Timbre and acoustics

A narrow vibrato needs no lock

A tuned close seventh binds and releases at the vibrato's swings only where its notes' timing swings by about the width of its onset tolerance, and the swing is the vibrato's width times how far the singers' vibratos disagree. Down to a ninety-five per cent lock the two trade one for one: a 71-cent vibrato locked ninety-nine and a half per cent is the same chord as a 7-cent one locked ninety-five. Below that the trade fails, in the singers' favour. With no lock at all the swing stops growing with the disagreement, and a vibrato of three cents over middle C — twelve over the bass — binds the chord at most of its swings with each singer's vibrato its own.

Assumes: Two octaves up, a sung chord can bind at every swing · A chord comes back into step too briefly to bind

A close seventh chord tuned to 4 : 5 : 6 : 7 makes the ear’s filters fluctuate together, at the rate of the chord’s own flutter, as long as its four notes stay in step — within a fraction of a millisecond of where a chord struck together would have them. A vibrato on every note moves each note’s waveform ahead of and behind where it would be, and unless the four vibratos agree exactly, the notes swing in and out of step with each other. Two octaves up, a sung chord can bind at every swing worked out how that swing scales. The swing in milliseconds depends on how far the vibratos disagree and not on the pitch; the tolerance for being in step halves with each octave; so a chord an octave higher behaves like the same chord with its vibratos disagreeing half as much. And because the chord’s flutter also speeds up an octave higher, a register exists — around middle C, with the vibratos locked ninety-nine and a half per cent — where the chord comes back into step at nearly every swing and stays long enough to bind.

That lock was the trouble. Four singers whose vibratos agree to a two-hundredth of what they would if each sang alone are not singers anyone has measured. The essay ended by pointing at the other factor in the swing. A vibrato’s swing is proportional to its width as well as to the disagreement, so a narrower vibrato should do what a tighter lock does: if width and disagreement enter only as their product, a 34-cent vibrato locked ninety-eight per cent is the same chord as a 71-cent vibrato locked ninety-nine. The question was whether the product is the whole story, and what width a close seventh around middle C would need for its singers to bind it at every swing with a lock they could actually keep.

The product is the whole story only for tight locks. For the locks singers might keep, it is wrong in a way that helps them.

The law, and where it holds

Fig. 1 The tuned close seventh over C4, exactly in tune, a vibrato on every note, its width chosen so that the width times how far short of complete the lock falls is the value on the axis: the share of eight seconds the chord spends within this register’s onset tolerance, for four locks. If width and disagreement entered only as their product the curves would coincide. The dial moves the chord to C2 and C3, with the axis scaled by the register. The buttons play the chord with a 71-cent vibrato locked 99.5 per cent and a 1.4-cent vibrato locked 50 per cent — the same product.

The model is the earlier essays’. Each note’s vibrato has a rate drawn from the spread measured in trained singers and a random phase; the lock pulls every rate and phase towards the bass’s by a stated share, so that a lock of ninety-nine per cent leaves each note a hundredth of its own disagreement. Each note’s timing is the integral of its frequency, read sample by sample; the chord is in step while all three upper notes lie within the register’s onset tolerance of the bass — 0.306 milliseconds over C2, 0.153 over C3, 0.070 over C4. The width runs from half a cent to seventy-one.

Over C4, with the product of width and disagreement at 0.355 cents, the chord is in step 62 per cent of the time with a 71-cent vibrato locked ninety-nine and a half per cent, and 61 per cent with an 18-cent vibrato locked ninety-eight. Across the whole axis those two curves lie on top of each other, to a per cent or two. At tight locks, width and disagreement trade exactly.

At ninety per cent the curve starts to come away. At half the lock it is in a different place altogether: a 1.4-cent vibrato locked fifty per cent, the same product of 0.71 cents that leaves the tightly locked chord in step 38 per cent of the time, is in step all the time. The narrow, loosely locked chord does not leave step until the product is twice as large again.

Turning the dial gives the same picture in the other registers, with the axis scaled by the factor each octave’s tolerance changes. The scaling law of the earlier essay and the trade between width and lock are one law: the swing measured against the tolerance. Where the trade fails, it fails in every register at the same lock.

Along one line of the law

Fig. 2 The tuned close seventh over C4, every bar a width and a lock whose product is 0.355 cents: the returns to step a second that last a whole cycle of the chord’s flutter, and the share of time in step. The buttons play the 18-cent vibrato locked 98 per cent and the 1.4-cent vibrato locked 75 per cent.

The binding measure tells the same story in the units the earlier essay used. With the product held at 0.355 cents, a 71-cent vibrato locked ninety-nine and a half per cent returns to step for a whole flutter cycle 8.71 times a second; 35.5 cents locked ninety-nine, 8.69; 17.75 cents locked ninety-eight, 8.54; 7.1 cents locked ninety-five, 8.02. Down to ninety-five per cent the chord binds at nearly every swing, and the singers can buy the lock they cannot keep with a vibrato a tenth as wide.

At ninety per cent, with a 3.55-cent vibrato, the chord binds a little more often — 9.56 returns a second — and is in step 72 per cent of the time rather than 61. Below that the chord stops leaving step at all: at eighty per cent with 1.8 cents, at seventy-five with 1.4, at fifty with 0.7, it is in step throughout, and with no departures there are no returns to count. The law says these chords should swing like the first; they do not swing out of the band at all.

Why the trade fails at loose locks

Fig. 3 The tuned close seventh over C4, a 10-cent vibrato on every note: the widest timing offset between an upper note and the bass over eight seconds, against the lock, with the line the tight-lock proportion carried on.

A note’s timing offset under a vibrato is the running sum of how far its frequency departs from steady, and for two notes whose vibratos are nearly alike, the difference between their offsets is nearly proportional to how far their rates and phases differ. That is the regime the earlier essays measured, and it is what makes the product law exact there: with a 10-cent vibrato, the widest swing is 0.030 milliseconds at ninety-nine and a half per cent, 0.061 at ninety-nine, 0.118 at ninety-eight, doubling as the disagreement doubles.

But a vibrato is a periodic wobble, and two of them cannot drift apart without limit. However different their rates and phases, each note’s offset from steady is a bounded oscillation whose size is set by its width and rate alone, and the difference between two bounded oscillations is bounded too. Loosen the lock and the swing stops growing: 0.240 milliseconds at ninety-five per cent, 0.298 at ninety, 0.393 at fifty, and 0.451 with no lock at all, where the proportional law would have it at 6.1. The swing saturates at about the size of one vibrato’s own excursion.

That saturation is what the singers can use. With no lock, the swing is set by the width alone, so a vibrato narrow enough that its own excursion is about the size of the onset tolerance swings the chord in and out of step at the vibrato’s rhythm without any agreement between the singers at all.

The width that binds with no lock

Fig. 4 The tuned close seventh, exactly in tune, every note with a vibrato of the width on the axis at its own rate and phase, nothing locked: the returns to step a second that last a whole flutter cycle, over C2, C3 and C4. The buttons play the chord over C4 with a 3-cent free vibrato and over C2 with a 12-cent one.

Every register has a width that binds best with the vibratos free. Over C4 it is three cents: the chord returns to step for a whole flutter cycle 7.4 times a second, against 8.7 for the 71-cent vibrato locked ninety-nine and a half per cent. Over C3 it is six cents, at 5.8 a second; over C2, twelve cents, at 3.3. Narrower than that and the chord never leaves step; wider and its returns grow too short to hold a flutter cycle.

The binding width halves with each octave up, for the reason the earlier essay’s scaling law gave: the tolerance halves, so the swing has to halve, and with no lock the swing is the width. And the number of binding returns rises with each octave, for the earlier essay’s other reason: a higher chord’s flutter is faster, so a return of a given length holds more of its cycles.

So the lead’s question gets an answer it did not expect. The width a close seventh around middle C needs for its singers to bind it at every swing, with a lock they can keep, is about three cents, and the lock they need to keep is none.

What a free three-cent vibrato does

Fig. 5 The tuned close seventh over C4, exactly in tune, a 3-cent vibrato on every note at its own rate and phase, nothing locked: how far ahead of or behind the bass each upper note’s waveform runs through the fourth second, for one draw. The band is the register’s onset tolerance; the shaded columns are the moments all three upper notes are inside it; the ticks along the top are the chord’s flutter cycles. The buttons play the chord with a free 3-cent vibrato and a free 12-cent one.

The trace shows the mechanism in one second of one draw. Each upper note’s timing swings about the bass by a little more than a tenth of a millisecond, at its own vibrato rate, and the three swings drift slowly in and out of phase with each other. The band is seven hundredths of a millisecond either side. Six times in the second, all three are inside the band together, and five of those returns last at least one of the chord’s fifteen-millisecond flutter cycles — most of them three or four. In this draw the widest swing is 0.18 milliseconds, two and a half times the tolerance; over eight draws the chord is in step 48 per cent of the time.

That is the in-between state the earlier essay identified as the binding one: a swing about the size of the band, so that the notes cross it at every half-cycle and spend a good part of each half-cycle inside. The earlier essay reached it by squeezing a 71-cent vibrato’s disagreement down to a two-hundredth. A free 3-cent vibrato reaches it by being small, and the slow drift between the singers’ rates, which at a tight lock would be the source of the swing, here only moves which swings line up — as the last half-second of the trace shows, where the seventh’s swing drifts against the others and the returns thin out until it comes round again. The pitch that does not wobble described a vibrato as a swing about a centre the ear hears as the pitch; in a tuned chord the size of that swing, measured against a fraction of a millisecond, decides whether the chord’s notes keep meeting.

Where binding lives, over middle C

Fig. 6 The tuned close seventh over C4, exactly in tune: the returns to step a second that last a whole flutter cycle, for every pair of vibrato width and lock in the grid, darker for more. The buttons play the chord with a 4-cent vibrato locked 90 per cent and a 35-cent vibrato locked 99 per cent.

Put width and lock on one map and the binding region is a band running diagonally from narrow and free to wide and tightly locked. With no lock, three and four cents bind at seven and six returns a second. At fifty and seventy-five per cent the same widths bind at eight to ten. At ninety per cent the band has moved to four cents; at ninety-five, to six and eight; at ninety-eight, to twelve and eighteen; at ninety-nine, thirty-five; at ninety-nine and a half, seventy-one. Off the band on one side the chord never leaves step; on the other it is never in step long enough.

The diagonal is the product law, and its lower end bending to the vertical is the saturation. Below about ninety per cent, the lock stops mattering: the band runs straight down at three to four cents to no lock at all.

What this does to the earlier picture

The earlier essays drew a sharp contrast between registers and between locks, and ended on a demanding conclusion: the chord that binds at the swing is a close seventh around middle C sung with a lock nobody has measured. Read with the width as the second lever, the conclusion changes shape. The binding chord is a close seventh around middle C sung with a vibrato of a few cents, and the lock is irrelevant.

A vibrato of a few cents is not what singers mean by vibrato. A trained classical voice’s vibrato is around seventy cents wide; a few cents is the residual wobble of a voice singing straight, the natural fluctuation that a note that is never at its pitch measured even in steady tones. So the arithmetic points at straight-tone singing — the habit of close-harmony groups, early-music ensembles and barbershop quartets, who sing their tuned chords with little or no vibrato — and says that the small irregular wobble such singing still contains is about the right size, near middle C, to make a tuned chord bind and release at the wobble’s own rhythm.

A shared vibrato cannot hold a tuned chord together found that the wider the vibrato, the tighter the lock it needs; a chord comes back into step too briefly to bind found the returns too short in the bass at the locks it tried. This essay closes the loop between them: narrow enough, the vibrato needs no lock, and the returns it makes are long enough to bind wherever the register’s flutter is fast enough to fit in them.

Why the bass needs more width than the treble

The registers’ binding widths — twelve cents over C2, six over C3, three over C4 — run the opposite way from the practice of singing low chords straight. The arithmetic does not contradict the practice; it measures something else. A wider wobble is needed in the bass to swing the chord out of step at all, because the bass’s tolerance is four times as wide; but the binding it then produces is weaker, 3.3 returns a second against 7.4, because the bass’s flutter is slow and few returns hold a cycle of it. A singer who wants a low tuned chord to sound steady sings it straight and keeps it in step; a singer who wants a chord around middle C to shimmer with the flutter of its own tuning can leave a little wobble in. A tuned close seventh chord low on the keyboard flutters at one rate began this whole line from the bass, where the flutter is slowest and the chord most easily heard as one sound.

The model under the timing

Vibratos as the earlier essays drew them: a rate of six a second plus a spread of half a hertz clipped to the measured range, a random phase, both pulled towards the bass’s by the lock, eight draws, every note with the same width. A real straight-tone singer’s wobble is not a sinusoid at six hertz; it is slower and irregular, and a wobble with energy spread over rates would swing the chord less regularly. The figures describe a sinusoidal vibrato narrowed, and the claim about straight singing is a claim about the size of the wobble, not its shape.

In step and binding as before: all three upper notes within the register’s onset tolerance, and a return counted as binding when it lasts a whole cycle of the chord’s flutter, a quarter of the bass frequency.

Exactly in tune. Every chord here is tuned to 4 : 5 : 6 : 7. A mistuning of a cent or two, which a tuned chord stays in step for half a cent-second priced, drifts the notes apart steadily; a narrow free vibrato swings them about a drifting centre, and the binding would last only as long as the drift kept the centre inside the band.

What the timing cannot settle

Whether the binding is heard. As before, a chord whose filters fluctuate together for a few tens of milliseconds, a few times a second, may be heard as pulsing between fused and unfused or as one slightly unsteady sound. The arithmetic describes the timing.

What singers’ straight tones contain. Whether the residual wobble of straight singing is two cents or ten, sinusoidal or not, and independent between singers, is a measurement of recordings that the figures assume rather than make. A choir’s vibratos must lock to two milliseconds is a reminder of how much of these essays rests on numbers no one has recorded singer by singer.

Still open: a wobble that is not a sinusoid

The saturation that frees a narrow vibrato from the lock comes from the vibrato being periodic: a sinusoid’s offset from steady is bounded. A straight-tone singer’s residual pitch fluctuation is closer to slow noise — a random wander of a few cents with most of its energy below two or three hertz — and a random wander’s timing offset is not bounded: it grows like a random walk. The computation is this one with each note’s frequency a smoothed random process of stated size and bandwidth in place of the sinusoid, and it would say whether straight singing’s wobble binds the chord as a narrow vibrato does, or whether, by wandering without returning, it drifts the chord out of step and leaves it there until a singer corrects.