Acoustics & Music Theory
why it sounds like that
Concept

Phon — where it appears

A unit of loudness level: a sound of N phons is as loud as an N-decibel tone at one kilohertz. It exists because the decibel is a fact about air and loudness is a fact about a listener, and the map between them bends with frequency and bends differently at every level.

Named by 7 essays across one field — 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 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 A hall fills the gaps a thinning ending leaves only slowly, so the written diminuendo reaches the seats nearly whole. Short-term loudness through the closing bars of a thirty-two-bar scheme of six parts, computed from each bar's part count on the excitation pattern, with a hall of 2 seconds' reverberation under it, its reverberant field in power six decibels over the direct sound in the steady state; the dashed line is the thinning ending with no room. thins to one: arriving -3.83 phons against the running impression; full final chord: arriving +0.13 phons against the running impression; unchanged: arriving +0.00 phons against the running impression; thins then full: arriving +0.43 phons against the running impression. With no room the thinning arrives at -4.46.

A hall leaves a thinning ending most of its diminuendo

A written diminuendo made by dropping parts arrives about four and a half phons under the listener's impression in a dry room. A hall's tail sounds under every bar and holds the level up as parts leave, and the obvious worry is that it fills the gaps a thinning leaves, as it fills a rest. It barely does. A concert hall leaves the thinning 86 to 100 per cent of its contrast and a stone church about three quarters, where the same rooms leave a two-second general pause three quarters and a half. A thinning steps down a decibel or two a bar, and a hall's tail falls nine decibels in the time a listener takes to register the arrival; only a cathedral is slow enough to blunt the last step, and it takes half.

perception · Loudness Three orders of leaving, one passage: the part left sounding sets the drop, and the room lifts it behind them all. Short-term loudness on the excitation pattern of six string parts, G3 C4 E4 G4 C5 E5 at 62 dB each, dropping one a bar in three orders, with a hall of 2 s in the middle bands, longer in the bass and shorter in the treble. top down: -9.39 phons from the steady texture to the last bar; bass up: -7.40 phons from the steady texture to the last bar; inner first: -8.05 phons from the steady texture to the last bar.

A hall forgets almost everything but the last part to leave

A hall's bass outlasts its treble, so a thinning ending ought to care which parts it drops first: top down should leave the tail's darkness to fill the gap, and bass up should remove what the tail holds longest. Searched over all 720 orders, it barely does. Give the tail a spectrum and the drop moves by under half a phon, in the direction that helps the order expected to suffer. What sets the ending is two notes: the one left sounding, and the one that went just before it. Everything earlier has gone.

perception · Loudness

Named alongside it

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

LoudnessDynamicsEqual-loudness contourSoneTextureCritical bandwidthOrchestrationDecibelReverberationHearing thresholdIntegration windowMasking

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