A chord comes back into step too briefly to bind

A dominant seventh tuned exactly 4 : 5 : 6 : 7 low on the keyboard flutters at one rate: every one of the ear’s filters from 60 hertz to a kilohertz and a half fluctuates at the same slow period, the common period of the four notes, and if the notes start 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 and found it needed the notes within about a third of a millisecond of each other. A shared vibrato cannot hold a tuned chord together then gave every note the typical vibrato of trained voices, 71 cents either side at about six a second, and found that unless the four vibratos were locked almost perfectly the chord came out of step within seconds, read through half-second windows.

That essay ended on a possibility its windows could not see. 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 an exactly tuned chord with a partly locked vibrato might come back into step for a moment every half-cycle. Half a second averages three vibrato cycles. Read finely enough, the chord might be binding and releasing several times a second — a different sound altogether from a chord that has drifted out.

The finer reading confirms the returns and denies the binding.

Where the half-second windows left it #

The chord is the earlier essays’ throughout: a close seventh over C2, its notes’ partials falling as one over the partial number, exactly in tune and struck together. Each note’s vibrato has 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, so that at one the four vibratos are a single vibrato and short of it they differ by what is left. The earlier figure read the binding as the correlation between filters’ envelopes in half-second windows. Here the same eight draws are read directly, as timing: how far ahead of or behind the bass each upper note’s waveform runs at every instant.

A slide that swings back #

An exactly tuned chord’s notes have exactly the right mean frequencies. A vibrato swings each note’s frequency up and back about that mean, and a note’s waveform runs ahead while its frequency is above the mean and falls back while it is below. With identical vibratos every note runs ahead and falls back together. With slightly different ones, each upper note runs ahead of the bass and falls back again in every cycle, and the gap between them oscillates — it does not accumulate, because the mean frequencies are equal.

Locked ninety-five per cent, the seventh swings about a millisecond either side of the bass, the fifth about half a millisecond, the third a fifth of one, all at the vibrato’s rate. Twice in each cycle all three pass through zero together, and the chord is back in step: thirteen times in the second drawn, in thin columns a few milliseconds wide. Turn the dial to ninety-eight and the swings shrink to two-fifths of their size, the columns widen until the chord is in step for about half of the second, in thirteen stretches. Turn it to ninety-nine and the chord stays in step for most of the second. Turn it to ninety and the swings are so large that in this second all three never pass through the band together.

The earlier essay’s drift was not a drift. Over all eight seconds and all eight draws, the largest gap between any note and the bass is 1.2 milliseconds at ninety-eight per cent lock, 2.3 at ninety and 3.3 at fifty. A mistuned note slides away for ever, as a tuned chord stays in step for half a cent-second found, but a vibrato’s slide is bounded. What grows over the first seconds is the size of the swing, as the vibratos’ small differences of rate walk their phases apart; once grown, it stays, and the chord settles into a state in which it is in step for some fixed share of the time and out of it for the rest.

The half-second figure, read again #

The bounded swing changes what the earlier figure was showing. Its half-second comodulation fell steadily over the eight seconds, and the natural reading was a chord drifting out of step, as a mistuned one does. The timing says something else. At the instant the chord is struck every note’s vibrato starts from where the draw put it, and the lock keeps the phases close, so the swings start small. The four vibratos’ rates differ by what the lock leaves of their natural spread — a hundredth of a hertz or so at ninety-eight per cent — and that small difference walks their phases apart, so each swing grows. The decline the half-second windows read is the swing growing towards its full size. The chord is not leaving step for good; it is settling into a pattern of brief returns.

The same arithmetic says the pattern is not permanent either. Two vibratos whose rates differ by a hundredth of a hertz come back into phase every hundred seconds or so, and as they do the swing between them shrinks again and the returns lengthen. A chord held at ninety-eight per cent lock would, on this model, bind and unbind on a cycle of a minute or two. That is the vibratos beating against each other, the same arithmetic as beats at a rate far too slow to hear, and no sung chord is held long enough for it to matter.

How long each return lasts #

The question the lead posed is whether those returns are binding. A filter can show the chord’s comodulation only by fluctuating with the chord’s flutter, and the flutter has a period: the four notes repeat together every 61 milliseconds, the common period of 4 : 5 : 6 : 7 over C2, about sixteen times a second. A return to step that lasts less than one cycle of the flutter gives the filters less than one shared fluctuation to show.

At eighty per cent lock each return lasts 6 milliseconds on average; at ninety, 9; at ninety-five, 22; at ninety-seven, 45. Only at ninety-eight per cent does the average return, 78 milliseconds, outlast one cycle of the flutter, and at ninety-nine it reaches 173. At ninety-nine and a half per cent the chord hardly leaves step and the average return is the length of the passage. The returns cross the flutter’s period at a lock within a point of the one at which the earlier essay found the half-second comodulation holding up for six seconds — two readings of the same fact, that a lock short of ninety-eight per cent gives the chord too little time in step.

The average flatters the returns, because a few long ones raise it.

Counted, the returns are frequent. At ninety-five per cent the chord comes back into step 8.5 times a second; at ninety-seven, 11.1; at ninety-eight, 9.9 — close to the twelve a second at which a six-a-second vibrato passes every note through its mean. Above that the returns merge into longer stretches and become fewer, 4.8 a second at ninety-nine; below it the three notes rarely cross zero together, and the count falls to 2.9 at ninety and 0.6 at eighty.

But the returns long enough to carry a whole flutter cycle are rare at every lock: 0.1 a second at ninety-five per cent, 0.4 at ninety-seven, 0.7 at ninety-eight, 0.5 at ninety-nine. The chord does come back into step at each swing, and almost never for long enough to bind. At the locks where the returns are most frequent, fewer than one in ten lasts a cycle of the rhythm the filters would have to share.

So the lead’s alternative — a chord heard as rhythmically binding and releasing several times a second — does not arise for this chord. To bind and release at the vibrato’s rate, the chord would have to hold step for a flutter cycle in each half-swing, which is 61 milliseconds out of 83; the vibrato’s swing is only one and a third flutter cycles long, and the chord is out of tolerance for most of it at any lock short of ninety-nine per cent.

That arithmetic does not depend on the windows chosen. Read in windows of a tenth of a second — about a flutter cycle and a half — the correlation between filters is unsteady even for a perfectly locked chord: it swings between about 0.49 and 0.62 with every 60-millisecond shift of the window, as the window catches one flutter peak or two. A window short enough to resolve the vibrato’s swings is too short to resolve the flutter it is supposed to be measuring. The binding is a property that takes a few flutter cycles to exist at all.

Time in step and comodulation are one measure #

The half-second comodulation and the share of time in step tell the same story by different means. At eighty and ninety per cent lock the chord is in step four and thirteen per cent of the time and its comodulation sits on the tempered chord’s 0.16–0.17: brief returns contribute nothing. At ninety-five per cent it is in step 29 per cent of the time and the comodulation is 0.20. At ninety-eight, 64 per cent and 0.33; fully locked, all the time and 0.57. The comodulation rises with the time in step, but it rises late: below about a third of the time in step, the returns are too short to register at all.

That is the same threshold seen from the half-second side. Short returns contribute nothing because each carries less than a cycle of the shared flutter; the comodulation a window reads is roughly the share of its flutter cycles that fall inside returns long enough to contain them.

A narrower vibrato returns for longer #

The swings are proportional to the vibrato’s width, so a narrow vibrato keeps the chord inside the tolerance for longer. At ninety-five per cent lock a 34-cent vibrato’s returns average 59 milliseconds, just short of a flutter cycle; a 71-cent one’s 22; a 123-cent one’s 9. At ninety-eight per cent they are 199, 78 and 29. A 34-cent vibrato locked ninety-five per cent is roughly as good as a 71-cent vibrato locked ninety-seven, and the widest vibratos in the measured range never return for a flutter cycle at either lock.

The earlier essay found the wider the vibrato, the tighter the lock, and this is the same result with its mechanism exposed: a vibrato’s width is the amplitude of the swing that carries the notes apart, and the lock sets only how far the swings disagree. A choir’s vibratos must lock to two milliseconds asked the same of a unison for a different reason, and ensembles that tune chords often narrow their vibrato: these figures give a reason that does not depend on taste.

What this does to a sung chord #

The arithmetic gives an ensemble two ways to keep a tuned chord bound while singing with vibrato, and neither is the one the earlier essay hoped for. One is near-perfect agreement: vibratos matched in rate and phase to ninety-eight or ninety-nine per cent, at which point the returns are long enough to carry the flutter and the chord spends most of its time in step. The other is a narrow vibrato, which makes every swing small enough that a modest lock keeps the notes within tolerance. What an ensemble cannot do is rely on the swings themselves — on the chord passing back through step twice a cycle — because each pass is too brief to hold the rhythm that makes the chord sound bound.

That fits what a vibrato hides a choir’s tuning above four hundred hertz found from the other side. There, a vibrato was useful because it moved partials across the ear’s frequency resolution faster than any beat could be followed, covering the singers’ mistuning. Here the same speed works against the chord: the swing is faster than the chord’s own flutter can be expressed, so the moments of agreement are swept away before the filters can register them. What a choir does that a soloist cannot — many slightly different versions of one note at once — is a gift to a unison’s warmth and a cost to a tuned chord’s binding, and the cost is paid in exactly the currency these figures count.

It also qualifies what the seventh has two places to be and its neighbours can expect of a sung harmonic seventh. Tuned exactly, the seventh at 7 : 4 is what gives the chord its single common period, and that period is the clock the returns have to outlast. Measured in time, though, a note’s swing does not depend on its pitch at all: a waveform runs ahead by the integral of its fractional frequency change, which is the same for a bass and a seventh singing the same vibrato. Every note is equally exposed, and the one that leaves step first is simply the one whose vibrato differs most from the bass’s — in the second drawn above, the seventh. The chord is only as bound as its least-matched voice.

The model under the swings #

The vibratos are the earlier essay’s draws. Each note’s rate is drawn as six a second plus a spread of half a hertz, clipped to the measured range, with a random phase, and both are pulled towards the bass’s by the lock; the same eight seeds give the same draws as the comodulation figures. Each note’s waveform is timed by integrating its instantaneous frequency, which is exact.

In step means all three upper notes within 0.3 milliseconds of the bass at once, the onset tolerance the binding essays found. A chord with two notes in step and one out was counted out; a looser rule would lengthen every return.

The flutter’s period is the chord’s common period, a quarter of the bass frequency, 61 milliseconds over C2. A chord built higher would flutter faster, and its returns would contain more cycles: an octave up, the same returns carry twice as many flutter cycles, and the lock at which they first carry one would fall.

What the timing cannot say #

Whether a listener integrates over returns. The figures ask whether each return is long enough to carry the binding cue on its own. A listener whose grouping accumulates evidence across many brief returns, rather than requiring a cycle within each, could hear a chord in step a third of the time as partly bound. Whether auditory grouping works that way for comodulation is not something the arithmetic settles.

How real singers’ vibratos move. Real vibratos drift in rate and depth through a held note and are not quite sinusoidal, and a singer’s upper partials do not follow the fundamental exactly. Each of those adds independent slide between notes and would shorten the returns.

Still open: the same chord an octave or two higher #

The flutter’s period is a quarter of the bass frequency, so the same close seventh an octave higher flutters twice as fast, and a return to step of a given length holds twice as many of its cycles. The vibrato’s swing in milliseconds, though, is set by its width and rate, not by the pitch: the notes slide by the same fraction of a millisecond whatever octave they are in. Higher up, the tolerance for being in step may also shrink, since the binding essays’ third of a millisecond was measured for this chord over C2. The computation is this one with the chord moved up the keyboard and the tolerance recomputed for each register, and it would say whether there is a register where a partly locked vibrato’s returns are long enough to bind — where the chord really would bind and release at each swing — or whether a shorter flutter period is always paid for by a stricter tolerance.