Acoustics & Music Theory
why it sounds like that
Generator

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.
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.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 65 of its 65 placements carry sound, built from the same numbers as the drawing, offering these buttons: 12 tones sharing it, 3 tones 5 semitones apart, same power, 3 tones on one pitch, 4 tones 13 semitones apart, same power, 4 tones on one pitch, 60 then 80 dB at C4, 60 then 80 dB at C6, 90 players — 20 dB louder and 33 more.

Called by 17 essays

the blast radius of changing it

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 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.

form · Loudness 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.

form · Loudness Four endings, and the loudness each produces from the page alone. Short-term loudness through the closing 6 bars of a thirty-two bar scheme, computed from the part count of each bar with no performance data of any kind — the parts are realised every way their ranges allow, every partial is placed in its critical band, and the sum is run through the two loudness smoothers. thins to one arrives at 0.764 of the running impression; full final chord arrives at 0.952 of the running impression; unchanged arrives at 1.000 of the running impression; thins then full arrives at 0.929 of the running impression. The result worth the figure is that full final chord is not the loudest: adding parts to a final chord adds power and almost no loudness, because the extra parts land in critical bands the chord already occupies. An ending is made loud by contrast with what preceded it, not by thickness.

A final chord is not made loud by adding to it

An earlier essay on closure said the loudest cue an ending has needs a corpus rather than an arithmetic. The arithmetic was built one essay ago, so it does not. Run four ending textures through it and two things come out backwards: a final chord three parts thicker than the rest arrives *quieter* against the running impression than the passage it ends, and a texture that drops a part a bar does not get quieter at all until the bar where there is one part left.

form · Closure Roughness and loudness do not rise together. One four-note chord, played at levels from 35 to 95 decibels, with both quantities drawn as multiples of what they are at the quietest. Roughness is quadratic in pressure, so 60 decibels multiply it by 1.0e+6. Loudness is compressive — about ten phons to a doubling of sones — so the same range multiplies it by 96. The gap between the two lines is the quantity: roughness per sone rises by a factor of 1.0e+4 between a pianissimo and a fortissimo of the same chord.

The ranking survives the dynamic and the chord does not

Roughness is quadratic in pressure and loudness is compressive, so sixty decibels multiply a chord's roughness by a million and its loudness by ninety-six. Roughness per sone therefore rises ten thousandfold between a pianissimo and a fortissimo of the same four notes — and yet the ranking of which doubling is smoothest, over four hundred and eighty voicings, does not move by a single place.

harmony · Orchestration The tempo turns, and almost nothing moves. The earlier arrival reading — what is sounding at the final chord over what the listener has been hearing — swept over bar lengths from 0.5 to 5 seconds, which is 480 down to 48 beats a minute, at 3 closing lengths. Every curve is nearly flat. Across a tenfold change of tempo one gesture's reading moves by a factor of 1.201 and the other's by 1.098, while the gap between the two gestures — which is what that essay was measuring — is 1.228. The expectation was that the tempo would decide the answer, on the grounds that a two-second bar against a two-second release is a comparable pair. The premise is wrong in a way the sweep makes obvious: the thing being compared with the release is not a bar, it is the WHOLE ENDING, which is 2 to 8 bars long and is therefore far longer than the release at every tempo anybody plays. The running impression has caught up with the closing texture before the final chord arrives, at 0.5 seconds a bar and at 5, and what is left is the last bar's own jump.

The parameter that did not decide the answer

An earlier essay on closure ended by naming the tempo as the thing every number in it was resting on, and said it was the kind of parameter that had caused trouble before by turning out to decide the answer. Turned across a tenfold range at a closing gesture of fixed length it moves the reading by four per cent, against a twenty-three per cent gap between the gestures it is distinguishing. The parameter beside it in the same figure — how many bars the gesture occupies — moves it by twenty, and nobody had named that one at all.

form · Closure A written dynamic is an instruction to the listener's impression. Every earlier scoring holds one chord still. A passage is a succession, and the running impression of loudness carries a chord into the one after it, so what a marking asks for and what playing the marking produces are different things. Here is a five-chord passage with a written shape. Playing each chord at its own written loudness gives the running impression 2.4, 3.0, 4.2, 5.6, 4.0 sones against the 2.4, 3.0, 4.2, 5.6, 2.0 that were asked for — right until the last chord, where it misses by 2.0. Solving for levels that make the impression arrive at the marking does not fix it: the last chord's target is I, two parts, and it is unreachable — the correction runs to silence and the impression still sits 1.1 sones above. A subito piano after a full chord is not a level a player can produce. It is a rate of change, and the smoother's two-second release is what refuses it.

A subito piano is a rate, not a level

All three earlier essays score one chord held still. An orchestration is a succession, and the running impression carries a chord into the one after it — so a written dynamic is an instruction to the listener's impression rather than to the instantaneous sound, and there are markings that cannot be produced at all. The correction runs to silence and the impression still sits above the target.

form · Orchestration One contrast survives every tempo anybody plays and the other does not. How much of each quantity's contrast between chords a listener still has at the end of each chord, against how long a chord lasts. The roughness curve is flat at one down to 45 milliseconds a chord and then falls off a cliff, because its window is 37 milliseconds and a boxcar either fits inside a chord or does not. The loudness curve is already losing at a second a chord and keeps 83 per cent at the slowest pace here, 39 at the fastest. Nothing in music is faster than the roughness window and a great deal of music is faster than the loudness one, so a passage delivers its dissonance and averages its dynamics.

The dissonance arrives and the dynamic does not

A scoring decides two things at once and both of them have to be integrated by a listener before they exist. The loudness smoother's release is two seconds and the roughness window is thirty-seven milliseconds, and that ratio of fifty decides which of the two survives at the pace music is actually played. Nothing anybody performs is fast enough to blur a dissonance, and a great deal of it is fast enough to average a dynamic.

form · Orchestration 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.

perception · Loudness A rest is a diminuendo, and a long one. How far a listener's running impression of loudness falls during a silence, converted into the diminuendo that would have taken it the same distance. Half a second of nothing is worth 3.2 decibels, a second and a bit is worth 8.1, and two and a half seconds is worth 17. The marked line is three and a half seconds, which is where a gap starts to be heard as an ending rather than as a pause: at that length the reference has fallen by 24 decibels, which is more than a fortissimo to a pianissimo. A tempo swept over a factor of ten, a deceleration over a factor of three and a gesture length over a factor of forty-eight all returned the same reading to four significant figures. This one moves it by twenty-four decibels.

A rest is a diminuendo

Three parameters swept over factors of ten, three and forty-eight returned the same reading to four significant figures. The one manipulation left unswept moves it by twenty-four decibels: a silence. A listener's running impression decays at the loudness smoother's two-second release, so a general pause is a diminuendo nobody wrote, and at the length that makes a gap an ending it is worth more than any marking a composer has.

form · Closure 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.

perception · Loudness 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.

perception · Loudness The top voice arrives whole and the bottom one arrives as a sine. 4 parts sounding together, each of 8 partials, with every partial tested against the summed masked threshold of every component in the texture. A filled mark is a partial the listener receives and an open one is a partial the part would have had alone and does not have here. The bass at C3 keeps 1 of 8, the tenor at C4 keeps 2 of 8, the alto at E4 keeps 5 of 8, the soprano at G5 keeps 8 of 8, every one of them at 70 decibels. Every part is at the same level and the difference is entirely where each one sits: masking spreads upward, so the part at the top of the texture has nothing above it to be masked by and the part at the bottom has everything.

The listener is given the top voice, and the bass as a sine

Four earlier essays put the masker and the probe in the same voice. Put them in different voices — a four-part texture at one level — and the soprano arrives with all eight of its partials, the alto with five, the tenor with two and the bass with one. Balancing the loudness, which is the constraint a scoring is solved under, changes none of that: equal loudness is not equal spectrum and cannot be made so.

perception · Masking The roughest chord on the page is at C1 and the roughest one heard is at A♭2. One close triad at 70 decibels through 17 registers, with its roughness computed twice: over every partial in the score, and over only the partials that stand above what the chord itself masks. The written curve rises all the way down and its maximum is the lowest register drawn, C1, which is the low-interval rule as it has always been computed here. The delivered curve turns over at A♭2 and falls to nothing below A♭1: a close triad down there is not rough, because it is not arriving as a chord — 1 of its 24 partials survives at C1 and there is almost nothing left to beat against anything.

A low chord stops being rough by stopping being a chord

Every count of audible partials until now is of a chord at middle C, and the three registers it did compare span C3 to C5 — a third of the range a chord is written in. Move the same triad down and the count collapses: 79 per cent of its partials arrive at E3 and 4 per cent at C1. So the roughest chord on the page is the lowest one and the roughest chord a listener receives is at G2, and where that maximum sits moves nearly two octaves with the dynamic.

perception · Masking 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 A bar of silence is worth 13.1 decibels written last and 6.0 written first. Four closing gestures, drawn against how many seconds of silence each contains. All four hold the same 12-part texture, write the same 6.0-decibel diminuendo over 4 seconds, and differ only in what order the diminuendo and the silence are written in. The quantity is how far the listener's running impression has fallen when the final chord arrives, in decibels of equivalent diminuendo. Written with the silence last, a general pause of a bar is worth 13.06 decibels and one of three and a half seconds is worth 28.3. Written with the silence first, both are worth 6.00 — exactly the diminuendo's own depth, because the music resuming after the silence puts the reference back at its own level. The silence written first is worth less than the silence written with no diminuendo at all, which reads 8.12 at a bar: a diminuendo placed after a general pause takes 2.12 decibels away and adds nothing.

A general pause is spent by the note after it

Whether a composer should write the pause before the diminuendo or after it looks like a question about how big the ensemble is. It is not. Forty decibels of ensemble are worth one decibel of silence, and the order is worth seven — because a running impression rises twenty times faster than it falls, so half a general pause is spent by ninety-seven milliseconds of sound.

form · Closure An entrance stops being a loudness event and never stops being a colour one. The same oboe entering on the same note at the same level, against how many players were already sounding. Its contribution to the loudness falls from 15.5 phons to 0.32 — a factor of 48 — and crosses the one-phon difference limen at 5 players already playing. Its contribution to the roughness rises by a factor of 12.3 over the same range, because roughness is a sum over pairs and the entrant makes one new pair with everybody. Both curves are drawn as a share of their own largest value, since a phon and a squared pascal have no exchange rate. The claim is the two directions, not the crossing point of two units.

An entrance is a change of colour

Eight essays on orchestration move the assignment and hold the ensemble still, and a score does the opposite: it brings players in and takes them out. Loudness is a sum over parts and roughness is a sum over pairs, so the player who joins adds one term to the first and one to the second for everybody already there. What the entrance is worth in phons falls by a factor of forty-eight across the range an ensemble spans and crosses the difference limen at five players; what it is worth in roughness rises by twelve, and by a further factor of ten for every ten decibels the passage is played at.

form · Orchestration

All figures · What can be heard