Acoustics & Music Theory
why it sounds like that
Timbre and acoustics

A straight tone drifts out of step and stays there

A tuned close seventh binds — its filters flutter together — only while its notes keep their timing within a fraction of a millisecond, and a narrow vibrato keeps them there because a sinusoid's timing offset swings inside a fixed range. A straight-tone singer's pitch is never quite straight: it wanders slowly, a cent or less, at random. That wander's timing offset is a random walk. Over C4 a wander of half a cent takes the chord out of step within a fifth of a second and leaves it in step four per cent of the time, where a three-cent vibrato holds it nearly half. What matters is the wander's slow part, and adding a vibrato on top never helps.
14 min read 5 figures What the listener suppliesSmall whole numbersThe instrument is not the model

Assumes: A narrow vibrato needs no lock · A tuned chord stays in step for half a cent-second

A close seventh chord tuned to the harmonic series, its notes in the ratios 4 : 5 : 6 : 7, has a property no tempered chord has: its notes’ partials coincide, and the auditory filters that hear them all flutter at one rate, as a tuned close seventh chord low on the keyboard flutters at one rate found. The flutter binds the chord into one sound only while the notes keep their timing together. A tuned chord’s flutter binds it only if its notes start together measured how closely: within about a third of a millisecond over C2, and within seventy microseconds over C4, two octaves up, where the flutter is four times as fast.

The essays that followed asked what keeps the notes in step once the chord is sounding. A chord sung exactly in tune stays in step forever. A chord sung a cent out drifts, and a tuned chord stays in step for half a cent-second put a number on how long. A vibrato warps each note’s timing back and forth, and a narrow vibrato needs no lock found that a vibrato of a few cents keeps the chord in step at most of its swings even when each singer’s vibrato is entirely their own: three cents over C4 holds the chord in step nearly half the time and lets it return to step seven times a second.

That essay ended on the difference between a vibrato and what a straight-tone singer actually produces. A straight tone is not a perfectly steady pitch. It wanders: slowly, by a cent or less, at random, with most of its energy below two or three hertz. And where a vibrato’s effect on timing is bounded — a sinusoid goes up and comes back — a random wander’s need not be.

A wander, drawn

A straight tone's slow wander drifts a chord's notes out of step and leaves them thereThe tuned close seventh over C4 held without vibrato, each note's pitch wandering on its own by 0.5 cent rms with its spectrum falling above two hertz: how far ahead of the bass each upper note's timing has run, in milliseconds, over eight seconds of one draw, with this register's onset tolerance of ±0.070 ms shaded. Over eight draws the chord is in step 4.3 per cent of the time and first leaves the tolerance after 0.16 s on average; a three-cent vibrato on every note, each its own, keeps it in step 48.2 per cent.0 s1 s2 s3 s4 s5 s6 s7 s8 s-0.500.000.50time since the chord was struckahead of the bass, msthe thirdthe fifththe seventhshaded: in stepwander 0.5 cent rms
Fig. 1 The tuned close seventh over C4 held straight, each note’s pitch wandering on its own with its spectrum falling above two hertz, the dial setting the wander’s size: how far ahead of the bass each upper note’s timing has run, over eight seconds of one draw, with the onset tolerance shaded. The buttons play the chord held straight and with a narrow vibrato on every note.

The model gives each of the four notes an independent pitch deviation that is a smoothed random process: a few tenths of a cent to a couple of cents root-mean-square, wandering with a correlation time set by the frequency above which its spectrum falls. A singer who drifts sharp tends to come back — that is what holding a pitch means — and the process does come back: it is mean-reverting, pulled towards true pitch on the time scale of its bandwidth.

What the figure draws is not the pitch but the timing. A note slightly sharp for a moment runs slightly ahead of the bass, and the amount it runs ahead is the integral of its pitch deviation over time. That integral is what the chord’s binding depends on. With a wander of half a cent, each upper note’s timing offset sets off from zero and does not come back: the third drifts a quarter of a millisecond behind, the fifth three quarters, the seventh oscillates near zero and leaves it again. The shaded band, the tolerance within which the chord binds, reaches only seventy microseconds either side of zero at C4. The chord is in it for the first sixth of a second.

Over eight draws, a half-cent wander keeps the chord over C4 in step 4.3 per cent of the time, and it first leaves step after 0.16 seconds on average. A three-cent vibrato on every note, each singer’s vibrato their own, keeps the same chord in step 48 per cent of the time. Turn the dial down to a quarter of a cent and the wander still does worse than the vibrato; turn it up to two cents and the chord is never in step after the first instant.

A vibrato swings; a wander walks

A vibrato's timing offset swings inside a fixed range; a wander's keeps growing with the square root of the time held. The tuned close seventh over C2, no lock: the root-mean-square timing offset of the upper notes against the bass, in milliseconds, against the time since the chord was struck, over eight draws, for pitch wanders of half a cent and one cent rms (spectrum falling above two hertz) and for vibratos three and twelve cents wide. a wander, 0.5 cent rms: 0.25 s 0.080, 0.5 s 0.104, 1 s 0.122, 2 s 0.203, 4 s 0.290, 7.9 s 0.492; a wander, 1 cent rms: 0.25 s 0.160, 0.5 s 0.208, 1 s 0.243, 2 s 0.407, 4 s 0.580, 7.9 s 0.983; a 3-cent vibrato: 0.25 s 0.065, 0.5 s 0.049, 1 s 0.070, 2 s 0.052, 4 s 0.050, 7.9 s 0.049; a 12-cent vibrato: 0.25 s 0.261, 0.5 s 0.195, 1 s 0.279, 2 s 0.206, 4 s 0.199, 7.9 s 0.195. The onset tolerance here is 0.306 ms.
Fig. 2 The tuned close seventh over C2, no lock: the root-mean-square timing offset of the upper notes against the bass, against the time since the chord was struck, for wanders of half a cent and a cent and vibratos of three and twelve cents, with the onset tolerance dashed. The buttons play the chord held straight and with a three-cent vibrato.

The difference is visible in one picture. A vibrato’s timing offset is the integral of a sinusoid, which is another sinusoid: it swings back and forth inside a range set by the vibrato’s width and rate, and never grows. Each note’s offset against the bass grows slowly as the vibratos’ rates disagree, but the swing itself is bounded, and averaged over the draws the offset stays where it began: 0.05 milliseconds for a three-cent vibrato at a quarter of a second and at eight seconds.

A wander’s offset is the integral of a mean-reverting random process, and over times longer than the process’s correlation time that integral is a random walk. Its spread grows as the square root of the time held: half a cent of wander gives 0.12 milliseconds at one second, 0.29 at four, 0.49 at eight. It crosses the C2 tolerance of a third of a millisecond at about four seconds, and nothing brings it back, because pulling the pitch back to true does not pull the timing back. A singer who has been a little sharp and has corrected is now in tune, but every cycle of their note is a little ahead of where it would have been, and stays ahead.

That is the mechanism the lead suspected, and it is as clean as arithmetic gets: a vibrato is bounded in pitch and therefore in time; a wander is bounded in pitch but not in time. The rest is how much that matters in each register.

How small a wander still unbinds the chord

Over C4 a wander of a quarter of a cent unbinds a chord that a three-cent vibrato holds half the time. The tuned close seventh held without vibrato, each note wandering on its own: the share of eight seconds it spends within the register's onset tolerance, over eight draws, against the wander's rms size, over C2 and C4, for wanders whose spectrum falls above one and above three hertz, with a three-cent vibrato on every note (no lock) as dashed lines. C2, 1 Hz: 0.125 97%, 0.25 62%, 0.5 27%, 1 10%, 2 3%; C2, 3 Hz: 0.125 100%, 0.25 93%, 0.5 54%, 1 20%, 2 8%; C4, 1 Hz: 0.125 23%, 0.25 9%, 0.5 3%, 1 1%, 2 1%; C4, 3 Hz: 0.125 49%, 0.25 18%, 0.5 6%, 1 2%, 2 0%; the vibrato: C2 100%, C4 48%.
Fig. 3 The tuned close seventh held straight, each note wandering on its own: the share of eight seconds it spends in step against the wander’s size, over C2 and C4, for wanders slower than one hertz and slower than three, with a three-cent vibrato on every note as dashed lines. The buttons play the chord over C4 held straight and with a narrow vibrato.

Over C2, where the tolerance is a third of a millisecond, a small wander is tolerable. An eighth of a cent keeps the chord in step 97 to 100 per cent of the time, a quarter of a cent 62 to 93, half a cent 27 to 54. By one cent it is in step a tenth to a fifth of the time, by two cents hardly at all. The three-cent vibrato keeps it in step throughout.

Over C4 the tolerance is a quarter as wide and the picture changes completely. Even an eighth of a cent of wander keeps the chord in step only a quarter to a half of the time; a quarter of a cent, 9 to 18 per cent; half a cent, 3 to 6. The three-cent vibrato, at 48 per cent, does better than any wander bigger than an eighth of a cent.

No published measurement gives the size of a trained straight-tone singer’s pitch wander in exactly these terms, but measurements of held sung notes commonly report slow fluctuations of a few cents rather than tenths of a cent. On this arithmetic that is far beyond what a tuned seventh over C4 can bear and well beyond what it can bear over C2. A straight-tone chord can be tuned perfectly and still never bind, because the singers cannot hold their pitches still enough for long enough.

Why the register decides so much

The two registers differ by more than their tolerances. Two octaves up, the chord binds at every swing found that a vibrato’s swing in timing is the same number of milliseconds in every register while the tolerance halves each octave, so that an octave higher is the same chord with its vibratos disagreeing twice as much. The same scaling governs a wander. Its timing offset after a given time is the same in every register, because the integral of a fractional pitch deviation does not depend on the pitch; the tolerance it has to stay inside is four times narrower over C4 than over C2. A wander that takes four seconds to leave the C2 tolerance takes about a sixteenth of that, a quarter of a second, for its typical spread to leave the C4 one, because a random walk’s spread grows as the square root of time — and the first excursion out of step comes sooner still, at 0.16 seconds on average for a half-cent wander. A tolerance four times narrower leaves a wander a sixteenth of the time in which to stay in step, not a quarter, and that square is the whole difference between the registers.

That is why the two registers come out so differently in the last figure, and why the comparison with the vibrato flips between them. Over C2 a three-cent vibrato and a quarter-cent wander are both nearly harmless, and the question is merely how much harm each does. Over C4 the vibrato still binds the chord at its swings and the wander leaves it out of step almost at once. The higher the chord, the more a held straight tone has to be held.

The vibrato essays traced the same pattern from the other side. A shared vibrato cannot hold a tuned chord together found a shared vibrato failing because a fraction of a cent of mistuning drifts the chord out of step in half a cent-second; a chord comes back into step too briefly to bind found that an imperfectly shared vibrato’s returns to step were too short to bind; the narrow vibrato essay found that a narrow enough vibrato fixes both. The wander is the case they all assumed away — a steady mistuning that is not steady — and it behaves like the first essay’s drift, only without a fixed direction.

The slow part is what costs

What a wander costs is its slow part: size over the square root of bandwidth decides the time in step. The tuned close seventh over C2 held without vibrato, each note wandering on its own: the share of eight seconds in step, over eight draws, against the wander's rms size in cents divided by the square root of the frequency above which its spectrum falls, for wanders falling above half a hertz, two hertz and eight hertz. 0.25 cent, 0.5 Hz (0.354): 47%; 0.5 cent, 2 Hz (0.354): 44%; 1 cent, 8 Hz (0.354): 42%; 0.35 cent, 0.5 Hz (0.495): 29%; 0.7 cent, 2 Hz (0.495): 26%; 1.4 cent, 8 Hz (0.495): 26%; 0.5 cent, 0.5 Hz (0.707): 20%; 1 cent, 2 Hz (0.707): 16%; 2 cent, 8 Hz (0.707): 16%.
Fig. 4 The tuned close seventh over C2 held straight: the share of eight seconds in step against the wander’s size divided by the square root of the frequency above which its spectrum falls, for wanders falling above half a hertz, two hertz and eight hertz. The buttons play the chord held straight and with a wide vibrato.

In the previous figure a wander slower than one hertz did more damage than an equally large wander slower than three. That is what the random walk predicts. A slow wander stays on one side of true for longer before it comes back, so each excursion carries the timing further; a quick wander reverses before the timing has gone far, and its excursions partly cancel. The rate at which the timing offset spreads is set by the wander’s size squared times its correlation time — equivalently, by its spectrum’s height at the lowest frequencies.

The figure tests that. Wanders of a quarter of a cent falling above half a hertz, half a cent above two hertz and a cent above eight hertz all have the same size over the square root of the bandwidth, and they keep the chord in step 47, 44 and 42 per cent of the time. Two more triples line up the same way at 29, 26, 26 and at 20, 16, 16. A sixteenfold change in bandwidth with the size scaled to match moves the time in step by five points or less.

So what a straight-tone singer’s wander costs a tuned chord is its slow part. The quick tremors a singer cannot help are almost harmless; the slow sag and recovery over a second or two are what drift the chord out of step. A singer trained to hold a straight tone is, in effect, trained to suppress exactly that slow part, and the arithmetic says that is the right thing to train.

A vibrato on top does not help

Adding a vibrato to a wander never brings a chord back into step. The tuned close seventh, no lock, a pitch wander of the stated rms size (spectrum falling above two hertz) and a vibrato of the stated width on every note, each its own: the share of eight seconds in step, over eight draws, over C2 and C4. wander alone: C2 44%, C4 4%; 3-cent vibrato alone: C2 100%, C4 48%; 3-cent vibrato and the wander: C2 42%, C4 3%; 12-cent vibrato and the wander: C2 24%, C4 1%; 3-cent vibrato, half the wander: C2 81%, C4 9%.
Fig. 5 The tuned close seventh, no lock, a pitch wander and a vibrato on every note, each its own: the share of eight seconds in step over C2 and over C4. The buttons play the chord over C2 held straight and with a three-cent vibrato.

The obvious remedy is to add a vibrato. A vibrato keeps the chord in step on its own; perhaps a vibrato over a wander does too. It does not. A half-cent wander alone keeps the chord over C2 in step 44 per cent of the time; with a three-cent vibrato on top, 42; with a twelve-cent vibrato, 24. Over C4 the wander alone gives 4 per cent, with a three-cent vibrato 3. The vibrato’s bounded swing rides on top of the wander’s unbounded walk, and adds to it rather than correcting it. Halving the wander helps far more than any vibrato: a three-cent vibrato over a quarter-cent wander keeps the chord over C2 in step 81 per cent of the time.

That settles the question the vibrato essays had been circling. Every earlier result about vibrato and binding assumed that each note’s pitch, apart from its vibrato, was exactly steady. The vibrato’s virtue was that it does not accumulate. Any part of a real voice’s pitch variation that does accumulate — any wander slower than the vibrato — undoes that virtue, and no vibrato restores it.

What straight-tone choirs are doing

Ensembles that sing just intonation in straight tone — some early-music consorts, some choirs specialising in tuned harmony — are often described as producing chords that ring, lock or bloom in a way vibrato singing does not. The binding these essays compute is one candidate for what that ring is: the flutter of a tuned chord heard as one sound. On the arithmetic here, a straight-tone chord is a worse candidate for binding than the same chord with a narrow vibrato, unless the singers hold their pitch steadier than a fraction of a cent over seconds.

That points to two possibilities, and the model cannot choose between them. One is that what such ensembles achieve is not this binding at all but something else — the absence of beating between mistuned partials, which a straight tone gives and a vibrato blurs, and which beats are arithmetic priced on its own terms. The other is that the singers do hold their pitch that steadily, perhaps by listening to the very flutter that tells them the chord is in step — a feedback the model does not have. If the second is true, the singers are not correcting pitch but timing, and that is a different skill from holding a note in tune.

The model under the wander

The chord and the tolerance are the earlier essays’. A close seventh tuned exactly to 4 : 5 : 6 : 7 over C2 or C4; the chord is in step while every upper note’s timing offset against the bass is within the register’s onset tolerance — 0.306 milliseconds over C2 and 0.070 over C4, re-measured at each bass as the register essay did.

Each note’s pitch wander is an Ornstein–Uhlenbeck process, independent across notes, with the stated root-mean-square size and a correlation time of one over two pi times the stated bandwidth, drawn from its own random stream so that the vibrato’s draws are the earlier essays’ exactly. A vibrato, where present, is the earlier essays’ with no lock.

Timing is the integral of pitch. Each note’s time is warped by two to the power of its deviation in cents over 1200, integrated sample by sample at 2 kHz; the offsets are the differences from the bass’s warped time.

What a model of a held chord cannot settle

How much real straight-tone singers wander. The figures sweep sizes from an eighth of a cent to two cents and bandwidths from half a hertz to eight. Where a given choir’s voices sit in that range is a measurement, and it decides everything.

Whether singers correct timing. The model’s singers correct pitch towards true and nothing else. A singer who heard the chord’s flutter slip and adjusted to restore it would be correcting the integral, and such a feedback could keep a chord in step indefinitely. Whether anyone does that, or could, is a question for listening and singing experiments.

Whether binding is what is heard as ring. The binding is a computed property of the auditory filters’ envelopes. That it corresponds to what listeners call a ringing chord is a hypothesis these essays have carried throughout, and it is no more tested here.

Still open: a singer who corrects the flutter

The model’s singers pull their pitch back towards true and let their timing go where it will. A singer listening to a tuned chord might instead hear its flutter slip out of step and nudge their pitch until it settles again — a correction aimed at the integral of their pitch rather than at the pitch. The computation is this held chord with each wandering note pulled not towards true pitch but towards zero timing offset against the bass, at a stated reaction time and gain, and it would say how quick and how strong that correction has to be to keep a straight-tone chord over C4 in step most of the time — and whether a correction a singer could plausibly make is enough, or whether a straight-tone chord bound by its flutter is something no voice can hold.

Part 26 of 26

One essay in the series on harmonic series. The essays either side of this one:

Over C4, width and disagreement trade one for one only while the lock is tight; loosely locked, a narrow vibrato stays in step far longer. 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, over eight draws, for four locks. If width and disagreement entered only as their product the four curves would coincide. 99.5%: 0.071 cents 100% (width 14.2), 0.142 cents 92% (width 28.4), 0.355 cents 62% (width 71), 0.71 cents 38% (width 142), 1.42 cents 20% (width 284), 2.84 cents 8% (width 568); 98%: 0.071 cents 100% (width 3.55), 0.142 cents 93% (width 7.1), 0.355 cents 61% (width 17.75), 0.71 cents 37% (width 35.5), 1.42 cents 19% (width 71), 2.84 cents 7% (width 142); 90%: 0.071 cents 100% (width 0.71), 0.142 cents 100% (width 1.42), 0.355 cents 72% (width 3.55), 0.71 cents 32% (width 7.1), 1.42 cents 15% (width 14.2), 2.84 cents 6% (width 28.4); 50%: 0.071 cents 100% (width 0.14), 0.142 cents 100% (width 0.28), 0.355 cents 100% (width 0.71), 0.71 cents 100% (width 1.42), 1.42 cents 65% (width 2.84), 2.84 cents 15% (width 5.68). A narrow vibrato needs no lock Part 25 — 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.

Where this goes next

Over C4, width and disagreement trade one for one only while the lock is tight; loosely locked, a narrow vibrato stays in step far longer. 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, over eight draws, for four locks. If width and disagreement entered only as their product the four curves would coincide. 99.5%: 0.071 cents 100% (width 14.2), 0.142 cents 92% (width 28.4), 0.355 cents 62% (width 71), 0.71 cents 38% (width 142), 1.42 cents 20% (width 284), 2.84 cents 8% (width 568); 98%: 0.071 cents 100% (width 3.55), 0.142 cents 93% (width 7.1), 0.355 cents 61% (width 17.75), 0.71 cents 37% (width 35.5), 1.42 cents 19% (width 71), 2.84 cents 7% (width 142); 90%: 0.071 cents 100% (width 0.71), 0.142 cents 100% (width 1.42), 0.355 cents 72% (width 3.55), 0.71 cents 32% (width 7.1), 1.42 cents 15% (width 14.2), 2.84 cents 6% (width 28.4); 50%: 0.071 cents 100% (width 0.14), 0.142 cents 100% (width 0.28), 0.355 cents 100% (width 0.71), 0.71 cents 100% (width 1.42), 1.42 cents 65% (width 2.84), 2.84 cents 15% (width 5.68). 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. A tuned chord struck in step drifts out of step as soon as it is mistuned. The tuned close seventh over C2, every note struck at the same instant and held, each note detuned by a draw with the stated spread in cents: the mean correlation between filters' envelopes, in half-second windows every quarter-second through eight seconds, averaged over eight draws. 0 cents: 0.61, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59; 0.1 cents: 0.61, 0.56, 0.50, 0.41, 0.36, 0.32, 0.28, 0.25 (halfway down by 4 s); 0.25 cents: 0.60, 0.43, 0.30, 0.23, 0.20, 0.18, 0.18, 0.17 (halfway down by 1.75 s); 0.5 cents: 0.57, 0.29, 0.19, 0.17, 0.17, 0.16, 0.18, 0.19 (halfway down by 1 s); 1 cent: 0.50, 0.18, 0.17, 0.18, 0.14, 0.18, 0.14, 0.17 (halfway down by 0.5 s), at whole seconds. The tempered chord's level is 0.16. Exactly in tune the chord stays in step; a tenth of a cent takes four seconds to lose half of it, a cent half a second. A tuned chord stays in step for half a cent-second A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates in every auditory filter at once, but only if its notes start within a millisecond of each other — and a tuned chord, it was said, keeps its phases for as long as it sounds. It keeps them only as long as it stays exactly tuned. Mistune its notes by a fraction of a cent and they slide against each other, and the chord drifts out of step in a time inversely proportional to the mistuning: about half a cent-second. A tenth of a cent holds it for four seconds, a cent for half a second. A chord that starts out of step never drifts into it. Locked 95 per cent, the vibratos carry the chord back into step 13 times in a second. The tuned close seventh over C2, exactly in tune, with the typical vibrato on every note — 71 cents at about six a second — the four vibratos 95 per cent locked to the bass's: how far ahead of or behind the bass each upper note's waveform runs, in milliseconds, through the fourth second, for one draw of rates and phases. The band is ±0.3 ms, within which the chord counts as in step. In this second the chord is in step for 180 ms in 13 stretches; over eight seconds and eight draws, 29 per cent of the time, in stretches averaging 22 ms after the first two seconds. One cycle of the chord's own flutter takes 61 ms. A chord comes back into step too briefly to bind A vibrato that is shared imperfectly does not carry a tuned chord's notes ever further apart: each note's time is warped back and forth, and the chord passes back through step twice in every swing. Locked ninety-seven or ninety-eight per cent, it does so about ten times a second. But each return lasts on average less than one cycle of the chord's own flutter, which is the rhythm the ear's filters would have to share to show the binding — fewer than one return a second lasts that long. The chord is in step often and never for long enough.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Auditory filterDominant seventhHarmonic seriesJust intonationPeriodicityVibrato