Acoustics & Music Theory
why it sounds like that
Series

Loudness — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · perception
  2. 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.

    part 2 · perception
  3. Crescendo, and what the impression does. A crescendo of 20 dB over 8 seconds, drawn as three loudnesses in sones. The pale line is what is physically sounding, the middle line is the short-term loudness of the moment and the heavy line is the long-term loudness, which is the passage's loudness as a listener would report it. The gap between the last two is the whole of the effect: at its widest the moment is 1.02 times the running impression, and the impression takes 2 seconds to come down against 99 milliseconds to go up.

    Loud is relative, and it comes down slowly

    The account of loudness had a model of a moment and the account of closure asked it for a model of a form. The published one exists and its content is a pair of numbers that are not the same: a listener's running impression of how loud the music is rises to meet a step in a fifth of a second and takes seven seconds to come back down. A twenty-decibel crescendo spread over eight seconds therefore buys almost no contrast at all, and the same twenty decibels taken as a step buys a factor of two.

    part 3 · form
  4. Adding parts adds power, and very little loudness. Each part is played at the same level, and the chord is realised every way its parts allow and averaged over them, so the quantity is a property of the texture rather than of one arrangement. Going from 3 parts to 8 adds 4.3 decibels of power and 0.1 decibels of loudness, because the extra parts land in bands that are already occupied — the count of occupied critical bands FALLS from 6.0 to 3.9 as the parts crowd into the same register.

    The dynamics are in the score already

    Count the parts in each bar, realise them in their ranges, put every partial in its critical band, sum the loudnesses and run the result through the two smoothers built earlier. What comes out is a dynamic curve for a piece with no performance in it anywhere — and it says that doubling the number of parts inside a fixed register adds three decibels of power and about one of loudness, because the extra parts land in bands that were already occupied. Let the register widen with the parts and the same arithmetic gives eight phon, which is what a tutti actually is.

    part 4 · form
  5. A page has two decibels and a player has sixty. Across, parts added to a final chord one at a time, each at the same level; up, the loudness that results, on a logarithmic scale. Going from one part to eight moves the total by 1.8 decibels and does not move it monotonically — four parts are louder than five and than eight. The faint line is what a naive power sum would give: 9.0 decibels. The band down the right is the same chord played by people, from forty to a hundred decibels, which spans 62. So a texture that thins from eight parts to one is not a diminuendo. It is a change of colour at constant loudness, and everything the closure figures call a dynamic belongs to the performance.

    A page has two decibels

    The account of closure called its dynamic component a corpus debt: it had no model of dynamics in a form. Loudness supplies one, and applying it answers the debt by refusing it. Adding parts to a final chord one at a time, each at the same level, moves the loudness by under two decibels and not monotonically — while a player has sixty. A texture that thins is not a diminuendo.

    part 5 · perception
  6. An entering part is worth 0.9 phons, in the middle of its range. A texture of 5 parts at 62 decibels each, with one more part added at the same level, tried at every semitone from C2 to C7. The vertical axis is what the addition is worth in phons, and a phon is a decibel here; the shaded strip is the difference limen for loudness, so an entry inside it is not heard as a change of level. The median entry is 0.89 phons and only 29 of 61 clear the limen — the lowest of them at A♭4, 415 hertz. The best available, at B♭6, is worth 4.2. The two lines are the two loudness models to hand: they agree everywhere above the tenor register and part company below it, where the greedy critical-band grouping reports 24 entries that make the texture quieter and the excitation pattern reports none.

    A part entering is not a change of level

    Five earlier essays measure a sonority that is already sounding. The one thing an orchestrator actually controls is the entry, and priced at every semitone it comes to 0.89 phons — under the difference limen — with only 29 of 61 available entries clearing it and none below A♭4. What an entry is instead is an object arriving, and the reason is that its own rise time is faster than the listener's.

    part 6 · perception
  7. Articulation is worth 2.6 phons, and nobody counts it. A passage of notes at 80 decibels, 2 to the beat, at 7 tempi and 3 articulations, scored against the same level held continuously. The variable is the fraction of each inter-onset interval that is sounding — 0.95 is a legato, 0.4 a staccato — and the vertical axis is what that costs the passage's running loudness in phons. Nothing here is anybody playing harder or softer. At 40 to the beat the span from legato to staccato is 1.47 phons; at 200 it is 2.61, because a staccato note there lasts 60 milliseconds and no longer reaches its own loudness either.

    A staccato is a dynamic mark

    Every loudness figure in this collection is of a sound that has been going on long enough, and no note in music has. Run the running-loudness model on notes with lengths in them and an articulation turns out to command 1.5 phons at a slow tempo and 8.1 at a fast one — more than the 1.8 decibels a whole texture commands, on the same page, written down in the same ink, and counted by nobody.

    part 7 · perception
  8. 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.

    part 8 · perception
  9. 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.

    part 9 · perception
  10. 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.

    part 10 · perception
  11. 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.

    part 11 · perception
  12. 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.

    part 12 · perception

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