Acoustics & Music Theory
why it sounds like that
Concept

Hearing threshold — where it appears

The quietest level at which a tone of a stated frequency can be detected, which varies by tens of decibels across the range. It is lowest around three kilohertz, which is where the ear canal's own quarter-wave resonance sits.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

Every point on one curve sounds equally loud. The equal-loudness contours of ISO 226:2003, evaluated from the standard's own parameters. The lowest curve is the threshold of hearing. Because the curves are not parallel — they crowd together in the bass and spread apart in the middle — the same change in decibels is a different change in loudness at every frequency, and a spectrum that was balanced at one level is not balanced at another.

The quietest thing audible, and why the volume knob is a tone control

A decibel is a fact about air. A phon is a fact about a listener, and the two do not line up — the map between them bends with frequency, and it bends differently at every level. One consequence is that turning a piece of music down does not turn all of it down equally, and the amount by which it does not is a number.

perception · Loudness Long-term average spectra: an orchestra, playing forte against a trained operatic soloist. Each source's mean spectrum over a long passage, in decibels below its own strongest region, on a logarithmic frequency axis. An orchestra, playing forte peaks at 250 Hz and is 30 dB down by 3,150 Hz; a trained operatic soloist peaks at 250 Hz and is 11 dB down by 3,150 Hz. The shapes are the same until about 1 kHz and separate above it: at 3153 Hz the difference is 19.0 decibels, which is the largest anywhere in the range. Nothing here is about level. Both curves are drawn against their own peaks, so what is being compared is shape.

One voice over ninety players

A soloist heard over a full orchestra is not louder than it and could not be. What the trained voice does instead is put a peak of energy at three kilohertz, which is where the orchestra's spectrum has already fallen away and where the ear's own threshold happens to be lowest. Nineteen decibels of advantage, in a place nobody is competing for.

timbre · The voice The top of a struck note's series falls while the note lasts. The highest partial still above the threshold of hearing, against time, for a note struck at 80 decibels on a fundamental of 130.8 hertz with a 1/n spectrum and a loss rising as the partial number to the power 1. It starts at partial 152 and it is falling from the first millisecond. The horizontal lines are the three tops computed from frequency alone, all of which assume a note that never ends: the note's own top drops past the difference-limen top after 0.03 seconds, past the semitone top after 0.32, and past the resolvable top after 0.64. After that the series is shorter than the ear could have resolved, and what stops it is the clock.

The top that falls while the note lasts

Four earlier essays have asked where the harmonic series stops, and all four answered with a number computed for a tone that never ends. A struck C3 has 152 audible partials at the strike and eight after two-thirds of a second, so the ear's own resolution limit governs the first ten per cent of the note and the decay governs the rest. Playing ten decibels louder buys the ear a further tenth of a second, and doubling its reign would take fifty-eight.

timbre · Harmonic series Both notes of a fifth, and the coincidences going out. The highest partial still above the threshold of hearing for each note of a fifth struck at 80 decibels on 130.8 hertz, against time, with a 1/n spectrum and a loss rising as the partial number to the power 1. Both curves come down from off the top of the frame — the lower note starts with 152 partials and the upper with 102, because twenty kilohertz is a ceiling in frequency and not in partial number. The rings are the interval's partial coincidences at the moment they stop existing — successive multiples of one ratio, which go out from the top down: 12:8 at 0.47 s, then 9:6 at 0.64 s, then 6:4 at 1.04 s, then 3:2 at 2.14 s. The lowest, 3:2, is the last, and after it the two notes have no partial in common that either of them can still supply.

A fifth on a piano is not a fifth a second later

Nine essays draw one note in silence, and a listener is given a texture. Put two struck notes an interval apart and consonance comes apart into two quantities that had agreed while nothing moved: the partial coincidence that names the interval outlives the strike in exactly the order common-practice theory ranks its intervals, and the roughness that scores it reorders itself inside a third of a second, with the fifth overtaken by four intervals every treatise calls harsher.

timbre · Harmonic series The thirds' difference-tone line, note by note, against the dynamic. Each note of the line the difference tone f₂ − f₁ draws under a scale in just thirds, as its level above the higher of the threshold of hearing and the primaries' masking, against the level of the primaries. The product sits 50 dB below the primaries at 60 dB and grows twice as fast as they do. C2 above the limit from 73.5 dB; A1 above the limit from 76 dB; C2 above the limit from 73.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; C3 above the limit from 66 dB.

A combination-tone bass needs a forte

A scale in just thirds draws a diatonic bass line through its difference tones, and in sixths the cubic product draws one. Given the two published level laws, with their constants swept, the thirds' bass is not heard at all below primaries of about 66 dB and is heard whole only from 71 to 81. The cubic products are a different kind of object: the primaries mask them decibel for decibel as they rise, so no dynamic changes whether they are heard. Most of the thirds' inner line never is, and the sixths' bass needs a forte and a gentle law.

intervals · Combination tone Every product of every pair of partials is a harmonic of one absent note. A 4 : 5 interval on 261.6 and 327.0 hertz, just, each note carrying 6 partials at one over n, with the fundamental at 80 decibels. Every pair of partials makes its own difference tone, and there are 19 distinct frequencies among them. All of them are exact multiples of 65.41 hertz — the note neither instrument is playing — because partial j of a p·f₀ note and partial k of a q·f₀ note differ by (kq − jp)·f₀ whatever j and k are. The heavy line is what each product has to clear — the threshold of hearing at its own frequency, or the masked threshold the two primaries cast there, whichever is higher. 11 of the 19 get through.

Three harmonics of the bass arrive before the bass

Every product priced until now was between two pure tones, and nothing that plays thirds is pure. Give each note a spectrum and the ear receives every pair of partials — and for a just interval p:q every one of their products is an exact multiple of the same absent fundamental. That crowd lands where the threshold of hearing is tens of decibels cheaper, so it names the bass at 71 decibels where the component at the bass's own frequency needs 74, and at 75 against 85 an octave lower. Tempered, the crowd still forms and names a note seventy cents flat.

intervals · Combination tone The ghost bass drops when the passage gets louder. The note the whole crowd of products names, as a multiple of the fundamental the interval implies, against how loudly the interval is played. a major third: 2.9999999999999996 times the fundamental below 70 decibels and 1 times above it, a drop of 19 semitones; a minor third: 4 times the fundamental below 62 decibels and 2 times above it, a drop of 12 semitones; a fourth: 2 times the fundamental below 72 decibels and 1 times above it, a drop of 12 semitones. Softly, only the cubic products clear their thresholds, and they are an exact series on (2p − q) times the fundamental with no gaps in it. Loudly, the difference tones fill in the low harmonics, no template on the higher note can explain them, and the fit falls. Nothing about the interval has changed; the listener is simply being given a different subset of the same harmonic series.

The ghost bass drops a twelfth at a forte

Both crowds arrive at once and every member of both is a multiple of the same absent fundamental, so a listener is never given a choice between them — only a different subset of one harmonic series at every dynamic. Softly, the subset is an exact gapless series on three times the fundamental. Loudly, the difference tones fill in the low harmonics and no template on the higher note survives them. Between 62 and 72 decibels, depending on the interval, the note the crowd names falls by an octave or a twelfth, and the two qualities of third cross at different levels.

intervals · Combination tone A listener whose ears differ by a decibel or two has a straight ahead that moves with the dynamic. Where the median plane sits for a listener whose two ears' thresholds differ, against the level of a 500 Hz tone, if the whole of the latency difference the published brainstem latency law gives reaches the comparator that reads interaural delay. 1 dB worse: 19.5° at 30 dB (172 µs), 12.0° at 60 dB (106 µs), 7.4° at 90 dB (65 µs); 2 dB worse: 40.6° at 30 dB (347 µs), 24.4° at 60 dB (214 µs), 14.9° at 90 dB (132 µs); 3 dB worse: 65.6° at 30 dB (524 µs), 37.7° at 60 dB (323 µs), 22.7° at 90 dB (200 µs). Every curve halves its offset every 43 dB and none reaches zero at any level a listener hears; the dashed line is one detectable step on the median plane, 1.68°.

A quiet bass is further off-centre than a loud violin

An ear whose threshold is a decibel worse than the other is a little later to report every sound, and the published brainstem latency law says how much later: a gap that halves every 43 decibels and grows toward the bass. Taken at full size, a one-decibel asymmetry would put straight ahead twelve degrees round at a moderate level — seven times what the ear's own two thresholds say a decibel is worth, so most of that latency cannot be reaching the comparator that reads direction. What survives any scaling is the shape. The displacement is not one number but a curve in level and in pitch, and it deforms an orchestra rather than moving it.

perception · Localisation A slow attack is heard where a much quieter note would be. How far a listener whose ears differ by 2 dB hears a 98 Hz note displaced toward the better ear, in detectable steps of 15 µs, against its dynamic, if 14 per cent of the brainstem latency law's difference reaches the delay comparator and the note's position is set by its onset, weighted over 3 ms from the moment it becomes audible. The dashed line is the note treated as a steady tone. struck (2 ms rise): 40 dB 5.73, 50 dB 5.00, 60 dB 4.32, 70 dB 3.70, 80 dB 3.16, 90 dB 2.69. plucked (5 ms rise): 40 dB 5.95, 50 dB 5.32, 60 dB 4.66, 70 dB 4.02, 80 dB 3.45, 90 dB 2.94. tongued (25 ms rise): 40 dB 6.46, 50 dB 6.07, 60 dB 5.52, 70 dB 4.87, 80 dB 4.23, 90 dB 3.63. bowed (100 ms rise): 40 dB 6.69, 50 dB 6.51, 60 dB 6.16, 70 dB 5.64, 80 dB 5.01, 90 dB 4.36. a soft bowed entrance (250 ms rise): 40 dB 6.76, 50 dB 6.67, 60 dB 6.46, 70 dB 6.07, 80 dB 5.52, 90 dB 4.87. Steady: 5.52, 4.70, 4.00, 3.40, 2.90, 2.47. At 90 dB a 250 ms entrance is displaced 1.98 times as far as the steady tone.

A bowed note is placed as if it were quieter

An ear a decibel or two worse than the other is later to report every sound, by an amount that grows as the sound gets quieter, and a listener takes a note's direction from its onset. A plucked or struck note reaches its dynamic before the onset is over and is placed almost where a steady tone would be. A bowed note is still near the threshold of hearing while its onset is being read, so it is placed as though it were 27 decibels quieter than it is. The prediction was that slow attacks would soften the displacement; they sharpen it, and an asymmetric listener's orchestra is deformed by its articulation as much as by its register.

perception · Localisation 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 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

Named alongside it

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

DynamicsMaskingHarmonic seriesInteraural time differenceLocalisationCombination toneDifference toneJust intonationAuditory filterCalibrationCounterpointCritical bandwidth

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