Acoustics & Music Theory
why it sounds like that
Concept

Auditory filter — where it appears

A band of the spectrum the ear analyses as one channel, with a rounded shape rather than a sharp edge and a width that grows with frequency. Its skirts decide what masks what and how much of a beat survives its neighbours, and the lower one shallows as the sound gets louder — so a loudly struck note is analysed through a wider filter than a quiet one.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

The channels a fluctuation is analysed into, and how wide they are. A modulation filterbank of quality factor 1, drawn every half octave, with two of the beats a mistuned octave at middle C produces marked at 0.89 and 1.26 a second. A channel of quality one has its half-power points at 0.618 and 1.618 of its centre, so two fluctuations closer than a factor of 1.618 never end up in different channels. The two marked rates differ by a factor of 1.42, which is inside that. They are one fluctuation and not two — which is the question an earlier essay asked and left open, answered by a bank rather than by a convention.

One fluctuation or two

Whether two members of a beat family are one thing or two was decided by asking whether their rates differ by a factor of two, and the factor was written down as a stand-in for a modulation filterbank nobody had run. Run, the bank gives 1.618 — the golden section, and not by accident, since a channel of quality one has its half-power points there. The stand-in was conservative rather than optimistic, and the recomputed counts do not change at a single register, because the criterion was never what was binding.

tuning · Beating A loudly struck note hides the beats it is being struck to reveal. The share of the fluctuation in its own auditory filter that belongs to each of the first five members of a mistuned octave's beat family, against how loud the note is. The filter's lower skirt shallows by about 38 per cent of its 51-decibel value every ten decibels, so more of the neighbouring partials get into the filter and the pedestal each member sits on grows. The mean share falls from 0.55 at 40 decibels to 0.07 at 100. Past about 90 decibels the curves are flat because the model's skirt is clamped rather than because anything stops changing — that clamp is the model's floor and not a measurement.

How hard the note was struck

The auditory filter is not a fixed shape: its lower skirt shallows by about 38 per cent of its reference value every ten decibels, so a loud note is analysed through a wider filter than a quiet one. Every share computed so far was quoted at a moderate level, and a tuner does not strike moderately. Recomputed, the mean share of a mistuned octave's filter falls from 0.55 at forty decibels to 0.07 at seventy, and the count of separable beats goes from two to none — which is a prediction too strong to be right, and the way it fails is the useful part.

tuning · Beating A struck octave becomes countable a second after the strike, or never. The number of separable beats a mistuned octave on A3 delivers, second by second after both notes are struck at 80 decibels, with every partial dying at its own rate (a 12-second fundamental, losses rising as frequency to the power 0.7). Read with the filter broadened by the level of the whole note, which is how the level-dependent count was first computed, the count is zero at every instant: the partials fall below audibility before the filter has narrowed enough to separate them. Read with the filter broadened by the level inside itself, which is what the published parameterisation was fitted against, the count is 0 at the strike, reaches 2 at 1.0 s and falls to nothing at 3.3 s.

Counted in the decay, or not at all

A mistuned octave struck hard delivers no countable beat at the strike, and the reconciliation offered for that was that a tuner listens to the decay. Computed through a real decay it fails on its own terms: the partials fall silent before the filter has narrowed enough to separate them. It succeeds only when the filter is broadened by the level inside it, which is what the published parameterisation was fitted against — and then the window opens at a twelfth of the note's life and shuts at a quarter.

tuning · Beating A string that decays twice opens its count at once and shuts it early. The count of separable beats a mistuned octave on A3 delivers after both notes are struck at 80 decibels, with each filter read at the level inside it, under one exponential decay of 12 seconds and under two stages — a prompt sound of 1.5 seconds carrying all but the last 20 decibels, and an aftersound of 12 seconds. One exponential: open from 1.00 s to 3.30 s, 11.8 beats. Two stages: open from 0.15 s to 1.77 s, 8.8 beats — and the single exponential struck 20 decibels softer closes at 1.77 s.

A string that decays twice is counted early

A mistuned octave's beats were found countable only between a twelfth and a quarter of a note's life, on a note decaying once. A piano string decays twice, a fast prompt sound over a slow aftersound, and the prediction was that this would open the count sooner and close it later. It opens sooner — at a seventh of a second rather than a second — and closes exactly where a single decay struck twenty decibels softer closes, so at 80 dB it holds 8.8 beats instead of 11.8. The count now rises with the strike to 90 dB, and a tuner who strikes hard is right.

tuning · Beating In the bass a bowed step keeps the entrance's displacement; in the treble every note is placed afresh. A listener whose ears differ by 2 dB, with 14 per cent of the brainstem latency difference reaching the delay comparator: how far the second note of a bowed legato pair at 60 dB is displaced, in detectable steps of 15 µs, against the interval it moves by, in three registers. A note with no onset of its own keeps the first note's position, drawn at the entrance's displacement; the dotted lines are each register's entrance from silence and the dashed its steady tone. G2, a double bass (entrance 6.16, steady 4.00): -12 6.19, -7 5.02, -4 4.35, -2 4.06, -1 no onset, +1 no onset, +2 no onset, +4 4.08, +7 4.26, +12 4.53. A4, a violin (entrance 3.73, steady 2.15): -12 3.72, -7 3.56, -4 2.82, -2 2.35, -1 2.19, +1 2.15, +2 2.22, +4 2.39, +7 2.67, +12 3.07. A6, a violin's top (entrance 2.13, steady 1.21): -12 2.13, -7 2.13, -4 1.77, -2 1.40, -1 1.26, +1 1.22, +2 1.29, +4 1.43, +7 1.66, +12 1.93. At the dial's up 2 semitones: G2 no onset, keeps 6.16; A4 2.22; A6 1.29.

A bowed bass line is placed by its entrance; a violin line by every note

A listener whose ears differ slightly places a note by its onset, and a bowed entrance is placed as though it were far quieter than it is. A legato line's later notes do not start from silence. Whether one of them has an onset at all depends on how much of the note before is already in its own auditory filter — and in the bass, where the filters are wide and their lower skirts shallow, a step of a tone or less has none, so the note keeps the entrance's displacement. In the treble nearly every note has its own onset and is placed near its steady position, except where the line leaps down.

perception · Localisation The tuned close chord flutters at one rate; the tempered one at several. The dominant seventh in close position over C2, tuned 4 : 5 : 6 : 7 and tempered: the modulation spectrum of the envelope in the auditory filter centred at 93 Hz, where the tuned chord's fluctuation stands furthest above the detection threshold, as a modulation index at each rate up to 40 Hz. The dashed curve is the published threshold for detecting amplitude modulation. Tuned: strongest at 16.4 Hz, the series fundamental 16.35 Hz, index 1.33, 42 times the threshold, 100 per cent of the modulation at that rate and its multiples. Tempered: strongest at 15.4 Hz, index 0.34, 45 per cent at one rate and its multiples.

A tuned close seventh chord low on the keyboard flutters at one rate

A dominant seventh tuned 4 : 5 : 6 : 7 in close position repeats exactly at a frequency two octaves below its bass, and below B2 that frequency is too slow to be a pitch. The question left was whether the repeat is heard as a flutter. In every auditory filter the chord reaches, its envelope pulses at that one rate, forty times over the threshold for detecting a fluctuation in the filter where it is strongest. The tempered chord fluctuates too, but a third as deeply and at rates that change from filter to filter. Tuning adds something here rather than sharpening it, and it adds it only within two cents of 7 : 4.

intervals · Harmonic series Read in its partials' filters, every step of a bowed bass line is an onset. The second note of a bowed legato pair on the double bass's G2 at 60 dB, for a listener whose ears differ by 2 dB: its displacement in detectable steps against the interval, placed by the filter at its fundamental alone (dashed, with pale dots where that filter sees no onset and the note keeps the entrance's position) and by the first of its eight partials' filters to see an onset (solid). By partials: -12 6.19 in partial 1, -7 5.02 in partial 1, -4 3.68 in partial 2, -2 3.12 in partial 3, -1 2.76 in partial 4, +1 2.61 in partial 5, +2 3.10 in partial 3, +4 3.56 in partial 2, +7 3.15 in partial 3, +12 3.51 in partial 2. The entrance, placed the same way, is 4.49 steps by its partial 4, against 6.16 by its fundamental. At the dial's +2: 3.10 steps, first in partial 3.

A bowed bass line has an onset at every step, in its upper partials

A legato note's onset was read in the auditory filter at its fundamental, and in the bass a step of a tone or less left that filter nearly as full as before: no onset, and the note kept its entrance's displacement. A bowed note is not a fundamental. Read in every filter its partials reach, every step of a bowed bass line has an onset, first seen in the third to fifth partial, where a tone moves a partial right out of its filter. The line is re-placed at every note after all — and nearer the centre, because a partial's filter is at a frequency where the ears' latency difference is smaller. The bowed bass entrance itself is displaced a quarter less than its fundamental said.

perception · Localisation One rate in every filter, and whether the filters keep time. The envelopes of four auditory filters — centred at 93, 170, 385, 822 hertz — over the first quarter-second of the dominant seventh in close position over C2, tuned 4 : 5 : 6 : 7. On the left every partial of every note starts together, as in a chord struck at one instant: all four filters pulse once every 61.2 milliseconds and their pulses line up. On the right each note is given its own starting phase, as four voices or four organ pipes have: every filter still pulses at the same rate, but each at its own moment in the period, and the lowest and highest filters' envelopes correlate at -0.34 rather than 0.56.

A tuned chord's flutter binds it only if its notes start together

A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates at one rate in every auditory filter, and one rate everywhere looked like the ear's evidence that a sound is one thing. It is only half of that evidence. The filters also have to rise and fall at the same moments, and whether they do depends on when each note starts — which tuning does not decide. Spread the notes' onsets by two milliseconds and the tuned chord's filters move together no more than a tempered chord's.

timbre · Harmonic series 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.

timbre · Harmonic series A vibrato short of perfect lock takes an exactly tuned chord out of step. The tuned close seventh over C2, exactly in tune and every note struck at the same instant, held for eight seconds: the mean correlation between filters' envelopes, in half-second windows, averaged over eight draws. Straight: 0.61, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59. With a 71-cent vibrato on every note, fully locked: 0.59, 0.57, 0.57, 0.57, 0.57, 0.57, 0.57, 0.57. Locked 98 per cent: 0.59, 0.55, 0.52, 0.47, 0.43, 0.39, 0.37, 0.33, halfway to the tempered chord's 0.16 by 6 s, at whole seconds. A straight tuned chord stays in step for as long as it is held; a vibrato can only keep it there by being locked completely.

A shared vibrato cannot hold a tuned chord together

A tuned close seventh chord binds its auditory filters together only while its notes stay within a third of a millisecond of each other, and mistuned by a fraction of a cent it drifts out of step in half a cent-second. A vibrato shared by every voice looked like a way to hold it: if all four notes swing together, they slide together. They do not stop sliding. A perfectly locked vibrato leaves the drift exactly as fast as before, because it changes every note's frequency by the same factor and leaves their differences alone; a vibrato locked ninety-eight per cent takes an exactly tuned chord out of step in six seconds, and one locked ninety per cent in one.

timbre · Harmonic series 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.

timbre · Harmonic series Locked 99 per cent, the higher the chord the more of its returns are long enough to bind. The tuned close seventh over C2, C3 and C4, exactly in tune, with the typical vibrato on every note, the four vibratos 99 per cent locked, each register judged against its own onset tolerance and its own flutter: how many times a second, after the first two seconds, the chord comes back into step, and how many of those returns last a whole cycle of its flutter, over eight draws. C2 (tolerance 0.306 ms, flutter 61.2 ms): 4.6 returns a second, 0.54 a flutter cycle long, in step 88 per cent of the time; C3 (tolerance 0.153 ms, flutter 30.6 ms): 9.7 returns a second, 3.83 a flutter cycle long, in step 65 per cent of the time; C4 (tolerance 0.070 ms, flutter 15.3 ms): 12.0 returns a second, 5.13 a flutter cycle long, in step 38 per cent of the time.

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.

timbre · Harmonic series 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.

timbre · Harmonic series A straight tone's slow wander drifts a chord's notes out of step and leaves them there. The 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.

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.

timbre · Harmonic series

Named alongside it

The objects these essays reach for when they reach for this one.

Dominant seventhHarmonic seriesJust intonationPeriodicityBeatingPartialVibratoMaskingCritical bandwidthDecayDynamicsHearing threshold

All concepts