Acoustics & Music Theory
why it sounds like that
Concept

Musical form — where it appears

The arrangement of a piece into sections that repeat, return and close. It is the level at which key plans, harmonic rhythm and endings are decided, and the one where a listener's memory is doing the most work.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

A listener has a quarter of an analyst's confidence and the same answer. The mean margin between a passage's best two key readings, against how many bars a listener's memory of the evidence takes to halve. The two-sided reading — the one that uses bars that have not happened yet — sits at 3.90; the forward pass with perfect recall at 1.79; a forward pass whose evidence halves every 6.6 bars at 1.00. The dots' size is how often that reading names the same key as the two-sided one: 100 per cent at perfect recall and 81 at a one-bar half-life. So forgetting costs a great deal of confidence and very little accuracy — the key is robust and the certainty is not.

The listener who forgets

Setting an analyst's reading of a key against a listener's measures what arrives late. Both passes assume perfect recall of their own half — which is as wrong going forward as knowing the future is going back. Put a decay on the forward pass and a listener with a memory of a few bars keeps a quarter of the confidence and nine tenths of the answers.

harmony · Key-relations 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 Which bars the key is decided by, and which bars it is believed on. Every bar of a 32-bar scheme removed in turn, with what its absence costs. The column is how far the passage's mean margin falls without that bar — how much of the model's certainty it supplies. The dot is how many bars are then read as a different key — how much of the answer it supplies. The 18 bars an analysis would point at — a section opening or closing, a dominant, the chord a dominant resolves to — average 0.079 bits of certainty and 0.50 bars moved; the 14 ordinary bars average 0.047 and 0.07. So the structural bars carry 1.7 times as much certainty as the ordinary ones, and 7.0 times of the answer. Those are different quantities, and the second is the one an analysis is about: an ordinary bar can carry a great deal of a passage's certainty and none of its reading.

The bars a key is made of

A discount that treats every bar alike is the wrong shape for a memory, so take each bar away in turn and see what it was worth. The bars an analysis points at carry seven times as much of the answer as the ordinary ones and slightly less of the certainty — and a memory built only of them reads the passage worse than a memory with no structure in it at all.

harmony · Key-relations Six named proportions, as blurred as the durations that make them. Six proportions between two parts of a piece — 1 : 1, 4 : 3, 3 : 2, golden section, 2 : 1, 3 : 1 — placed on one axis by the logarithm of the ratio of the longer part to the shorter, and drawn as bars one criterion wide (d′ = 1) for a listener timing both parts with a Weber fraction of 7%, 15%, 35%. Bars that overlap are proportions that listener cannot tell apart. At 7%, 4 of 5 neighbouring pairs stay apart; at 15%, 3 of 5 neighbouring pairs stay apart; at 35%, 0 of 5 neighbouring pairs stay apart.

A proportion is only as fine as its two durations

Analyses of form measure proportions in bars and report them to three figures — a climax at 0.618, a section in the ratio 3 : 2. A listener has each part only as an estimate of how long it lasted, and a ratio of two estimates is blurred by both. Timed as well as anyone times a single second, eleven proportions fit between 1 : 1 and 3 : 1; timed from memory over minutes, two do. The golden section is told from 3 : 2 only below a Weber fraction of 5.4 per cent.

form · Proportion The same forms by the clock and by what is stored. Six forms, each drawn twice: its sections sized by their share of the bars, and sized by their share of what a listener has to store when a bar counts only if it is recognised from 1 bar of context. Returns are drawn pale with a dashed edge. twelve-bar blues: returns take 67 per cent of the clock and 13 per cent of the storage; thirty-two-bar AABA: returns take 25 per cent of the clock and 5 per cent of the storage; rondo, ABACA: returns take 40 per cent of the clock and 10 per cent of the storage; verse and chorus: returns take 50 per cent of the clock and 13 per cent of the storage; two eight-bar phrases: returns take 0 per cent of the clock and 0 per cent of the storage; a four-bar ostinato: returns take 88 per cent of the clock and 0 per cent of the storage.

A return is shorter than its first hearing

A rondo's refrain takes three fifths of the clock and a verse-and-chorus song is balanced to the bar. Count instead the bars a listener could not have predicted when they arrived, and the returns shrink to between a tenth and a quarter of what is kept — so a song equal by the clock is between three and seven times heavier in its first half. A coder that learns repeats one bar at a time says the halves are equal, and the two memories disagree by more than any proportion a listener could confuse.

form · Proportion A count is least reliable at both ends and best in the middle. How precisely a listener knows the length of a section they are counting, against how many units long it is, at three kinds of timing judgement. A slip — one unit miscounted, at 2% a unit — accumulates as a random walk, so its relative cost FALLS as the section lengthens. A lapse — the count lost altogether, at 1% a unit — compounds, so the chance of still having the count falls geometrically and a long enough section is certain to lose it. A listener who has lost the count is back to timing, so the two failures mix into a floor. Against a Weber fraction of 7.5% the count is worth most at 21 units, where it is 1.7 times finer than timing, and falls back under a quarter better by 98 units; Against a Weber fraction of 15% the count is worth most at 10 units, where it is 2.4 times finer than timing, and falls back under a quarter better by 101 units; Against a Weber fraction of 38% the count is worth most at 4 units, where it is 3.7 times finer than timing, and falls back under a quarter better by 102 units. The length at which it stops being worth much is nearly the same in all three, because it is set by the lapse rate alone.

A count is not an estimate

Both established routes to a proportion are estimates — a duration timed, blurred by a Weber fraction, and a duration stored, biased by what was new. A listener who has induced a hypermetre has a third, and it is exact until it fails. It fails two ways that pull opposite: a slip miscounts one unit and its relative cost falls as the section lengthens, while a lapse loses the count entirely and its chance compounds. The mixture has a floor at about four units, where counting is 3.7 times finer than timing, and it is worth almost nothing past a hundred.

form · Proportion Timing blurs a whole form evenly; counting sharpens it downward. A piece of 480 seconds divided 7 times, each level half the length of the one above, with how many proportions between 1 : 1 and 3 : 1 a listener can tell apart at each. Timed, the answer is 2.1 at the top and 5.2 at the bottom, a spread of 2.4 — because a timing judgement's Weber fraction is a step function of duration and almost every level of a piece falls in one step of it. Counted, in units of 2 seconds, the answer runs 2.2 to 10.3, a spread of 5.5. At no level does timing separate 3 : 2 from the golden section.

A form is sharp at the bottom and vague at the top

A movement is divided into sections, each into phrases, each into bars, and every level is a ratio of two estimates. Timed, the hierarchy is almost uniformly blunt — 2.1 distinguishable proportions at the top and 5.2 at the bottom, because a Weber fraction is a step function of duration and six of a piece's seven levels fall in one step of it. Counted, the same hierarchy runs from 2.2 to 12.2 and sharpens monotonically downward. At no level of either does timing separate 3 : 2 from the golden section.

form · Proportion The golden section and an equal division are one judgement. Where a boundary falls in a piece, as a share of its length, with the band a listener cannot tell from the golden section shaded. A stretch of minutes is judged with a Weber fraction of about 38%, so one criterion's worth of ratio spread around 0.618 covers everything between 0.492 and 0.730 — a quarter of the piece wide, and containing the halfway point. 1 : 1 and 4 : 3 and 3 : 2 and golden section and 2 : 1 are inside it. A claim that a climax falls at the golden section rather than at the middle is, at this resolution, not a claim about anything a listener could hear.

A golden section is a coin toss with six coins

An analysis that reports a climax at 0.618 of a piece has not tested one prediction; it has looked at a piece with several defensible boundaries and reported whichever landed nearest. The rate at which that happens under no hypothesis is one line of arithmetic, and the tolerance it needs is not a number chosen on the page — it is the blur a listener's own timing puts on the judgement. Over a stretch of minutes that blur covers everything from 0.492 to 0.730 of the piece, which contains the halfway point, and six candidate boundaries produce a hit eighty per cent of the time.

form · Proportion The breath is the looser ceiling nearly everywhere. How long a trained singer can hold a phrase on one breath, across a compass and at four dynamics, against the 8-second ceiling the psychological present puts on the same phrase. The flow through the folds rises with pitch and with loudness, so the breath ceiling falls both ways: at 60 decibels it runs 32.6 seconds at the bottom of the compass to 21.2 at the top; at 70 decibels it runs 23.1 seconds at the bottom of the compass to 15.0 at the top; at 80 decibels it runs 16.4 seconds at the bottom of the compass to 10.6 at the top; at 90 decibels it runs 11.6 seconds at the bottom of the compass to 7.5 at the top. The shaded line is the listener's ceiling and it does not move. The breath binds only where the two lines cross — 1 of the 40 cells drawn, all of them loud and high. So the constraint everybody names when asked why a phrase is the length it is, is almost never the constraint that decides it.

The ceiling everybody names is the loose one

Ask why phrases are the length they are and the answer given is the breath. It is arithmetic — usable lung volume over the air a note costs per second — and it comes out between fifteen and twenty-three seconds at a comfortable dynamic and between seven and twelve at a loud one. The ceiling the present moment imposes, the two-to-eight seconds inside which a stretch is heard as one thing rather than as a series, is two to three times tighter at almost every note and dynamic. A singer in an adagio is not running out of breath at the phrase end. They are running out of present.

form · Phrase Checkpoints sharpen the middle of a form and leave its top vague. A piece of 480 seconds divided 7 times, with how many proportions between 1 : 1 and 3 : 1 a listener tells apart at each level: timed, counted in 2-second units, and counted with a second count of 16-second phrases that can mend a lapse in the first. 480 s: 2.1 timed, 2.2 counted, 3.9 with 0 per cent of lapses shared and 2.5 with 50 per cent of lapses shared; 240 s: 2.1 timed, 2.5 counted, 7.1 with 0 per cent of lapses shared and 3.1 with 50 per cent of lapses shared; 120 s: 2.1 timed, 3.1 counted, 13.2 with 0 per cent of lapses shared and 4.0 with 50 per cent of lapses shared; 60 s: 2.1 timed, 4.1 counted, 19.4 with 0 per cent of lapses shared and 5.4 with 50 per cent of lapses shared; 30 s: 2.1 timed, 5.4 counted, 18.1 with 0 per cent of lapses shared and 7.2 with 50 per cent of lapses shared; 15 s: 5.2 timed, 12.2 counted, 12.2 with 0 per cent of lapses shared and 12.2 with 50 per cent of lapses shared; 7.5 s: 5.2 timed, 10.3 counted, 10.3 with 0 per cent of lapses shared and 10.3 with 50 per cent of lapses shared. With the two counts failing independently, the level of 60 seconds goes from 4.1 to 19.4, and the whole piece only from 2.2 to 3.9.

Checkpoints sharpen the middle of a form, not its top

A listener who counts bars loses the count somewhere in a long section and is thrown back on timing the whole of it. A listener who also counts phrases can mend the lapse at the last phrase. If the two counts fail independently, the level a minute long goes from four distinguishable proportions to nineteen; the whole eight-minute piece goes only from two to four, because thirty phrases are long enough to lose a count as well. And if a fifth of lapses take both counts at once, three quarters of the gain is gone.

form · Proportion

Named alongside it

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

DurationProportionMemory decayPhraseWeber fractionHypermetrePerceptual presentCadenceDynamicsHidden markov modelInferenceKey-finding

All concepts