Singers who hear each other late lock their vibratos in opposition

A vibrato hides a choir’s tuning above four hundred hertz found that two singers a section’s usual spread apart in tuning beat slowly enough to be followed when they sing straight, and that their vibratos, swinging seventy cents either side six times a second, sweep the beat rate through and past the range a listener can follow — so at the singers’ formant, near three kilohertz, a vibrato section exposes about a third of what a straight one does. A choir’s vibratos must lock to two milliseconds then asked what happens when the vibratos entrain. Half-entrained, they hide the tuning as well as independent ones; ninety per cent locked, they give back three-quarters of it; to come within a tenth of a straight section they must be locked about ninety-eight per cent, which is vibrato peaks about two milliseconds apart.

That lock was imposed. The essay drew each pair’s vibratos and then pulled one towards the other by a stated share. Real singers who entrain do it by listening, and what each hears is the other’s pitch arriving late — a few milliseconds through the air, then the time the ear and the brain take to register a pitch change, then the time it takes to act on it. The question the essay left was what a coupling through hearing would actually lock: whether two singers listening to each other can, even in principle, reach the two milliseconds the formant asks for, or whether the delay keeps them apart.

The answer is a sharper version of the second. The delay does not merely loosen the lock. Past about a quarter of a vibrato cycle it reverses it.

Two vibratos that hear each other #

Each singer’s vibrato is treated as an oscillator with a phase that advances at its own rate — drawn, as in the earlier essays, around six a second with the half-hertz spread measured in trained tenors — and each is pulled towards the phase it hears from the other. The pull is proportional to the sine of the phase difference, so it is strongest when the two are a quarter-cycle apart and vanishes when they agree or are exactly opposed; its strength is a single number in radians a second, roughly the rate at which a singer would close a phase error. What each hears is the other’s phase a stated delay ago. This is the simplest model of mutual entrainment there is, the one used for fireflies, pendulum clocks and clapping audiences, with a delay added.

Two such oscillators with different rates lock — settle to one common rate and a fixed phase between them — if the pull is strong enough to overcome the difference in their rates. Where they lock is the question.

At a pull of ten radians a second, two vibratos at 6.0 and 6.2 a second that hear each other instantly lock with their peaks 1.6 milliseconds apart — inside the band the formant asked for. Heard twenty milliseconds late, 2.4 apart. At thirty, a little further. Then, at about forty milliseconds, the picture changes all at once: the pair locks with its peaks 62 milliseconds apart, nearly opposite, and as the delay grows to 120 milliseconds the gap grows with it, to 96. At 140 milliseconds the pair is back near in step, 1.6 apart, and at 200 it is opposed again.

The pattern repeats with the vibrato’s own period, about 165 milliseconds, and the reason is the delay’s share of a cycle. A singer pulled towards a peak heard late is being pulled towards where the other singer was. When the delay is under a quarter of a cycle, that is close enough to where the other singer is that the pull draws the two together. When it is between a quarter and three-quarters of a cycle, the heard peak is closer to the other singer’s trough, and pulling towards it pulls away from the real peak: the only stable arrangement is the one in which each singer’s heard peak coincides with its own, which is antiphase. Past three-quarters, the heard peak is nearly a whole cycle old and points at the real one again.

The arithmetic of the boundary #

The boundary can be written down. Two oscillators locked at a common rate Ω with a phase ψ between them, each pulled with strength K towards the other’s phase heard a time τ ago, must satisfy

$$\sin\psi = \frac{\omega_2 - \omega_1}{2K\cos(\Omega\tau)}$$

where $\omega_1$ and $\omega_2$ are their own rates. The lock near $\psi = 0$ is stable when $\cos(\Omega\tau)$ is positive and the lock near $\psi = \pi$ when it is negative, so the switch comes where the delay is a quarter of the locked cycle, $\Omega\tau = \pi/2$. At six a second that is about forty milliseconds, which is where the figures switch. The same equation says why the in-step lock is never exact: the phase between the two is the rate difference over twice the strength, divided again by a cosine that shrinks towards the boundary, so a pair heard thirty milliseconds late locks further apart than a pair heard instantly, and near forty milliseconds the cosine vanishes and no pull is strong enough.

It also says the delay and the rate interact. A slower vibrato has a longer cycle, so the same delay is a smaller share of it. A singer whose vibrato runs at 5.5 a second reaches the boundary at about forty-five milliseconds; one at 7.5, at thirty-three. In a choir with a spread of rates the boundary is a band, which is why the crowd figure splits over a range of delays rather than at one.

An imposed lock and an entrained one are different things #

The earlier essay’s lock pulled a second voice’s vibrato rate and phase a stated share of the way towards the first’s. At ninety per cent the rates still differed by a tenth of what they had, so the phases slowly walked apart and back. An entrained lock is the opposite: the rates become exactly equal, and the phase stays wherever the pull and the rate difference balance it. Sixteen sweeps against sixteen and a section against another section both treated a singer as a sweep with its own rate; entrainment takes that freedom away and replaces it with a fixed relation.

That matters for the beats a listener hears. A walking phase spends some time near agreement in every cycle of its walk, and the earlier lock’s partial credit came from those stretches. A fixed phase spends all its time at one value. At four degrees that is better than any walking lock could be; at a hundred and eighty it is worse than no lock at all. Entrainment sharpens the outcome in both directions, and the delay chooses which.

Two milliseconds is a hard pull #

Even inside the friendly range the band is narrow.

A pair that locks in step does not lock exactly in step. The faster singer has to be held back and the slower pushed on, and the phase difference that does that is set by the difference in their rates over twice the strength of the pull. With rates drawn as tenors’ are, the median locked pair is 14.6 milliseconds apart at a pull of two radians a second, 6.8 at six, 4.5 at ten and 2.4 at twenty. Half the pairs come within two milliseconds only at thirty radians a second — a pull that would close a phase error in about a thirtieth of a second. Eighty per cent need sixty.

A thirtieth of a second is far quicker than any correction a singer is known to make to a heard pitch. So even with no delay at all, a pair of singers pulling towards each other with a plausible strength would lock several milliseconds apart: close enough to give back some of the section’s tuning, not close enough for the formant’s two milliseconds.

Locked in step and locked in opposition #

Drawn as pitch swings, the two locks look like opposites, and they are. In the upper pair the two voices rise and fall almost together; in the lower pair one rises while the other falls. Both pairs are locked in the full sense — they share a rate and keep a fixed phase — and nothing distinguishes the two singers or their effort. The delay alone decides which arrangement they find.

A crowd of pairs with tenors’ spread of rates does what the single pair does. Heard within twenty milliseconds, every pair locks in step, and about a quarter come within two milliseconds. At forty milliseconds, the boundary, the crowd splits: sixty per cent in step, thirteen opposed and twenty-eight not locked at all, because near the boundary the pull points sideways and barely holds them. From fifty to a hundred milliseconds nearly every pair locks in opposition, and none within two milliseconds. At 120 the crowd splits again, and by 140 it is back in step.

What each lock lets a listener hear #

The earlier essays’ measure was the beats a listener can follow at the formant when two voices of a section, a draw of fifteen cents apart, sing together. Sung straight, 2.33 beats a second at 3,200 hertz; with independent vibratos, about 0.68.

Locked within four degrees — about two milliseconds — the vibratos give back nearly all of the straight section’s beats at every partial: 2.19 of 2.33 at the formant. Eight degrees gives 1.97, fifteen 1.95, thirty 1.26. Then the curve falls steeply: sixty degrees, 0.72, about what independent vibratos give; ninety, 0.48; and in opposition, 0.19 at the formant — less than a third of what independent vibratos let through, and a twelfth of a straight section.

The reason is the vibrato’s own arithmetic, the same that makes a note that is never at its pitch of every sung tone, and like any beat the rate is a difference of frequencies. Two voices swinging in opposition have instantaneous frequencies that differ by twice the vibrato’s swing at every instant except two moments a cycle, so the beat between them races through rates no one can follow almost all the time. Independent vibratos at least drift through stretches where the swings agree. Locked in opposition, they never do.

So the coupling through hearing turns the earlier essay’s result upside down for the range of delays in which it locks in opposition. A choir that entrains with a delay of a third to two-thirds of a vibrato cycle does not give back the section’s tuning, as the imposed lock did. It hides it more thoroughly than a choir that does not entrain at all.

Where real singers sit #

Which range a choir is in depends on the delay, and the delay is mostly not acoustic. Sound crosses a metre in three milliseconds, so two singers side by side hear each other within ten. But responses to a change in a heard pitch — studied mostly by shifting the pitch of a singer’s own voice in headphones — are reported to begin roughly a tenth to a seventh of a second after the change. If the pull between two singers’ vibratos runs through that loop, its delay is 100 to 150 milliseconds, which is two-thirds to nine-tenths of a six-a-second vibrato’s cycle: on the boundary between the opposed lock and the return to step, and squarely in the opposed range for the slower vibratos in the measured span.

The figures do not settle which loop a singer’s vibrato actually runs through, and that is the important caution. A vibrato is not a pitch a singer corrects cycle by cycle; it is an oscillation the larynx produces, and whatever entrains it may act on its timing far faster than a deliberate pitch correction does, or may act through the singers watching each other or breathing together rather than through hearing. What the figures do establish is that the delay is not a detail. A coupling with the latency of a pitch correction would not merely fail to bring two singers within two milliseconds; it would push them to the opposite arrangement.

The pull drags the rate as well #

A locked pair sings at one rate, and with a delay that rate is not the mean of the two singers’ own. At no delay it is 6.10, their mean. Twenty milliseconds late, the pair slows to 5.14, each singer holding its peak back to meet a peak heard late; at forty, now opposed, it races to 7.58; at a hundred, 5.56. The rate can move by up to the pull’s strength divided by 2π — about 1.6 a second at ten radians a second — and at several delays it leaves the range in which singers’ vibratos are measured.

That is a prediction recordings could test, and a warning about the model. If singers’ vibratos were coupled as strongly as two milliseconds demands, ensembles should show vibrato rates shifted away from their singers’ solo rates by an amount that depends on how far apart they stand and how they hear. If no such shift appears, the coupling is weaker than the lock requires — and the two milliseconds is then out of reach by a margin the delay only widens.

The model the figures rest on #

Each vibrato is a phase oscillator — a sinusoidal swing whose timing is all that the coupling acts on — and the coupling is the sine of the heard phase difference. A vibrato note keeps a single heard pitch through its swing, as the pitch that does not wobble found of the period a pitch extractor recovers, and nothing here changes that: entrainment moves the swings’ timing, not their centre. Real vibratos vary in rate and extent through a note, and the larynx’s oscillation has its own dynamics that this reduces to a single phase.

The delay is fixed and the same both ways. In a choir, singers stand at different distances and some hear others mainly by reflection; a spread of delays would blur the boundaries between the in-step and opposed ranges without moving them, since they are set by the delay’s share of a cycle.

The beats count is the earlier essays’: two voices a draw of fifteen cents apart, each null in their summed envelope counted as a heard beat while the instantaneous beat rate is at or below fifteen a second.

The pull’s strength is swept, not fitted. No measurement used here gives it, and the essay’s conclusions are stated across its range.

What these figures cannot show #

What actually entrains a singer’s vibrato. Hearing is one candidate; seeing, breathing and the conductor are others, each with its own delay. The model takes the hearing loop as given and asks what follows; whether vibratos in a real choir entrain through it at all is a measurement.

And a section rather than a pair. Sixteen singers each hearing fifteen others at different delays is a network, not a pair, and networks of delayed oscillators can settle into patterns — waves, clusters, splayed phases — that two oscillators cannot. What a choir does that a soloist cannot found that a section’s many pairs beat at many rates at once; a section locked into clusters would beat at a few.

Still open: what a section of sixteen settles into #

The pair has two answers, in step or opposed. A section has many: with every singer hearing every other at the delay their positions give, the lowest-energy arrangement may be two clusters in opposition, or a phase that turns steadily along the row, or no lock at all. Each arrangement sets which pairs beat followably at the formant, and so how much of the section’s tuning a listener hears, and the earlier essays’ sixteen-voice counts could be run on it directly. The computation is this one with sixteen oscillators and a matrix of delays drawn from a choir’s layout, and it would say whether a section that listens to itself ends up exposing its tuning, hiding it, or — in a splayed lock with every phase represented — hiding it as completely as a section that never listened at all.