Timbre and acoustics

Two octaves up, a sung chord can bind at every swing

A vibrato's swing between two notes is the same number of milliseconds in every register; the tolerance for being in step halves each octave, and so does the period of the chord's own flutter. The first two make a scaling law — an octave higher is the same chord with its vibratos disagreeing half as much — and the third breaks the tie. Matched for time in step, the higher chord's returns carry twice as many flutter cycles: over C2 fewer than one return a second is long enough to bind, over C4 with the vibratos locked ninety-nine and a half per cent nearly every return is.

Assumes: A chord comes back into step too briefly to bind · A tuned chord's flutter binds it only if its notes start together

A chord comes back into step too briefly to bind followed a dominant seventh tuned exactly 4 : 5 : 6 : 7 over C2 through eight seconds of singing, every note carrying the typical trained vibrato — 71 cents either side at about six a second — with the four vibratos locked to one another to some degree. It read the chord as timing: how far ahead of or behind the bass each upper note’s waveform runs at each instant. A partly locked vibrato does not carry the notes ever further apart; it swings them back and forth, and twice in every vibrato cycle all three pass back within the third of a millisecond that a tuned chord’s flutter binds it only if its notes start together had found the chord’s binding could bear. The chord comes back into step about ten times a second. But each return was too short to carry even one cycle of the chord’s own flutter — the 61-millisecond period at which every one of the ear’s filters fluctuates for this chord — and fewer than one return a second lasted that long at any lock.

It ended on the obvious next question, because two of its quantities move with the register and one does not. The flutter’s period is a quarter of the bass frequency, so an octave higher the chord flutters twice as fast and a return of the same length holds twice as many cycles. The vibrato’s swing in milliseconds is set by its width and rate, not by its pitch. But the tolerance for being in step had been measured only over C2, and higher up it might be stricter. Is there a register where the returns are long enough to bind — where the chord would really bind and release at each swing — or is a shorter flutter always paid for by a stricter tolerance?

There is such a register, and the reason is a scaling law that makes the question nearly exact.

The tolerance, measured again

The first thing to settle is the one quantity the earlier essay could not carry up the keyboard.

Fig. 1 The tuned close seventh over C2, C3 and C4, each note displaced in onset by a uniform draw up to the spread on the axis: the mean correlation between the ear’s filters as a share of the chord struck together, with the filters spanning the same range relative to the bass in each register. The dots are where it has fallen five per cent — the onset tolerance. The buttons play the chord over C2 and over C4, its vibratos fully locked.

Over C2, displacing the notes by up to a tenth of a millisecond leaves the comodulation at 99 per cent of the chord struck together, and it has fallen five per cent at 0.306 milliseconds — the third of a millisecond the earlier essays used, recovered from the same computation. Over C3 the same five per cent is reached at 0.153. Over C4, at 0.070.

The tolerance halves with each octave, and the reason was already in the binding essay. What decides whether the filters fluctuate together is the relative phase of the partials that meet inside each filter, and displacing a note in time turns each of its partials by an angle proportional to the partial’s frequency. Move the chord up an octave and every partial that meets in a filter is an octave higher, so the same displacement turns it twice as far. The tolerance is a fixed fraction of the period of the partials doing the work, and that period halves with the register.

The swing does not move

The second quantity is the vibrato’s swing, and here the arithmetic is short enough to state in full. A note’s waveform runs ahead of where it would be by the integral of its fractional change of frequency, and a vibrato of 71 cents changes a note’s frequency by the same fraction whether the note is at 65 hertz or at 262. So two notes whose vibratos differ by a given amount drift apart and back by the same number of milliseconds in every register. The swing depends on how far the vibratos disagree, and on nothing about pitch.

Put those two facts together and they make a law.

Fig. 2 The tuned close seventh over C2, C3 and C4 with a vibrato on every note, each judged against its own onset tolerance: the share of eight seconds the chord spends in step, against how far the four vibratos are locked, drawn as how far short of complete the lock is on a logarithmic axis.

The three curves are one curve, shifted. Over C2 the chord is in step 64 per cent of the time at ninety-eight per cent lock; over C3 it is in step 65 per cent of the time at ninety-nine; over C4, 62 per cent at ninety-nine and a half. Down the curve the matches are as close: C2 at ninety-nine, 88 per cent; C3 at ninety-nine and a half, 88; C4 at ninety-nine and three quarters, 86.

An octave up, the chord is the same chord with its vibratos disagreeing half as much. The swing in milliseconds is proportional to the disagreement, how far the lock falls short of complete; the tolerance halves with each octave; so halving the disagreement puts the swing back inside the band exactly as often as before. At any one lock the higher chord is in step less: at ninety-eight per cent, 64 per cent of the time over C2, 39 over C3 and 19 over C4. A shorter flutter really is paid for by a stricter tolerance, and the payment is made in time in step.

The flutter is what breaks the tie

If the tolerance were the whole story, the higher registers would simply be harder, and an ensemble would need a tighter lock to do the same thing. But the thing it is trying to do is carry the chord’s flutter through a return, and the flutter has halved too.

Fig. 3 The tuned close seventh over C2, C3 and C4, each register judged against its own tolerance and its own flutter: how many times a second the chord comes back into step after its first two seconds, and how many of those returns last a whole cycle of its flutter. The dial sets the lock. The buttons play the chord over C4 and over C2 with the vibratos as drawn.

At ninety-nine per cent lock the chord over C2 comes back into step 4.6 times a second, and 0.54 of those returns a second last a flutter cycle of 61 milliseconds: most of the time it is simply in step, and when it leaves, its returns are brief. Over C3 it returns 9.7 times a second and 3.8 of the returns last one of its 31-millisecond flutter cycles. Over C4 it returns twelve times a second — once at every pass of the vibratos through their mean — and 5.1 of those returns last a cycle of its 15-millisecond flutter.

Turn the dial to ninety-nine and a half and the picture is the one the earlier essay looked for and did not find. Over C4 the chord returns 10.2 times a second and 8.7 of the returns last a flutter cycle or more: nearly every swing of the vibrato carries the chord back into step for long enough to bind and lets it go again. Over C2 at the same lock the chord hardly leaves step at all, and over C3 it is between the two.

Fig. 4 The same three registers with the vibratos locked ninety-nine and a half per cent. Over C2 the chord is in step nearly all the time and returns only when it has strayed; over C4 it returns at nearly every swing of the vibrato, and nearly every return lasts a flutter cycle.

Turn the dial down to ninety-seven and nothing binds anywhere: 0.44 long returns a second over C2, 0.65 over C3, 0.25 over C4. The returns are too short in every register, and shortest in milliseconds where the tolerance is tightest.

Matched for time in step, the higher chord binds

The scaling law turns these three readings into one comparison, and the comparison is the essay’s answer. Take each register at the lock that gives it the same share of time in step — C2 at ninety-eight per cent, C3 at ninety-nine, C4 at ninety-nine and a half, all about 64 per cent in step. The returns are then the same length in milliseconds in every register, because they are the same swings measured against a proportionally scaled band: 80 milliseconds on average over C2 and C3, and 65 over C4. But a return of 80 milliseconds is 1.3 of C2’s flutter cycles and 2.6 of C3’s, and the 65-millisecond returns over C4 are 4.3 of its own.

So matched for time in step, the higher chord’s returns are long enough to bind several times as often per octave: 0.7 a second over C2, 3.8 over C3, 8.7 over C4. The lead asked whether a shorter flutter is always paid for by a stricter tolerance. It is paid for in time in step, exactly; it is not paid for in binding, because binding is measured in the flutter’s own cycles and the cycles shrink faster than the returns do once the lock has been tightened to compensate.

Fig. 5 The tuned close seventh over C4, exactly in tune, every note with the typical vibrato, the four vibratos locked ninety-nine and a half per cent: how far ahead of or behind the bass each upper note’s waveform runs through the fourth second, for one draw. The band is C4’s onset tolerance; the shaded columns are the moments in step; the ticks along the top are the chord’s flutter cycles. The buttons play the chord at this lock and at ninety-eight per cent.

The trace shows what that looks like. Through one second, the three upper notes swing either side of the bass by a few hundredths of a millisecond to a few tenths — the band is seven hundredths — and the chord passes into step thirteen times, each time for several of the ticks along the top. The columns are not slivers, as they were over C2 at the locks the earlier essay drew; on average each holds about four flutter cycles. Over eight draws the chord is in step 62 per cent of the time, in returns averaging 65 milliseconds. That is a chord binding and releasing in time with the vibrato: about twelve times a second it comes together long enough for its filters to fluctuate in common, and twelve times it comes apart.

Why the higher chord breaks its time into returns

It is worth seeing why a higher chord at a tight lock produces many returns rather than a few long ones, because that is what makes it bind where the lower one simply stays in step.

The returns happen where all three upper notes pass through the band together, and each note’s offset swings at the vibrato’s rate about a slowly wandering centre. When the band is wide compared with the swing — the bass register at a tight lock — the offsets spend most of each swing inside it, and the chord leaves step only occasionally, when the slow wandering pushes one note’s swing out at its peak. Those are few and long: over C2 at ninety-nine per cent, 4.6 returns a second, averaging 181 milliseconds. When the band is narrow compared with the swing — any register at a loose lock — the offsets cross it quickly and the returns are slivers. The binding register is the one in between, where the band is about the size of the swing: the notes are inside it for a good part of every half-cycle and outside it at every peak. Then the chord leaves and returns at the vibrato’s own rhythm, and each return is a fraction of a vibrato cycle long — about 65 milliseconds out of the 83 between passes.

Whether that fraction contains a flutter cycle depends on the register, because the flutter’s period does and the vibrato’s does not. Over C2 a 61-millisecond flutter barely fits in a return of that length; over C4 a 15-millisecond flutter fits four times. That is the sense in which the flutter breaks the tie. The scaling law makes every register reach the in-between state at a lock of its own, and only the higher registers have a flutter fast enough to be carried through it.

A tuned chord stays in step for half a cent-second found the comparable law for mistuning: the time a chord stays in step is inversely proportional to how far it is out of tune. Here the corresponding quantity is the vibratos’ disagreement, and the register enters as a scale on the tolerance rather than as a separate effect. In both, the chord’s coherence is a matter of milliseconds, and the register decides how many of the ear’s flutter cycles those milliseconds contain — the point a tuned close seventh low on the keyboard flutters at one rate began from, carried up the keyboard.

What it costs to get there

The register where this happens is not an easy one.

Fig. 6 The tuned close seventh over C2, C3 and C4: the mean length of a return to step after the first two seconds, in cycles of that register’s own flutter, against how far the vibratos are locked, on logarithmic axes. The line is one cycle. The buttons play the chord over C3 locked ninety-nine per cent and over C2 locked ninety-eight.

At any fixed lock, the higher chord’s average return is no longer even in its own faster cycles: at ninety-nine per cent, 2.96 cycles over C2, 2.63 over C3, 1.50 over C4. The higher chord wins only because, at the locks where it binds, it breaks its time in step into many returns, each long enough, where the lower chord at a lock that tight would hardly leave step at all. A lock of ninety-nine and a half per cent means the four singers’ vibrato rates and phases differ by a two-hundredth of what they would differ by if each sang alone. The choir essays — a choir’s vibratos must lock to two milliseconds — found locks of that order demanding for a unison, and there is no measurement here of whether any ensemble achieves it.

So the register question has a two-part answer. Higher is not easier: at any lock an ensemble can actually hold, a higher chord spends less time in step. But higher is the only place where binding at the swing is possible at all, and the earlier essay’s hope — a chord heard as rhythmically binding and releasing with the vibrato — is a description of a close seventh around middle C sung with a very tight lock, and of nothing in the bass.

The surprising part

The surprising thing is the scaling law, because it says something that a listener could not easily guess: that a vibrato’s effect on a tuned chord’s coherence depends on register only through the product of the register and the vibratos’ disagreement. A chord an octave higher with vibratos half as different is, in every timing quantity, the same chord. That is the same kind of statement as the equal-temperament ratio in the octave that fails both checks least, where a balance turned out to belong to the temperament rather than to the piano; here the swing turns out to belong to the vibrato’s disagreement rather than to the pitch, and the register only sets the scale it is judged on.

It also connects with a vibrato hides a choir’s tuning above four hundred hertz, which found the vibrato’s speed working against the ear’s resolution in the treble, and with what a choir does that a soloist cannot: many slightly different vibratos are a warmth to a unison and a cost to a tuned chord, and the cost falls hardest in the bass. And it gives a reason, from the timing side, for a habit often described in close-harmony singing: matching vibratos within a section, and singing low close chords with little or none. The earlier essays found the wider the vibrato, the tighter the lock needed to keep the chord in step; this one adds that the lower the chord, the less a lock can do for its binding once it has one. Neither essay shows that singers do this for these reasons. They show that the reasons are available.

The models under the timing

The tolerance is a five-per-cent fall in comodulation. Each register’s tolerance is the spread of onsets at which the mean correlation between filters has fallen five per cent below the chord struck together, with the filters spanning 60 hertz to 1.5 kilohertz scaled by the register. That reproduces the earlier essays’ third of a millisecond over C2. A different criterion would move every tolerance by the same factor and leave the halving per octave alone.

The vibratos are the earlier essays’ draws, each note’s rate six a second plus a spread of half a hertz clipped to the measured range, with a random phase, both pulled towards the bass’s by the lock, over the same eight seeds in every register. The scaling law depends on the swing being linear in the disagreement, which it is to the accuracy shown here because the vibrato rates differ by hundredths of a hertz at these locks.

In step means all three upper notes within the tolerance at once, and binding means a return lasting a whole flutter cycle, the period of 4 : 5 : 6 : 7 over the bass. A listener might need two or three cycles rather than one, which would raise every lock in this essay without changing which register can reach it first.

What the timing cannot show

Whether the binding is heard as a rhythm. Twelve returns a second, each holding a few flutter cycles, is a chord whose filters fluctuate in common for a few tens of milliseconds at a time. Whether a listener hears that as a chord pulsing between fused and unfused, or as a single slightly unstable sound, is a question about grouping at a time scale the arithmetic can describe and not settle.

Real voices higher up. The chord over C4 sits in the middle of a mixed choir’s range, where real vibratos are often narrower than the typical 71 cents and where the singers’ formants shape which partials reach the filters. A narrower vibrato, as the earlier essay found, shrinks the swing in proportion and would bring the binding register down to locks an ensemble might hold.

Still open: the vibrato’s width as the second lever

The scaling law has two factors in it, the register and the vibratos’ disagreement, and the earlier essay showed a third that enters the same way: the vibrato’s width, which multiplies the swing directly. If width and disagreement enter only as their product, then a 34-cent vibrato locked ninety-eight per cent over C4 is the same chord as a 71-cent vibrato locked ninety-nine — a lock an ensemble might plausibly hold. The computation is this one with the width swept alongside the lock in each register, and it would say 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 can actually keep.