A row that hears its neighbours moves its tuning, it does not hide it

Singers who hear each other late lock their vibratos in opposition found something a duet can do that nobody would design: two singers who each pull their vibrato towards what they hear of the other, hearing it a third of a vibrato cycle late, settle with one voice’s peak on the other’s trough, and that arrangement hides their tuning from a listener better than if they had not listened at all. Sixteen singers cannot all oppose each other then put the same coupling into a section where every singer hears every other about equally, and found the trick gone. If A opposes B and B opposes C, then A and C are in step, and the coupling between A and C wants them apart. No arrangement of sixteen satisfies the delay everywhere, and at the latencies where a pair opposes, the section does not lock at all: sixteen vibratos at sixteen rates, the tuning exactly as exposed as if no one had listened.

That essay ended on its own assumption. Everyone hearing everyone alike is what made opposition impossible. A singer in a long row in a dry room hears the voices beside it much more than the far end, and a chain of neighbours has a consistent opposed arrangement: every singer opposite the ones beside it, which puts every second singer in step. If the row could find that arrangement, it would be a line of duets, each neighbouring pair hidden as the pair was. The question was at what point a row stops behaving like a crowd that cannot oppose and starts behaving like that.

The answer turns out to have two parts that pull against each other. A row that hears its neighbours does move towards alternation, and it does hide the tuning between neighbours as well as a duet. It also exposes the tuning between every second singer by almost exactly the amount it hides, so that a listener hearing the whole row hears what independent voices would have given.

The coupling, falling with distance #

The model is the section essay’s, put in one row. Sixteen singers stand sixty centimetres apart; each vibrato is a phase oscillator with its own rate drawn from the spread measured in trained voices; each is pulled towards what it hears of every other, after the sound’s travel and a common latency, with a total pull of ten radians a second — the value the pair and section essays used. The one change is how that pull is shared out. In the section essay every voice got an equal share. Here each voice’s share is weighted by its distance to a power, from nought, which is the equal sharing, through two, the way sound energy falls in a free field, to eight and beyond, where a singer effectively hears only the voices beside it. Four draws of rates and starting phases are averaged.

What a listener can hear of the tuning is the earlier essays’ count: two voices a draw of fifteen cents apart with 71-cent vibratos, and the beats a second a listener can follow at 3,200 hertz, where the singers’ formant puts the energy, given the phase gap between the two vibratos at each moment. Sung straight, two such voices offer 2.33 followable beats a second; with independent vibratos, 0.68; locked in opposition, 0.19.

Neighbours hidden, every second singer exposed #

At sixty to a hundred milliseconds — the band where a pair locks in opposition — a row that hears mainly its neighbours lets a listener follow 0.28 to 0.30 beats a second between neighbouring singers. That is far below the 0.68 of independent vibratos and close to the opposed pair’s 0.19: neighbours in such a row are nearly as well hidden as a duet. Between singers two places apart the count goes the other way: 0.94 at sixty milliseconds, 1.21 at eighty, 1.15 at a hundred, nearly double what independent voices give.

And over every pair in the row, the third line sits on the independent voices’ line: 0.67 at sixty milliseconds, 0.67 at eighty, 0.69 at a hundred, against 0.68. The row has moved its tuning from one set of pairs to another and left the total where it was.

Turn the dial to nought and the picture is the section essay’s again: at eighty milliseconds neighbours give 0.67, every second singer 0.63 and every pair 0.65, because a row that hears every voice alike is a crowd that cannot oppose, whatever shape it stands in. At a falloff of two, the free-field rate, the split has begun — neighbours at 0.35, every second singer at 0.79 — and by four it is nearly complete, at 0.30 and 1.15.

The pattern along the row #

Drawn against distance along the row, the alternation shows as a zigzag. Hearing mainly its neighbours, the row gives 0.28 beats a second at one place apart, 1.21 at two, 0.37 at three, 0.79 at four, 0.51 at five. The odd distances are hidden and the even ones exposed, and the contrast fades with distance — by six or seven places the counts are near the independent line — then comes back weakly towards the far end of the row, where the ends of the chain constrain each other through everyone between. Heard alike, the row gives 0.62 to 0.74 at every distance: no pattern, only the crowd.

So the lead’s picture is right in shape. Opposition between neighbours is consistent along a chain, and a row that hears mainly its neighbours finds it: each neighbouring pair is a duet, opposed, and each pair two apart is a pair of singers both opposite the one between them, which puts them in step and exposes their tuning as a straight-sung pair’s would be exposed, less what the drift costs. It is the frustration argument run backwards. Where everyone hears everyone, the arrangement the delay favours for each pair is one the whole cannot adopt; where each hears its neighbours, the arrangement the delay favours is available, and the cost is paid by the pairs the delay never asked about.

The row half-alternates #

The settled phases make the arrangement visible. Heard alike, the row’s phases scatter: most dots sit near the centre, drifting against the first singer, and the few on the rim are on no particular side. Heard mostly beside, every dot is on the rim — each singer has settled into a fixed relation with the first — and the odd places gather in one arc of the cycle and the even places in the arc opposite. That is alternation, but not exact: each group is spread over about a quarter of the cycle, because the phase twists slowly along the row as well as flipping between neighbours.

A single number summarises it. The staggered order turns every second singer half a cycle and measures how far the row then lines up, one for exact alternation and near nought for none. In this draw it is 0.80 heard mostly beside and 0.25 heard alike.

Averaged over four draws, the staggered order climbs from 0.21 with equal hearing to 0.37 at a falloff of two, 0.52 at four and 0.60 at eight, levelling there. R1 stays under a tenth throughout: the row never comes into step. And the beats over every pair stay between 0.64 and 0.67 at every falloff, which is the figure’s real point. Hearing mainly its neighbours rearranges the row completely and changes its overall exposure by nothing measurable.

Why the total does not move #

The cancellation is not an identity of the model, and it is worth saying what it does and does not rest on. A row in exact alternation would hide every odd-distance pair and expose every even-distance pair fully, and since a row of sixteen has about as many of one as the other, its overall count would be halfway between the opposed pair’s 0.19 and the straight pair’s 2.33 — about 1.26, far above independent voices. The row is not in exact alternation. Its neighbours are opposed and its next neighbours are only partly in step; beyond a few places apart the relation dissolves into drift, and the many distant pairs sit near the independent value. The gains near the singer and the losses just beyond roughly balance, and the long tail of distant pairs carries the rest.

That balance is a measurement, not a law: it holds to the hundredth across the falloffs and the latencies drawn here, and it is the kind of coincidence that a different beat count, or a formant at a different frequency, might break. What is robust is the shape underneath it: local hearing cannot make a row hide its tuning the way a duet does, because the same coupling that opposes neighbours puts the next pair in step, and the next pair is as numerous as the first.

How long the chain can be #

The row’s length decides how far the alternation holds. Four singers in a row alternate almost exactly, with a staggered order of 0.91; eight reach 0.73; sixteen, 0.60; twenty-four, 0.39. A short chain can settle into one alternating pattern from end to end; a long one alternates locally and lets the pattern twist along its length, so that singers far apart bear no fixed relation to each other.

The neighbours do not notice. In every length the tuning between neighbouring singers is followable at 0.28 or 0.29 beats a second, and over every pair the row gives 0.67 or 0.68. A quartet of voices standing in a line, each hearing mainly the ones beside it, is the nearest thing in this model to the lead’s line of duets — and it is still, over its six pairs, exactly as exposed as four singers who did not listen.

What the duet keeps, and what the row gives away #

The comparison with the pair is worth making exactly, because the row does reproduce the pair’s result where it can. Two singers opposed eighty milliseconds late leave a listener 0.19 beats a second; neighbours in the row leave 0.28. The difference is the row’s twist: a neighbouring pair in the row is not held at exactly half a cycle, because each of its members is also being pulled by its other neighbour, which is pulling it a little towards some other phase. The duet has nobody else to answer to.

What the row gives away is in the pairs the duet does not have. A duet has one pair and it is opposed. A row of sixteen has fifteen neighbouring pairs and fourteen pairs two apart, and the arithmetic of a chain means the second set is pulled towards being in step exactly as hard as the first is pulled towards opposition. Beats between two voices in step are beats as arithmetic describes them, a difference of frequencies heard plainly, and the vibrato no longer smears them. That is the price of the chain, and nothing about the coupling can avoid it: opposition between neighbours is agreement between next neighbours.

The same logic is what sixteen sweeps against sixteen met from the other side, counting the pairs between two whole sections an interval apart: in a choir the number of pairs grows much faster than the number of singers, and any effect that acts pair by pair is dominated by the pairs nobody arranged.

What local hearing does without latency #

The lead’s question was about the opposing band, but the coupling’s falloff matters at least as much outside it.

With no latency, a row that hears every voice alike locks in step: R1 0.95, and a listener can follow 1.72 beats a second per pair, three quarters of what a straight-sung row would give. That is the section essay’s quick-loop regime, in which listening exposes a section’s tuning. Let the pull fall with distance and the lock fails. At a falloff of two, R1 is 0.78; at four, 0.55; at eight, 0.52. The beats fall with it, to 1.29, 1.00 and 0.96: local hearing hides nearly half of what a whole-row lock would show.

The mechanism is ordinary for coupled oscillators. A chain whose links are each pulled towards their neighbours can carry a slow twist of phase along its length — each singer a little ahead of the one beside it — that a fully connected crowd cannot, and the twist is enough to keep sixteen vibratos from settling into one. The row’s neighbours are still nearly in step, at 1.62 beats a second, and its distant pairs are not.

So local hearing has opposite effects at the two ends of the latency range, and neither is what the lead expected. In the opposing band it rearranges the tuning without hiding any of it. With a quick loop it prevents the row from locking and hides a good deal.

What it means for a choir standing in a line #

A choir that stands in a single long row and hears mainly its neighbours, if its vibratos are entrained through hearing at all, is in one of two states. If whatever moves a vibrato acts within a few tens of milliseconds, the row does not lock as a whole; it carries a slow twist from one end to the other, and a listener hears a good deal less of its tuning than if it had locked — less, in this count, than a section standing in a block where everyone hears everyone. If the loop takes sixty to a hundred milliseconds, the row half-alternates, neighbours opposed and every second singer in step, and a listener anywhere far enough away to hear every pair hears the row exactly as exposed as independent voices.

That puts a qualification on what a choir does that a soloist cannot. Many slightly different versions of one note are a choir’s warmth, and the earlier essays found that listening could only ever expose a section’s tuning or leave it as it was. A row that hears locally is a third case: at short latency it hides more than a block would. A choir’s vibratos must lock to two milliseconds priced the lock that exposes tuning; a row’s geometry is one way that lock fails to form even when every singer is listening quickly.

It also matters where the listener is. A listener standing close to two neighbours in the opposing band hears them as a duet hides itself, at 0.28 beats a second; a listener close to two singers one apart hears them nearly twice as exposed as independent voices. A vibrato hides a choir’s tuning above four hundred hertz found how much the vibrato alone conceals; how much of that concealment survives depends, in a row, on which pairs a listener is close enough to separate.

The model under the row #

One row, a falloff and a total. Each singer’s pull is shared among the others in proportion to their distance to a power, and always totals ten radians a second, so a singer who hears mainly its neighbours is pulled by them as hard as a section singer is pulled by everyone. A singer whose total pull also fell with the room’s absorption would be pulled less in a dry room and would lock less, which would push every result towards independent voices.

Phase oscillators. Only the vibratos’ timing is coupled, never their width or their centre, as in the pitch that does not wobble and every essay since. A real singer’s vibrato varies in rate and width from cycle to cycle, as a note that is never at its pitch measured, and that variation would loosen every pattern drawn here.

The beats count is the pair essays’, read from the pair’s phase gap moment by moment, at one formant frequency and one detuning. The cancellation in the whole row’s count is a result of that count’s shape, not a general law, and the essay says so where it reports it.

What a model of hearing cannot settle #

Whether vibratos are entrained by hearing. The pair and section essays’ caution stands: sight, breath and the conductor are other channels with other delays, and nothing here shows that singers’ vibratos follow each other through the ear at all.

What a choir actually hears of itself. The falloff is swept rather than measured. In a reverberant hall a singer hears the whole choir through the room as well as the neighbours directly, which is closer to the equal sharing; on a dry stage the neighbours dominate. Recordings of a row of singers with a microphone on each would show which regime a choir is in — one rate, a twisting chain, or an alternation — which no arithmetic can.

Still open: a block of rows, each hearing its neighbours #

A row is one line of neighbours. A section standing in two or three rows has neighbours beside it and in front, and a lattice of neighbours cannot always alternate: three singers in a triangle are a small crowd, frustrated as the section was, while four in a square can alternate like a chess board. The computation is this one with the singers in a block and the pull falling with distance in two dimensions, and it would say whether the staggering of a choir’s rows — singers placed in the gaps of the row in front, which makes triangles — frustrates the alternation a square block could find, and whether that is one more reason the arrangement of a section on the risers changes what a listener hears of its tuning.