half survives the arithmetic. A felt channel strong enough to be certain everywhere would need the reverberant drum to clear the body’s threshold, which would make it a very loud drum; and a mark that is certain is worth 19.8 points under a loud crowd, not the quiet room’s 23.4, because the crowd’s real damage is to the soft strokes, which the felt channel does nothing for.

What the felt channel does do is extend the drum’s existing lead into crowds the heard drum cannot survive. In a crowd thirty decibels over the band, a dancer near a stage that feels the drum recognises the dance with eight more points of help than one who only hears it. That is the situation of a dancer in front of the speakers at a loud party, and it is a real, if narrow, advantage for the low drum that no other mark can match: the bass, which carries energy at the same frequencies, would be felt too only if it were as loud as a kick drum.

Each mark more robust, and each worth less #

Set beside the earlier essays, the felt drum completes a sequence. A dancer who comes in late needs the downbeat marked found the mark worth 23.4 points in a quiet room, and the accent buys two units however loud it is that in quiet the mark only has to be distinguishable. In a loud room the accent had to be loud; in a crowd it had to be low; and now, felt as well as heard, it can be made certain. Each step made the mark more robust, and each step’s gain was smaller than the last, because the crowd’s damage moved off the mark and onto the stream around it. By the time the mark is certain, the downbeat has been protected completely and the dance is still harder to recognise than in quiet, because the soft strokes that said how the bar divides are gone.

That is the general shape of the result, and it is not particular to drums or dancing. A signal made of a strong marker and weak context is protected first by protecting the marker, and the protection has a ceiling set by the context. One sound hides another priced the masking that removes the context; no amount of strength in the marker returns it. A band that wants a crowd to recognise its dances has, past a point, to make the inner beats survive, not the downbeat.

The model under the room #

The room has one critical distance and a perfectly diffuse reverberant field. Real rooms have a field that decays somewhat with distance, which would make the far end of the room a little quieter for both the music and the felt drum, and leave the comparison between them unchanged.

The felt channel follows the drum’s airborne level. Vibration carried through the floor from the stage follows a different law — it depends on the floor’s construction and falls with distance through the structure — and would favour dancers on the same floor as the drum. That is not modelled.

The crowd is local and spectral, as in the earlier essay: a fixed speech spectrum at a stated level, the same everywhere in the room. In a real room people near the band may talk louder to be heard, which would narrow the stage’s advantage.

Recognition is the earlier essays’ — ten dances, a random entry, a listener who hears the crowd and counts silences only in quiet stretches, as a dancer who hears the crowd can trust the quiet set it up.

What these figures cannot show #

What the body’s threshold is. The felt radius stands in for a threshold for perceiving low-frequency sound as vibration, which depends on frequency, on the listener and on how the vibration is attended to. The figures report what follows from each radius rather than choosing one.

Whether a felt beat counts the same as a heard one. The figures add the felt channel to the heard one as a second way for the same mark to arrive. A dancer might weight a felt downbeat more than a heard one, as the thing the body moves to, or less, as a sensation without a pitch. A crowd costs a late dancer the silences, not the steps is a reminder that how a listener interprets what reaches it matters as much as what does.

Still open: the inner beats felt too #

The ceiling is set by the soft strokes, and a felt channel could carry those too. A band whose drummer plays every beat on a kick drum — the four-on-the-floor of much dance music — puts every landmark, not only the downbeat, into the channel talk cannot mask. Felt that way, the stream would arrive whole in the noisy half of the time, and the downbeat mark’s worth would return to the quiet room’s 23.4. But a stream in which every beat is marked alike has no downbeat at all, and the figures above found a marked downbeat worth most when it stands out from a thinned stream around it. The computation is this one with a felt channel for every stroke, the downbeat louder than the rest by a stated margin, and it would say whether felt inner beats with a felt accent beat a felt downbeat over unheard inner beats — or whether, once everything is felt, the accent is too small a difference in vibration to mark anything.