A shared vibrato cannot hold a tuned chord together

A dominant seventh in close position, tuned exactly 4 : 5 : 6 : 7 low on the keyboard, flutters at one rate and makes every auditory filter from 60 hertz to a kilohertz and a half fluctuate at one slow rate, and if its notes begin together the fluctuations rise and fall together across the filters. A tuned chord’s flutter binds it only if its notes start together measured that comodulation — a correlation of 0.59 between the filters’ envelopes, against 0.16 for the same chord tempered — and found it needed the notes within about a third of a millisecond of each other. A tuned chord stays in step for half a cent-second then found that a chord struck in step does not stay there unless it is tuned exactly: a note a cent sharp slides 0.58 milliseconds a second against the others, and the chord leaves step in a time inversely proportional to its mistuning — four seconds at a tenth of a cent, half a second at a cent.

That essay ended on a hope. The drift is the enemy because each note slides at its own rate. Voices that shared a vibrato — swinging their pitch together, in rate and in phase — would still slide, but would slide together, and the relative timing the filters need would be kept to the extent that the vibratos agreed. A choir’s vibratos must lock to two milliseconds had priced such a lock for a unison. The question was whether a lock as tight as that could hold a tuned chord in step for longer than an exact, straight chord could.

It cannot, and the reason is short enough to give before any figure. A vibrato is a change in frequency shared by a note’s partials. If every note shares it exactly, every frequency in the chord is multiplied by the same factor at every instant. The differences between the notes’ frequencies — which is what a mistuning is, and what the drift is made of — are multiplied by that factor too, and on average are exactly what they were. A perfectly shared vibrato warps the chord’s time and leaves its tuning alone. An imperfectly shared one adds a slide of its own.

The chord, now with a vibrato on every note #

Everything is held from the earlier essays: the tuned close seventh over C2, its notes’ partials falling as one over the partial number, a bank of auditory filters one equivalent rectangular bandwidth apart from 60 hertz to 1.5 kilohertz, and the comodulation read as the mean correlation between every pair of filters’ envelopes in half-second windows, averaged over eight draws. Each note now carries the typical vibrato of trained voices, 71 cents either side of the note at about six a second, with its own rate drawn from the measured range and its own phase. The lock pulls every note’s rate and phase towards the bass’s: at nought the four vibratos are independent, at one they are a single vibrato. Each partial runs at its note’s instantaneous frequency, so it also sweeps across the filters’ skirts, and that sweep is counted in each filter’s envelope.

Sung straight and exactly in tune, the chord stays at 0.59 for all eight seconds, as the earlier essay found. With a fully locked vibrato it stays at 0.57: the sweep of every partial across the filters’ skirts adds a little fluctuation of its own, at the vibrato’s rate, which correlates imperfectly across filters and costs two hundredths. So a perfectly shared vibrato is almost free — and that is the most it can be.

Drag the lock down. At ninety-eight per cent the chord starts at 0.59 and sinks steadily, halfway to the tempered chord’s 0.16 by six seconds and at 0.33 by eight. At ninety-five per cent it is halfway out of step by 2.25 seconds, at ninety by one second, at eighty by half a second, and half-locked it leaves step almost at once. The chord is exactly tuned throughout. Nothing in it slides except what the vibratos put there.

What the vibrato does inside a filter #

The two hundredths a perfectly locked vibrato costs are worth a paragraph, because they show what a vibrato is to the ear’s filters as well as to the chord’s timing. A filter centred between two partials sees each of them through its skirt, at a weight set by how far it lies from the centre. When a vibrato swings every partial up by 71 cents and back, the partial below the centre climbs towards it and grows, and the partial above climbs away and shrinks, six times a second. Every filter’s envelope therefore carries a fluctuation at the vibrato’s rate on top of the chord’s own slow one. The fluctuation is not the same in every filter — in one the stronger partial is below the centre, in the next above — so it correlates only partly across the bank, and the mean correlation falls a little.

That is the same mechanism by which a vibrato hides a choir’s tuning above four hundred hertz: a swing of tens of cents moves partials across the ear’s frequency resolution far faster than any beat between them could be followed. There it was useful, covering the voices’ mistuning. Here it is mildly harmful, covering the chord’s own common rhythm with a faster one that the filters do not share. Either way, what a choir does that a soloist cannot — produce many slightly different versions of one note at once — is precisely what a tuned chord’s binding cannot tolerate, and a vibrato is one more way of making the versions differ.

The lock is a second clock #

Exact tuning was the best case. A real chord is mistuned by a fraction of a cent as well, and the earlier essay’s clock is still running.

With the notes a tenth of a cent apart, the straight chord is halfway out of step at four seconds and the chord with a fully locked vibrato at 3.75. At a quarter of a cent, 1.75 against 1.5; at half a cent, one second against 0.75. The two curves lie almost on top of each other, the vibrato a little ahead. A perfectly shared vibrato does not slow the drift at all, and the small acceleration is the same sweep that cost two hundredths in the exact chord: it adds a fluctuation that has nothing to do with the chord’s own period.

Put the two clocks together and the chord leaves step on whichever runs out first. Exactly in tune, only the lock’s clock runs: half a second at eighty per cent, one at ninety, 2.25 at ninety-five, 4.25 at ninety-seven, six at ninety-eight, still in step at ninety-nine. A tenth of a cent out, the mistuning’s clock stops the chord at four seconds sung straight, and the vibrato can only shorten that: 3.75 fully locked, 3.25 at ninety-eight per cent, one second at ninety. A quarter of a cent out, the mistuning stops it at 1.75 seconds straight and the vibrato at 1.5 from ninety-seven per cent upward. At no lock and no mistuning does the vibrato keep the chord in step longer than singing it straight.

So the lead’s last clause had the answer the wrong way about. It asked whether the lock a held tuned chord needs might be beyond any ensemble, as the unison’s was. The lock is the lesser problem: an ensemble that achieved perfect lock would gain nothing over singing straight, and one that fell short of it would lose. The vibrato is not a clock that keeps the voices together. It is a second way for them to come apart.

The first clock, for comparison #

The mistuning’s clock is the earlier essay’s law, and it is worth seeing again beside the lock’s, because the two have different shapes.

Sung straight, the time in step is inversely proportional to the mistuning: halve the error, double the time. The product is about half a cent-second, which is a slide of a third of a millisecond at a cent’s rate of 0.58 milliseconds a second — the onset tolerance reached by accumulation. The lock’s clock is not proportional in that way. It is set by the spread of the vibratos’ rates, which makes their phases walk apart, and by the extent, which turns a phase difference into a slide. From ninety to ninety-eight per cent, the time multiplies by six while the lock’s shortfall falls by five: roughly inverse too, but with the extent as a multiplier that the mistuning’s clock does not have. A chord sung with vibrato by voices whose vibratos agree to two per cent, and whose tuning agrees to a tenth of a cent, has two clocks of a few seconds each, and like any beat the faster of the two decides what is heard.

Why a few degrees of vibrato are a third of a millisecond #

The earlier essays’ unit was a slide in time, and the vibrato’s cost is easiest to see in that unit.

Two notes with the same vibrato a phase apart have instantaneous frequencies that differ by a swinging amount, and the integral of that difference is how far one waveform slides against the other. Unlike a mistuning, which accumulates, it swings back: the slide oscillates at the vibrato’s rate with an amplitude set by the extent and the phase. For the typical 71-cent vibrato at six a second, notes ten degrees apart in vibrato phase slide 0.19 milliseconds each way, notes thirty degrees apart 0.56, and notes in opposite phase 2.18. The tolerance the onsets were held to is about a third of a millisecond, and the 71-cent vibrato passes it at sixteen degrees of phase — seven milliseconds of vibrato timing at six a second. A narrow 34-cent vibrato allows 33 degrees; a wide 123-cent one, nine.

That is the same order as the two milliseconds the choir essay asked of a unison’s vibratos, reached from a different requirement. The unison needed its vibratos close so that the slowest beats between voices stayed followable at the singers’ formant. The chord needs them close so that its notes’ waveforms stay within the onset tolerance of each other. And a lock short of complete does not hold a fixed phase: the voices’ rates differ slightly, so the phase between them grows, and a vibrato that begins two degrees apart is sixteen degrees apart a few seconds later. The six seconds of the ninety-eight-per-cent chord are that walk.

The wider the vibrato, the tighter the lock #

The slide is proportional to the extent, so the lock a chord needs depends on how wide the singers’ vibrato is. At ninety-five per cent lock a 34-cent vibrato keeps the exact chord in step for 5.25 seconds, a 71-cent one for 2.25 and a 123-cent one for a second. At ninety-eight per cent, vibratos of 34 and 50 cents keep it in step for the whole eight seconds, 71 for six, 123 for 3.25. Fully locked, none of them takes the chord out of step, and the cost of the sweep across the filters rises only from 0.59 to 0.56 over the whole range of extents.

So a narrow vibrato is the forgiving one, as it was for the unison, and for the same reason: everything a vibrato does to the timing between voices scales with how far it swings. Singers in ensembles that tune chords carefully often narrow their vibrato or drop it, and these figures give one reason why that is not merely taste. A chord that is meant to bind needs either very tightly matched vibratos or very small ones, and the second is far easier to sing.

What it means for a held chord #

Put together, the arithmetic leaves the exact, straight chord as the only one that stays in step for as long as it sounds. Every departure from it costs time: a fraction of a cent of mistuning costs seconds, and a vibrato that is not perfectly shared costs seconds too, whichever is the more expensive deciding. A shared vibrato is at best a way to sing with a vibrato without paying for it, and only an ensemble that locked perfectly would get even that. The same holds for instruments. A string quartet playing a tuned chord with vibrato faces the same two clocks, its players’ vibratos rarely matched in rate; an organ, whose pipes hold their pitch without vibrato and can be tuned to the chord, faces only the first, and on an organ a tuned chord could stay in step for as long as the pipes stay in tune with each other — which, since organ pipes pull one another’s frequencies when they sound together, may be longer than the arithmetic of independent sources would allow.

The seventh has two places to be and the series is not a chord treated the tuned seventh as a matter of where the notes sit. These figures add that a tuned chord’s distinctive behaviour — one rate in every filter, rising and falling together — is also a matter of how still the notes are, and that a listener could only ever hear it in a chord sustained by voices or instruments holding their pitch steady to a fraction of a cent and a few degrees of vibrato phase.

The model the figures rest on #

The chord’s partials are steady: no decay, so that any change through the eight seconds is the drift and nothing else. A sung chord’s partials change with the vowel and the voice, and that would add fluctuation of its own to every filter.

The vibrato is a sinusoid of fixed rate and extent on each note, every partial of a note swinging by the same number of cents. Real vibratos vary in rate and extent through a note and are not quite sinusoidal, and a singer’s partials do not all follow the fundamental exactly; each of those would add independent slide between notes and make the requirement on the lock stricter, not looser.

The lock is the choir essay’s convention: each note’s vibrato rate and phase drawn as a singer’s would be, then pulled towards the bass’s by a stated share. It is a description of agreement, not a model of how singers come to agree.

The tolerance of a third of a millisecond is read from the earlier essay’s onset figures, and the slide figure uses it only to put the phase in familiar units.

What these figures cannot show #

Whether a listener hears the comodulation. The correlation between filters’ envelopes is a cue the ear is known to use for grouping in laboratory sounds; whether a listener hears a held tuned chord as more fused while its filters are in step, and notices when they drift apart, has not been measured with chords like these.

And whether singers can hold a vibrato in step. Recordings of ensembles show vibratos partly entraining, and the choir essay put ninety-five per cent lock at peaks about two milliseconds apart. Whether trained singers reach ninety-eight per cent and hold it for six seconds is a measurement.

Still open: whether a chord re-binds at each swing #

A slide caused by a vibrato swings back. At the instants when two notes’ vibratos cross, their waveforms are back where they started relative to each other, and in an exactly tuned chord with a partly locked vibrato the filters may come back into step for a moment every half-cycle before the walk of the phases carries them away again. The half-second windows used here are too long to see that: they average three vibrato cycles. Read in windows of a tenth of a second, the comodulation might show a chord that is in step several times a second rather than one that has drifted out — and whether a listener integrates over the long window or catches the short ones would decide whether a vibrato chord is heard as unbound or as rhythmically binding and releasing, which would be a different sound altogether.