A tuned chord stays in step for half a cent-second

A tuned close seventh chord low on the keyboard flutters at one rate found the one place in these chords where exact tuning adds something rather than sharpening it. A dominant seventh in close position, tuned exactly 4 : 5 : 6 : 7 over a bass below B2, repeats as a whole at a rate too slow to be a pitch, and every auditory filter from 60 hertz to a kilohertz and a half fluctuates at that one rate. A tuned chord’s flutter binds it only if its notes start together then asked whether the filters rise and fall together — comodulation, one of the cues the ear treats as evidence that a sound is one thing — and found that it depends on when each note begins. Struck at the same instant, the filters’ envelopes correlate at 0.59 on average. Spread the notes’ onsets over half a millisecond and that falls to 0.51; over two milliseconds, to the tempered chord’s 0.16.

That essay said one thing in passing that this one tests. Once begun, a tuned chord’s relative phases stay where they were for as long as it sounds, since its notes share a period exactly. That is true of a chord tuned exactly. Nothing played by voices, bows or breath is tuned exactly, and the obvious question is what a fraction of a cent does to a relationship that was found to need a fraction of a millisecond.

A cent is a slide in time #

A note detuned by d cents has a period shorter or longer than its tuned period by a factor of two to the power d/1,200. Over one second, its waveform slides against a perfectly tuned partner by that factor less one, of a second. For one cent that is 0.00058 — 0.58 milliseconds of slide every second. For a tenth of a cent, 0.058 milliseconds a second; for five cents, nearly three.

That turns the earlier essay’s finding about onsets into a finding about tuning. The chord’s notes have to be within about half a millisecond of each other for the filters to stay locked together, and a note a cent out of tune moves half a millisecond out of place in under a second. A chord struck in step and held with its notes a cent apart should therefore leave step in about the time it takes the slide to reach the tolerance the onsets were found to need.

The computation holds the chord rather than letting it decay, so that any change in the windows is the drift and nothing else. Each of the four notes has its partials at whole-number multiples of its own slightly mistuned fundamental, so each note is internally coherent and the notes slide against each other. The envelopes in every filter one equivalent rectangular bandwidth apart from 60 hertz to 1.5 kilohertz are compared in half-second windows every quarter-second, and the mean correlation over every pair of filters is the comodulation, as before. Eight draws of the four notes’ mistunings are averaged.

Half a cent-second #

Struck together and exactly in tune, the chord’s comodulation sits at 0.59 for all eight seconds. With its notes spread by a tenth of a cent it is still 0.56 after one second, 0.41 after three, 0.32 after five, 0.25 after seven. Spread by a quarter of a cent it is 0.43 after one second and 0.23 after three. By half a cent, 0.29 after one second and at the tempered chord’s level by three. By a cent, 0.18 after one second.

Read as the time to fall halfway from 0.59 to 0.16, the figures obey a law. The time in step is inversely proportional to the mistuning, and their product stays near half a cent-second: four seconds at a tenth of a cent, three at 0.15, 1.75 at a quarter, 1.25 at 0.35, one second at half a cent, three-quarters at three-quarters, half a second at a cent. On logarithmic axes the points lie on a line of slope minus one.

The constant has a meaning. Half a cent-second is half a cent sustained for one second, a slide of 0.29 milliseconds; or a tenth of a cent for five seconds, the same slide. The chord leaves step when its notes have slid about a third of a millisecond against each other, whatever the mistuning that slid them. That is the same order as the onset tolerance the earlier essay found: half a millisecond of spread at the start cost the chord a seventh of its comodulation, one millisecond half of it. The drift and the onset are one tolerance, reached two ways.

Put the two figures side by side and the correspondence can be read off them. An onset spread of one millisecond halves the chord’s advantage over the tempered chord; so does a drift that has slid the notes by about a third of a millisecond on average against each other, since the drawn mistunings and the drawn onsets spread the notes differently — a draw of onset times within a millisecond puts the average pair of notes about a third of a millisecond apart, and a drift of half a cent-second does the same. The comparison is not exact, because a drift and an onset spread distribute the notes’ offsets differently, but the order of magnitude is the same, and it is set by the partials that meet in the filters that matter: the third to the tenth of each note, with periods of one to five milliseconds, which a third of a millisecond turns by a large fraction of a cycle.

Who holds a chord that well #

The law makes a stringent demand. A sustained chord of four seconds — not a long one in a slow choral piece — stays in step for its length only if its notes are within about an eighth of a cent of exact. The essays on choral tuning here, a vibrato hides a choir’s tuning above four hundred hertz among them, have used a spread of fifteen cents between voices in a unison, from measurements of real choirs, and even an exceptional ensemble holding its tuning to two or three cents would be out of step within a fifth of a second.

An organ tuned in just intonation is the nearest acoustic candidate, since a pipe holds its pitch far better than a voice. But an eighth of a cent is a demanding standard even for a tuner. At C3, the second partial of a chord on C2, an eighth of a cent is a difference of a hundredth of a hertz: two pipes that far apart beat once every hundred seconds. A tuner setting the chord’s pipes by ear would have to leave no beat between their coinciding partials faster than about one in a minute and a half, and then hope the temperature of the building did not move any rank against the others while the chord was held.

This sharpens the irony the earlier essay ended on. The instruments that can tune a seventh to 7 : 4 — voices, bowed strings, trombones — are the ones whose notes start independently, so they lose the chord’s comodulation at the start. It now turns out they would lose it anyway within a fraction of a second, unless they could hold their tuning to a tenth of a cent. The one sound source that does both, starting its notes together and holding them exactly, is an electronic instrument generating 4 : 5 : 6 : 7 from one clock.

There is a parallel in the voice essays that is worth putting beside this. A choir’s vibratos must lock to two milliseconds found that a section’s vibratos would have to peak within about two milliseconds of each other for a listener to count its mistuning at the singer’s formant as if the voices were straight. Here a tuned chord’s notes have to stay within a third of a millisecond for its filters to stay comodulated. Both are timing tolerances on the order of a millisecond, imposed on performers whose natural timing varies by tens. Both describe structure the arithmetic finds in an ideal ensemble and an actual ensemble cannot deliver.

Out of step stays out of step #

The earlier essay’s lead hoped for the opposite: that a chord started out of step, if its notes were slightly mistuned, would walk its notes’ timings around their periods and pass through alignment, so that a sustained chord would be comodulated some of the time whatever its onsets.

It does not. Started within five milliseconds and held exactly in tune, the chord’s comodulation is 0.20 throughout — the same as the earlier essay’s asynchronous chords. With half a cent of drift the mean is 0.15 and the highest any draw reaches in any half-second window is 0.36. With one cent, 0.34; with two, 0.32. Struck in step, the same chord is at 0.59.

The reason is that alignment is a coincidence of four timings at once. Each note slides around its own period, of nine to fifteen milliseconds, at its own rate, and coming back into step requires all four to pass within about a millisecond of one another at the same moment. Even with three of them free to wander, that coincidence is rare enough that eight seconds of eight different chords never produce it. A drifting chord samples its alignments; it does not seek them.

That echoes the grouping cue the earlier essay set beside comodulation. The cue that settles it put the onset cue’s window at about twenty milliseconds, and what makes two partials one note found thirty milliseconds of head start enough to take a harmonic out of its note: shared onsets bind within tens of milliseconds and need no later help. Comodulation binds within one millisecond and cannot be found again once lost.

So the slight mistuning that every real chord has does not give back the comodulation a spread onset took away. It takes away what an exact onset gave. Drift moves a tuned chord towards the tempered chord’s level, from either direction, and nothing in the drift moves it the other way.

One note out of tune #

A chord whose mistuning is all in one note behaves differently from one whose mistuning is shared. With only the third half a cent sharp, the comodulation falls from 0.60 to about 0.36 within two seconds and then stays there for the rest of the eight. With every note spread by half a cent, it falls to 0.18. The three notes still in tune with each other stay in step with each other, and the filters they dominate — the ones the third contributes little to — stay correlated. The one drifting note decorrelates the filters it shares, and no more.

That is the shape of the earlier findings about which notes do what in these chords. A tuned seventh chord does not beat; it buzzes followed the chord’s fluctuation filter by filter, and each filter’s envelope is set by the few partials that fall in it. The chord’s comodulation is not a property of the chord as a whole but of each filter’s mixture of notes, and a filter that one in-tune note dominates goes on fluctuating in step with the others however the rest of the chord drifts. A chord held by three exact players and one less exact one is three-quarters bound. A chord held by four slightly inexact players is not bound at all.

Whatever the window, at every bass #

The half-second window is a convention: comodulation over a shorter stretch is a noisier estimate, over a longer one a blurrier one. Read through windows of a quarter-second and of a whole second, the time in step at a quarter of a cent is 1.6 and 1.75 seconds; at half a cent, 0.9 and 1.0; at a cent, 0.6 and 0.75. The law does not belong to the window.

Nor does it belong to C2. On basses from C2 up to A♭2 — the range in which the earlier essays found the flutter, below the point where its rate becomes a pitch — the in-step value is 0.54 to 0.60 and the time in step at half a cent is between three-quarters of a second and a second. That is what the arithmetic predicts. A cent shortens a period by the same fraction whatever the period, so the slide in milliseconds per second is the same at every bass, and the tolerance it has to reach, set by the filters the chord’s partials fall in, changes little across a fifth.

The model the numbers rest on #

The chord is the earlier essays’ close dominant seventh, 4 : 5 : 6 : 7 over the bass, each note a series of partials at whole multiples of its own fundamental with the amplitudes of a struck string at the strike, held at those amplitudes for eight seconds. Each note’s fundamental is moved by a stated number of cents or by a draw from a normal distribution of the stated spread, and, where stated, begun at a random moment within a stated number of milliseconds. Each auditory filter is a rounded-exponential one equivalent rectangular bandwidth wide, its envelope the magnitude of the partials it passes summed as phasors, sampled four hundred times a second. Comodulation is the mean zero-lag correlation between every pair of filters’ envelopes in a window. Eight draws are averaged at each point, on fixed seeds, so every mistuning is the same chords.

What the picture cannot show #

Whether drift is heard as the loss of a bound sound. The arithmetic says when the filters stop rising and falling together. Whether a listener holding a tuned chord in attention hears it come apart after a second — a sound that started as one pulsing object and became three or four fluctuating regions — is the experiment the earlier essay named and could not run, now with a clock attached: the prediction is that the change, if it is heard at all, comes at a time inversely proportional to the mistuning.

Many voices on a note. Each note here is one source. A chord sung by a choir has several voices on each note, each drifting by its own amount, and what a choir does that a soloist cannot found that a section’s own unison already spreads its fluctuation across a band of rates. A chord of four sections would be four such bands, drifting; nothing in it could stay in step.

And real sustained notes. The notes here hold perfectly steady amplitudes and perfectly steady mistunings. Real voices and bows waver in pitch by far more than a cent, and their wavering is itself a drift, of varying rate, which on this arithmetic would take any chord out of step faster still. The figures are therefore an upper bound on how long a sung or bowed tuned chord could stay comodulated.

Still open: whether a vibrato can be the clock that keeps four notes together #

The drift is the enemy of comodulation because each note slides at its own rate. A shared vibrato would change that. If all four voices of a chord swung their pitch together, in rate and phase — the lock the vibrato essay priced for a unison — their notes’ timings would still slide, but they would slide together, by the same fraction of each period, and the relative timing that comodulation needs would be preserved to the extent the lock was exact. The computation is this one with a common vibrato applied to every note, partly locked as the voices were, and it would say whether a vibrato shared as precisely as the formant asked could hold a tuned chord in step longer than an exact straight tone could — or whether, as with the unison, the lock it needs is beyond any ensemble.