Acoustics & Music Theory
why it sounds like that
Concept

Equal-loudness contour — where it appears

A curve through the sound pressure levels at which tones of different frequencies are judged to match one another in strength. The contours are not parallel, so turning a passage down removes more of the bass than of the middle.

Named by 8 essays across 4 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 The most a 6 cm cone can make of a low note. The maximum sound pressure level at one metre from a circular radiator of effective radius 3.2 cm, moving 1.5 mm at its limit, in a system resonating at 250 Hz. Below resonance the cone is already at that limit and the pressure a piston makes goes as the square of frequency, so the curve falls at twelve decibels an octave: 66 dB at 40 Hz, 78 dB at 80 Hz, 94 dB at 200 Hz. The 40 Hz figure is 28 decibels below the 200 Hz one, and that gap is arithmetic about a radius and a displacement rather than a property of any particular loudspeaker. The dots are the harmonics of a 41.2 Hz note with a one-over-n source spectrum; the loudest of them is the 6th.

The bass a small loudspeaker does not make

A three-inch cone at its excursion limit produces sixty-six decibels at forty hertz, which the ear converts to seventeen phons — barely above nothing. The note is heard anyway, because its harmonics are radiated and its fundamental is supplied by the listener. Computing what arrives turns the residue from a curiosity into a design decision, and finds that the fundamental of a low note is not the loudest part of it on any system a listener is likely to own.

instruments · Missing fundamental Roughness and loudness of one interval against level. A minor third on C4 evaluated at every level from 20 to 100 dB SPL per note, with both quantities drawn relative to their own value at 60 dB. Roughness is the Plomp–Levelt sum over the partials that are above ISO 226's threshold at that level; loudness is the same partials in sones. Between 50 and 95 dB the interval grows 3.2 × 10⁴ times rougher and 24.9 times louder, so roughness grows like loudness raised to the power 3.2.

The same chord is harsher when it is louder

Every roughness number so far was computed at a level nobody stated. Roughness is the product of two partial amplitudes, so it is quadratic in pressure, while loudness is compressive — which makes a minor third at middle C thirty-two thousand times rougher at fortissimo than at pianissimo and only twenty-five times louder. A chord has no single consonance to report.

intervals · Consonance A dynamic mark is an instruction about the spectrum. Six dynamic markings, given a hammer velocity each in a stated sequence of factors of two, with what the string then does. The level rises 35.1 decibels from pp to ff, which is the part everybody means. The contact time falls from 2.26 to 0.95 milliseconds, so the first null of the hammer's own pulse moves from partial 2.5 to partial 6.0 and the spectral centroid rises by 56 per cent. The partials between those two nulls are not quieter at pp; they are not there.

The mark that is not a level

There are six of them, they carry no units, and a performer has to turn one into a number before it means anything. What they instruct is not loudness. On a struck string a harder blow shortens the hammer's contact from 2.26 milliseconds to 0.95, which moves the first null of its own pulse from the third partial to the sixth: the partials between those are not quieter at pianissimo, they are gone. A fortissimo is a different sound, and the page has one word for both things it changes.

timbre · Notation 3 tones, one power, and the interval between them. 3 tones of fixed total power, spread symmetrically about 440 hertz, drawn against the interval between neighbours. Piled on one pitch they are one sound of that power; separated by more than a critical band — 4.5 semitones here — they are 3 sounds whose loudnesses add, and the same power reaches 2.08 times the loudness at 5 semitones. The two lines are two models of the same rule and they disagree about how abrupt the change is, not about where it goes.

A chord is not as loud as its notes

The first essay on loudness said what a tone's loudness is and recorded that it had said nothing about a chord's. Here is the missing rule, and it has a musical consequence nobody would predict from it: the same three notes, at the same power, are twice as loud in the treble as in the bass — because the critical band that makes a low triad five times rougher also makes it one sound instead of three.

perception · Loudness A 20-decibel crescendo is 27 phons on a bass note and 20 on a high one. The same change of level, from 60 to 80 decibels, converted to loudness at each register through ISO 226's equal-loudness contours rather than at one kilohertz. The heavy curve gives each note a string spectrum, so its partials are converted in their own bands and summed; the pale one is the fundamental alone. On the spectrum-aware curve the crescendo is worth 27.3 phons at C1 and 20.3 at C7. On the fundamental alone it is 77 at C1, which is not a finding but an artefact: a 60-decibel tone at 33 hertz sits 1.8 decibels above the threshold of hearing and is very nearly nothing. The honest correction is the smaller one, and it is still a difference of 7.1 phons across the compass for a mark written in the same ink.

A subito piano is four seconds longer in the bass

Every loudness figure with time in it converts level to loudness at one kilohertz, and the equal-loudness contours say that no other frequency works that way. Joining the two sorts the published numbers into those that were about the treble and those that were not. Three move a great deal — a twenty-decibel crescendo is worth 27 phons on a bass note and 20 on a high one, and the seven seconds a subito piano takes becomes eleven and a third. Three do not move at all, and the reason they do not is the same reason in every case.

perception · Loudness An ending built up part by part, counted two ways. A final chord built up one part at a time from middle C to eight parts, every part a string tone at 62 decibels, with the loudness of each stage as a loudness level: the level of a 1 kHz tone judged equally loud. Grouped into critical bands one after another, as the earlier essays counted it, the chord runs 76.2, 80.9, 79.4, 81.9, 76.4, 77.9, 74.2, 74.5 phons — a span of -1.8, and not monotone. Integrated as an excitation pattern it runs 73.2, 77.9, 79.5, 81.6, 82.5, 83.9, 84.5, 85.6 — a span of 12.4, rising at every step. The dashed line is the power sum, 9.0 decibels from one part to eight; spread across the compass, the parts are louder than their power.

A page has twelve phons, and its entries always said so

An ending built up one part at a time was found to move its loudness by under two decibels, and five essays have leaned on that number since. It came from a way of grouping partials into critical bands that sorts an eight-part chord's sixty-four partials into two bands. Counted along the ear instead, the same ending gains twelve phons at every step, which is what the collection's own price for a single entry always implied.

perception · Loudness From the page alone, a thinning ending is a diminuendo and a fuller final chord is barely louder. Short-term loudness through the closing bars of a thirty-two-bar scheme whose texture is 6 parts until the ending, computed from each bar's part count with no performance in it: the parts realised every way their ranges allow, each bar's partials summed as an excitation pattern, and the result passed through the two loudness smoothers. The dashed line is the running impression under the thinning ending. thins to one: final bar 73.2 phons, arriving at 0.690 of the running impression (-4.46 phons); full final chord: final bar 84.7 phons, arriving at 1.004 of the running impression (+0.05 phons); unchanged: final bar 83.3 phons, arriving at 1.000 of the running impression (+0.00 phons); thins then full: final bar 84.2 phons, arriving at 1.014 of the running impression (+0.17 phons).

The page can write a diminuendo and not a climax

Counted on the excitation pattern rather than in grouped bands, the four ending textures the closure essays compare come out the right way round at last: a thinning ending is a ten-phon diminuendo written into the score, and an ending that adds parts arrives louder than what came before rather than quieter. But the gain is tiny. Three more parts on a six-part final chord buy under a phon and arrive a twentieth of a phon above the listener's running impression; only a full chord straight after a thinning stands out, and only at a quick tempo. A page can take loudness away. It cannot make a climax.

perception · Loudness

Named alongside it

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

LoudnessSonePhonCritical bandwidthDecibelDynamicsOrchestrationMaskingSpectral balanceTextureBrightnessHearing threshold

All concepts