A choir's vibratos must lock to two milliseconds

A vibrato hides a choir’s tuning above four hundred hertz counted the beats a listener can follow between two voices of a section singing one note. Sung straight, two voices fifteen cents apart on average give 2.3 followable beats a second at 3,200 hertz, the band where a trained voice carries. With a vibrato of seventy cents at about six cycles a second, the same voices give 0.7. The beats are still in the signal, but the vibrato carries the difference between the two voices’ frequencies past fifteen hertz for most of every cycle, and a fluctuation that fast is heard as roughness rather than followed as a beat.

The arithmetic under that figure is as old as tuning by ear. Beats are arithmetic: two tones a few hertz apart swell and fade at their difference. Every partial beats at its own rate, so two voices a fixed number of cents apart beat faster and faster up their spectra, and at some partial the beating passes fifteen a second and becomes roughness. Sung straight, the pairs of a fifteen-cent section are spread from nearly unison to thirty cents apart, and at 3,200 hertz the closer ones still beat slowly enough to follow. Sixteen sweeps against sixteen found what a vibrato does to that: the difference between two vibrato voices sweeps through a wide range six or seven times a second and spends most of its time far above the line.

That count drew every voice’s vibrato independently: its own rate, somewhere between 5.5 and 7.5 cycles a second, and its own phase. The essay ended on the obvious exception. Singers listen to each other, and two voices whose vibratos moved in step would sweep their partials up and down together. Their difference would stay near its steady value, and the beats would come back. How far do two vibratos have to lock before that happens?

What locking means here #

The pair is drawn exactly as before: each voice detuned from the note by a draw of fifteen cents, each with a vibrato of seventy-one cents, a rate drawn about six hertz with a spread of half a hertz, and a random phase. Then the second voice’s vibrato is pulled towards the first. Its difference of rate from the first voice’s, and its difference of phase taken the short way round the cycle, are each multiplied by one minus the lock. At no lock the pair is the earlier essay’s. At full lock the two vibratos have one rate and one phase, and the only difference between the voices is their fixed mistuning.

That is the simplest coupling with the right two ends, and it is a stand-in rather than a model of how singers entrain. It treats rate and phase alike, and it makes a partly locked pair a pair whose vibratos are partly similar, not one that drifts in and out of step. What it buys is a single number that can be read against a recording. A lock of nine-tenths means the two vibratos’ rates differ by a tenth of what two independent singers’ would, typically 0.07 hertz instead of 0.7, and their phases by a tenth of a typical independent difference, nine degrees instead of ninety.

Half-locked is still hidden #

At the singer’s formant the curve is nearly flat for most of its length. Independent vibratos give 0.68 followable beats a second against a straight section’s 2.33. Locked a quarter of the way, 0.70; half, 0.80; seven-tenths, 1.13. A section whose vibratos were half entrained — which would be a strong and easily measured tendency in any recording — hides its tuning at 3,200 hertz almost exactly as well as one whose vibratos have nothing to do with each other.

Then the curve turns up. Locked to eight-tenths, 1.10; to nine-tenths, 1.70, three-quarters of the straight section’s; to ninety-five per cent, 1.98; to ninety-eight per cent, 2.12, within a tenth of straight. Full lock gives 2.23, the straight value to within the sampling.

Read as a whole, the figure has the shape of a threshold rather than a slope. Up to about seven-tenths locked, the section is as hidden as an independent one. Past nine-tenths, it is nearly as exposed as a straight one. Between the two lies the band in which a recording would have to place a real choir for the question to be interesting, and it is narrow: a pair of singers whose vibratos are clearly correlated, but no better, sits on the hidden side of it.

Lower in the spectrum the turn comes earlier. At 1,217 hertz the pair goes from 1.60 unlocked to 2.88 at eight-tenths and 4.41 at ninety-five per cent, against 4.66 straight. At 660 hertz, near the crossing below which the vibrato adds beats instead of hiding them, it goes from 2.77 to 4.45 at eight-tenths and reaches the straight value of 5.14 at about nine-tenths. The higher the partial, the closer to complete the lock must be, and at the formant, where the earlier essay found the vibrato hiding most, the section must be almost perfectly entrained before its tuning is exposed.

Why the lock must be so tight #

A single pair shows the mechanism. Two voices six cents apart at 3,200 hertz beat at 11.1 a second sung straight, comfortably under the fifteen-hertz line, so every one of their beats is followable. Give them independent vibratos and the instantaneous beat rate swings up to 273 hertz, spending 9 per cent of the second under fifteen. Lock them to ninety-five per cent and it still swings up to 49 hertz, spending 44 per cent of the second under fifteen. Only fully locked does it stay on its straight value, breathing slightly with the vibrato, for the whole second.

The reason is a multiplication. What matters to the beat is the difference between the two voices’ instantaneous frequencies, in hertz, at the partial being compared. A vibrato of seventy-one cents at 3,200 hertz swings a partial by about 130 hertz either side. Two such swings slightly out of step differ by roughly 130 hertz times the phase difference in radians. A phase difference of five degrees, about a tenth of a radian, is therefore a difference of about eleven hertz between the two partials, which on its own is nearly the whole of the fifteen-hertz band. A few degrees of vibrato phase, multiplied by the partial’s frequency and the vibrato’s width, is the beat rate.

The same multiplication explains the earlier essay’s crossing. At 330 hertz the vibrato’s swing is about fourteen hertz either side, so even independent vibratos rarely carry a difference far past the fifteen-hertz line, and the sweep through zero adds slow beats. At the formant the swing is ten times as large, and so is the precision the phase has to have.

In time rather than angle, the figures are severe. A vibrato at six cycles a second takes 167 milliseconds per cycle, and five degrees of it is 2.3 milliseconds. Ninety-five per cent locking, at which the formant shows 85 per cent of a straight section’s beats, means two vibratos whose peaks are on average about two milliseconds apart, and whose rates differ by about three-hundredths of a hertz, so that they drift a whole cycle apart only every half-minute.

Up the spectrum, and with wider vibratos #

The multiplication says what the lock depends on: the partial’s frequency and the vibrato’s width. The figure finds the lock at which a pair gets back nine-tenths of what it would have sung straight, across the spectrum, for the narrowest, typical and widest vibratos the earlier essays took from measurements of operatic soloists.

At 330 hertz no lock is needed for any of them. That is below the crossing the earlier essay found near 400 hertz, where a vibrato sweeps a slow difference faster and so adds followable beats rather than removing them. At 440 hertz the narrow vibrato still needs none, the typical one about a third, the wide one eight-tenths. From 880 hertz up the typical vibrato needs to be locked to nine-tenths or more, the widest to ninety-five per cent or more, and the narrowest to eight-tenths or more. At every frequency from 660 hertz, the wider the vibrato, the tighter the lock it needs. The curves do not rise smoothly all the way, since sampling three hundred pairs leaves a few hundredths of scatter at the top, but none of them needs less than nine-tenths at the formant.

At the formant itself the three extents separate as the multiplication predicts. Unlocked, the narrow vibrato leaves 1.16 followable beats a second, the typical one 0.68, the wide one 0.55. Each recovers as the lock tightens, and each reaches the straight section only at full lock, but the narrow vibrato is at 2.03 by nine-tenths while the wide one is at 1.33. Halving a vibrato’s extent buys the same exposure at roughly twice the phase difference, which is the same as saying it halves the precision the lock must have.

So a choir that wants its tuning heard has two levers, and they are not equally available. It can narrow its vibratos, which is a matter of style and is done routinely in some repertoire. Or it can lock them, which nothing in the training of singers is aimed at, to a precision of milliseconds.

The better a section is tuned, the more the vibrato hides #

Everything so far is for a section spread by fifteen cents. Change the spread and the conclusion does not merely shift; it gets worse for the better section.

Sung straight, a section spread by five cents beats slowly — at 3,200 hertz its pairs differ by a few hertz — so nearly all its beats are followable: 4.86 a second. A vibrato carries those slow differences far past fifteen hertz, and unlocked the section keeps only 18 per cent of them. It needs its vibratos locked to about ninety-eight per cent to show its tuning as a straight section would; at ninety-five per cent it shows three-quarters of it. A section spread by thirty cents is in the opposite position. Sung straight, most of its pairs already beat faster than fifteen a second at the formant and only 0.97 beats a second are followable; its vibrato takes little from that, leaving 71 per cent, and a lock of about 0.85 restores it.

The irony is plain. The vibrato’s hiding power is greatest for exactly the section whose tuning most deserves to be heard. A well-tuned choir singing with vibrato sounds, by this count, much more like a badly tuned one at the formant than either does sung straight, and the lock that would separate them again is tightest for the good choir. It sits beside what what a choir does that a soloist cannot found about the choral sound: sixteen voices fluctuate as much as two, but their fluctuation is spread across a band of rates with no single beat in it. A vibrato does the same thing to a pair, and does it hardest to the pair whose single beat was clearest. The rate that does not rise with the partial found the vibrato’s own six-hertz fluctuation standing in that band; what is lost is the slower one that would have said how far apart the voices are.

The front of the note #

The earlier essay found the straight tone at the start of a sung note — the quarter-second before the vibrato is established — to be the one stretch where an unlocked section’s tuning shows, and found it short.

Unlocked, the first tenth of a second carries 1.27 followable beats a second at the formant, about twice the settled vibrato’s 0.6, and that is the only part of the note in which a listener has much to count. Locked fully, every tenth of a second carries between about 2.0 and 2.6, around the straight section’s 2.33, from the first to the last. Locked to nine-tenths, the note starts near the straight rate and drifts down through the held note as the small differences of rate accumulate into differences of phase.

So locking does not merely add to what the onset already gives. It removes the onset’s special status. The earlier essay asked whether the straight front of a note lets a listener judge a section’s tuning, and found less than one beat to judge by. A locked section makes that question moot, because the held note itself becomes as informative as the front was meant to be.

The model the counts rest on #

The pairs, the vibratos and the counting rule are the earlier essay’s: two pure partials at the compared frequency, each detuned by a draw from a normal distribution of the section’s spread, each with a sinusoidal vibrato whose extent grows linearly over a quarter-second and whose rate is drawn about six hertz with a spread of half a hertz, clipped to the measured range; a null in the envelope of their sum counted as a beat only while the instantaneous difference of their frequencies is at or below fifteen hertz; three hundred pairs at each point, on a fixed seed so that every lock is the same pairs. The lock is applied after the draws, by shrinking the second voice’s differences of rate and phase from the first’s by a stated factor, so a partly locked pair is a pair partly alike rather than one that is alternately locked and free.

What the counts leave out #

How singers actually entrain, if they do. A real coupling between two singers would act through hearing, with a delay of tens of milliseconds and a strength that varies with the singers, the room and the music. It might lock rate and leave phase to wander, or pull phase together only near the vibrato’s peaks. The single number here treats all of that as one factor; a model of the coupling itself is a different computation, and what it would be modelling is not established here.

Whether a section of sixteen locks as a pair does. Every count is for two voices, and a section of sixteen is a hundred and twenty pairs whose beats a listener hears together, not one at a time; a roughness with a rate of its own and the mean survives the window are the account of what a listener’s window makes of the moving sum. Sixteen voices all locked to one another would have to share a vibrato, which is what a single singer has, and a listener would hear it as the whole section swinging together. Whether a partial lock among sixteen is closer to the pair figures or to something worse is not computed.

What none of these figures show #

Whether choirs lock at all. These are counts of what a lock would do, not a claim that any choir achieves one. What the figures supply is a target for measurement: in a recording of two singers on one note, the difference between the times of their vibratos’ peaks. If it is a few milliseconds, sustained, their tuning is as exposed at the formant as if they sang straight. If it is tens of milliseconds, they might as well be independent. A half-entrained pair, which a measurement might well find and report as entrainment, is on this arithmetic no better than none.

And whether a listener uses the formant band to judge a choir’s tuning. The earlier essays’ caution stands: a listener may judge intonation by roughness, by the spread of pitch, or by the fundamental region where a vibrato adds beats rather than hiding them. The formant band matters here because it is where one voice stands clear of ninety players, and so where a choir’s sound reaches a hall; it is not established that it is where a listener reads tuning. And the pitch of each voice survives its vibrato — the pitch that does not wobble found the period intact at a cost — so a listener comparing the voices’ pitches rather than counting their beats is outside these figures altogether.

Still open: what a coupling through the ear would actually lock #

The lock here is imposed, not grown. Two singers who entrain do so by hearing each other, and what each hears is the other’s pitch arriving a few tens of milliseconds late through the air and the auditory system. A coupled pair of oscillators with that delay, pulling each other’s vibrato phase at a strength a recording could fit, would settle at some phase difference and some residual wander, and both are the quantities the figures here turn into followable beats. The computation is two vibrato oscillators coupled with a delay, swept over the delay and the strength, with the steady state fed into this count — and it would say whether hearing each other can, even in principle, bring two singers within the two milliseconds the formant asks for, or whether the delay alone keeps them tens of milliseconds apart.