Sixteen singers cannot all oppose each other

Singers who hear each other late lock their vibratos in opposition gave two singers of a section the typical vibrato of trained voices and let each pull its vibrato towards the one it heard from the other, heard late by the time sound and the auditory and motor loops take. With little delay the pair locked nearly in step; with a delay of forty to a hundred and twenty milliseconds, a third to two-thirds of a vibrato cycle, it locked in opposition, each singer’s peak where the other’s trough was. Locked in opposition, the two voices’ instantaneous frequencies differ by twice the vibrato’s swing almost all the time, and the beats between them race past any rate a listener can follow: at the singers’ formant a listener could follow 0.19 beats a second, against 0.68 with independent vibratos and 2.33 sung straight. The pair, by listening, hid its tuning better than if it had not listened at all.

That essay ended on the question its pair could not answer. A section is not a pair. Every singer hears fifteen others, each at the delay their positions give, and a network of delayed oscillators has more arrangements open to it than two in step or two opposed: clusters, a phase turning steadily along a row, or no lock at all. Each arrangement decides which pairs of voices beat followably, and so how much of the section’s tuning a listener hears.

The answer is the simplest of the possibilities, and it removes the pair’s most surprising result.

Where the pair left it #

The figure is the pair essay’s measure, and it is the one every result below is read through. Two voices a draw of fifteen cents apart, each with a 71-cent vibrato, hold a fixed phase between their vibratos; a beat counts as followable while the instantaneous rate between their partials is under fifteen a second. At the 3,200-hertz formant a pair in step gives back 2.23 of a straight pair’s 2.33 beats a second; thirty degrees apart, about 1.0; ninety, about 0.4; in opposition, 0.19. For a section, the same table is read for every one of its 120 pairs at every moment, at the phase gap that pair actually has.

Sixteen singers who hear each other late #

The section stands in two rows of eight, sixty centimetres apart, the rows eighty centimetres apart — a choir’s ordinary spacing. Each singer hears every other after the sound’s travel, three milliseconds a metre, and after a common latency: the time a heard pitch takes to reach whatever moves a singer’s vibrato, which the pair essay found decides everything and which no measurement used here fixes. Each vibrato is a phase oscillator with its own rate, drawn from the measured spread of trained voices, and is pulled towards what it hears of the fifteen others with a total strength shared equally among them. Four draws of rates and starting phases are averaged.

Two numbers summarise what the section does. R1 is the length of the average of the sixteen vibratos’ phases, one when every vibrato is in step and near nought when they are spread round the cycle. R2 is the same for twice the phases, which is one both when every vibrato is in step and when the section has split into two halves in exact opposition. A section split in two would show R2 high with R1 low.

With the pull at ten radians a second, the pair essay’s value, the section is in step with no latency (R1 0.96), and stays so to twenty milliseconds (0.89). By forty it is half-ordered, and from fifty to a hundred milliseconds R1 is between 0.07 and 0.19 and R2 never above 0.24: the section is neither in step nor split in two. From 110 milliseconds it comes back into step, at 0.68, then 0.90 at 120 and 0.97 at 150, where the latency is nearly a whole vibrato cycle and in step is in step again.

At no latency on the dial does the section ever settle into opposed halves. Turn the pull down to four and the section never orders much at all; turn it up to thirty and it stays in step from nought to forty milliseconds and from a hundred upwards, leaving a narrower gap around sixty to eighty — but the gap is still disorder, never opposition.

Why the opposed lock cannot form #

The pair’s opposed lock is a solution to a two-body problem: each singer’s peak sits where the other’s trough was a delay earlier, and with two singers that is consistent. With three it is not. If A is opposite B and B opposite C, then A and C are in step, and at a delay that favours opposition the coupling between A and C pulls them apart. Every arrangement of sixteen leaves most pairs either in step or at some angle neither in step nor opposed, and none satisfies the coupling everywhere. The section is frustrated: the arrangement the delay favours for each pair is one the whole cannot adopt.

A frustrated network has a few ways out. It could split into two clusters, which satisfies the coupling for the 64 pairs that cross between clusters and violates it for the 56 inside them; it could spread its phases evenly round the cycle, a splayed state; or it could fail to lock and let every vibrato run at its own rate. In this section, with every singer hearing every other at nearly the same delay, it takes the last.

With no latency the sixteen phases gather in one arc at the top of the circle; at 150 milliseconds they gather again. At eighty milliseconds most of the dots sit near the centre, because their phase against the first singer’s turned steadily over the five seconds and its average came to almost nothing, and the few on the rim are singers whose rates happened to sit close to the first singer’s. There is no cluster, no splay, and no opposed pair of groups: just sixteen vibratos drifting past one another.

The rates make the same point in another unit. Where the section locks it sings one rate — 5.61 at no latency, 6.94 at 120 milliseconds, 6.22 at 150 — and, as the pair essay found, the rate is dragged by the delay, at twenty milliseconds down to 4.89, below anything measured in trained singers. Where it does not lock, from forty to 110 milliseconds, each singer keeps a rate near its own, spread over a hertz: at eighty milliseconds, 5.51 to 6.35. A recording of a section in that state would show sixteen vibratos at sixteen rates, indistinguishable from singers who did not listen at all.

What a listener hears of the tuning #

At the formant, a section pulled at ten radians a second and hearing itself with no latency lets a listener follow 1.74 beats a second per pair, three-quarters of what a straight section gives; at a pull of thirty, 2.13, nearly all of it. In the disordered band the count falls to 0.65 to 0.68, which is the independent vibratos’ 0.68 to the hundredth. It never approaches the pair’s 0.19.

So the pair’s surprising result does not survive the step to a section. The pair essay found listening worse than not listening, because a delayed pull locked the two voices in the one arrangement that hides most. A section has that arrangement taken away by its own size, and the worst it can do is to fail to lock and leave the tuning exactly as exposed as if no one had listened. A section that listens to itself can expose its tuning or leave it as it was, but it cannot hide it. The lead’s third possibility — a splayed lock hiding the tuning as completely as a section that never listened — is right about the outcome and wrong about the mechanism: there is no splayed lock, only no lock.

How many singers it takes #

Between the pair and the sixteen the change is gradual. At eighty milliseconds two singers lock in opposition in every draw (R2 0.96) and let a listener follow 0.26 beats a second. Three singers lock in half the draws, in arrangements that are not opposition — R2 falls to 0.43 — and give 0.42. Four give 0.51, six 0.58, eight 0.62, twelve 0.64, sixteen 0.65. Every singer added makes the frustration worse and gives back some of the tuning the pair hid, until the section is indistinguishable from independent voices.

That makes the pair essay’s result a property of duets. Two singers sharing a part, or a solo voice with one other close beside it, are the only arrangement in which a delayed lock can hide tuning. What a choir does that a soloist cannot — produce many slightly different versions of one note — turns out to include this: a choir cannot, by listening, conceal its intonation in the way a duet can.

The rate a locked section sings at #

Where the section does lock, it locks at a rate none of its singers chose. With no latency it sings at 5.61 a second in the draw drawn above, a rate set by the whole section’s coupling rather than by any one singer’s own, and in that draw towards the slow side of the spread the singers were drawn from. At twenty milliseconds it is dragged to 4.89, below anything measured in trained singers, each singer holding back its peak to meet a peak heard late. At 120 milliseconds it races to 6.94. The pair essay found the same drag for two singers — 5.14 a second at twenty milliseconds for the pair it drew, 4.89 for this section — and the section shows it as clearly as the pair.

That is a second prediction a recording could test, and in a sense a sharper one than the pair’s. A section that locks through hearing should sing a vibrato rate that differs from its singers’ solo rates by an amount that depends on the latency, and the whole section should differ together. A section that does not lock should keep sixteen rates, each near the singer’s own. A note that is never at its pitch measured how much a single voice’s vibrato varies; whether a section’s sixteen vibratos converge on a rate none of them sings alone is a measurement nobody needs a model to make.

Opposition is a duet’s resource #

The two results — the pair’s opposed lock and the section’s failure to lock — are not in tension; they are the same coupling meeting different numbers of voices. A duet is the only arrangement in which every pair of voices can stand opposite every other, and so the only one in which a delayed coupling can find the arrangement it favours. Add a third voice and the favoured arrangement is no longer available; add enough and the voices stop trying.

That says something about the beats that are arithmetic in a choral unison, and about where they come from. In a section, a listener hears the tuning of many pairs at once, and sixteen sweeps against sixteen counted what happens when two whole sections’ vibratos sweep past each other an interval apart, where the pairs number in the hundreds rather than one. The coupling would act across that interval too, and every argument above about frustration applies to it with more voices, not fewer. The coupling result adds one more difference: whatever the singers do by listening, a section’s tuning stays at least as audible as independent voices would leave it, while a duet’s can vanish. Two singers sharing a line in a small ensemble are the one case where hearing each other late could make their intonation harder to judge.

What it means for a choir #

The arithmetic gives a choir that listens to itself two regimes, set by how fast whatever entrains a vibrato acts. If the loop is quick — under about thirty milliseconds, or longer than about 110 — the section’s vibratos lock in step, and the stronger the pull the more of the section’s tuning a listener can follow at the formant: three-quarters to nearly all of a straight section’s beats. That is the arrangement a choir’s vibratos must lock to two milliseconds priced as the condition for exposing tuning, arrived at here by the section’s own coupling rather than imposed. If the loop takes forty to a hundred milliseconds — the range the pair essay found for a deliberate pitch correction — the section does not lock, and a listener hears its tuning exactly as through independent vibratos, which a vibrato hides a choir’s tuning above four hundred hertz found already conceals most of it.

Either way, the section’s listening never makes its tuning harder to hear than not listening would. A conductor who wants a section’s vibratos out of each other’s way is better served by asking for less vibrato than by hoping the singers’ ears will arrange it, and a section whose vibratos do lock is one whose intonation will be heard.

The model under the section #

The coupling is all-to-all and equal. Every singer hears every other with the same weight, the delays differing only by the sound’s travel across a few metres, which is a few milliseconds against a latency of tens. A real singer hears the neighbours more than the far end of the row, and a coupling that fell with distance would make the section more like a chain of pairs, in which local opposition might form; that is not computed here.

The pull is swept, not measured, and the latency is the pair essay’s unknown. The conclusions are stated across both.

The beats count is the pair essays’: two voices a draw of fifteen cents apart, followable while the instantaneous rate is under fifteen a second, read from a table of fixed phase gaps every five degrees. A pair whose gap drifts is read at its gap moment by moment, which treats a slowly drifting pair as a sequence of fixed ones.

Each vibrato is a phase oscillator, as in the pitch that does not wobble and the pair essay: only its timing is coupled, never its width or its centre.

What the section cannot settle #

Whether vibratos entrain through hearing at all. The pair essay’s caution stands: a vibrato may be entrained by sight, breath or the conductor, each with its own delay, and the figures take the hearing loop as given.

What recordings of sections show. The model predicts a sharp division: one shared vibrato rate, or sixteen different ones, with nothing in between that looks like a split section. Sections recorded singer by singer — one microphone each — would show which regime real choirs are in, and whether their shared rate, where there is one, is pulled away from the singers’ solo rates by the amount the delay predicts.

Still open: a row that hears only its neighbours #

Everything here assumed each singer hears the whole section about equally. In a long row in a dry room, a singer may hear mainly the two or three voices nearest, and a chain of nearest neighbours is a different network: opposition between neighbours is consistent along a chain, every other singer in step with each other and opposite the ones between. A row that hears only its neighbours might then settle into alternation — the pair’s opposed lock repeated down the line — and hide the tuning between neighbours while exposing it between every second singer. The computation is this one with the coupling falling with distance, from equal weights to nearest neighbours only, and it would say at what point a section stops behaving like a crowd that cannot oppose and starts behaving like a line of duets that can.