Acoustics & Music Theory
why it sounds like that
Concept

Closure — where it appears

The property of an arrival that settles what came before it, supplied by several independent signals rather than by one quantity. Harmony, rhythm, register, dynamics and silence each contribute, and a weak ending is one where they disagree.

Named by 19 essays across 3 fields — each of them below, with the objects they name alongside it.

Five signals, computed separately, and no total. The five components of closure for 6 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted.

What makes an ending an ending

Cadences are ranked. The authentic one is strong, the plagal weaker, the deceptive weaker still — and the solver here measured the quantity that ranking is usually explained by and found it says something else entirely. What survives is not a weaker version of the ranking but a different kind of object, with five components and no total.

form · Closure Five signals, computed separately, and no total. The five components of closure for 4 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted.

An ending that exists so a bigger one can

Half of the cadences in tonal music are built to fail. A phrase that stopped convincingly at bar four would be a piece four bars long, so the ending at bar four is engineered to arrive and not to settle — and the components it withholds are exactly the ones its partner at bar eight supplies. Closure is nested, and the nesting is what turns two phrases into one thing.

form · Closure How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire.

The chord that did not come

A deceptive cadence is described as a surprise, and the explanation offered is that the wrong chord arrived. Measured against the tonal hierarchy already in use, the wrong chord is the second best-fitting triad in the key — and two of its three voices do exactly what they would have done in the right one. The surprise is not statistical. It is one voice, and it is the bass.

harmony · Tonal-expectation A cycle has no ending to compute, so it uses density instead. 5 layers over 32 cycles of a 12-step pattern, with each layer's entry and exit marked, and the onsets per step summed underneath. No chord changes and no cadence occurs; the closure vector is zero throughout, and every change a listener hears is a change in how many things are playing.

A cycle cannot cadence

Every component of closure is defined by a first time and a last time. Music built on a repeating cycle has neither, so the whole apparatus returns zero on it — not a small value, zero, at every setting. What such music uses instead is how many things are playing, and that is a curve which can be computed from the onsets and nothing else.

form · Cyclic rhythm A tracker following a tempo that will not stay still. The tracker's error as a fraction of a beat, against beat number, for 3 rates of tempo change with a period-correction gain of 0.2. It settles at 0.025 of a beat behind at 0.5 per cent a beat, 0.092 of a beat behind at 2.0 per cent a beat, 0.206 of a beat behind at 5.0 per cent a beat. It does not lose the beat; it lags, by very nearly the rate divided by the period-correction gain, and the lag reaches a quarter of a beat at 6.3 per cent a beat — at which point the tracker is nearer the wrong onset than the right one.

A metre has to be able to change its mind

Replace the scoring function with a phase-corrected oscillator and the two failures reported earlier separate. The beat now survives two silent bars, drifting 23 milliseconds of a 560-millisecond beat. But it does not lose a moving tempo so much as lag behind it, by the rate over the correction gain — and a cadential ritardando that halves the tempo in eight beats is faster than the model can follow.

form · Metre induction Three ways to arrive at the same final tempo. Tempo against position in the closing passage, ending at 35 per cent of the opening tempo, for curvature exponents 1, 2, 3. All three begin and end at the same tempo, so what separates them is the middle: at the halfway point they read 68 per cent for linear in score position, 75 per cent for constant deceleration, 80 per cent for q = 3. The straight line is the one nobody plays. Measured ritardandos fit the decelerating curves, which is the whole of Kronman and Sundberg's argument: a closing gesture has the shape of a body stopping rather than of a dial being turned, and the parameter that varies between performances is the final tempo rather than the shape.

An ending is a deceleration

Every performance slows down at the end and the slowing has a shape. Tempo read against score position is the velocity of a body stopping — a square root rather than a straight line — and the three candidate curves agree at both ends by construction, so the whole audible difference is in the middle, where they part by fifteen per cent of the passage's length.

form · Closure The boundary operator run over only what has been heard. Foote's checkerboard novelty on thirty-two-bar AABA at a kernel width of 4 bars, computed twice: once with the whole piece available, and once using only the bars heard up to and including each bar. The kernel reaches 4 bars forward, so every cell it needs has been heard 3 bars after its centre — the retrospective curve replotted 3 bars to the right lands on the causal one, and the operator turns out to be causal at a fixed delay rather than blind. The dashed verticals are the encoding's real section boundaries and are not an input.

An ending that can be heard coming

Two measurements are both called hearing an ending coming and they point in opposite directions. By the halfway mark of an ordinary form almost nothing new arrives — and the cost of coding each bar has not fallen at all. Neither statistic says anything is about to stop, because no statistic over content can: predicting the next event well is not predicting that there will not be one.

form · Closure A silence measured in two ways that were not chosen to agree. How far a carried beat drifts during a silence, in fractions of a beat, for beat periods of 200 ms, 550 ms, 2000 ms and a tempo estimate 5 per cent wrong — which is the published discrimination limen rather than a figure chosen here. Half a beat of drift is where the metre coming out of the silence is no longer the one that went in, and it is reached after 10 beats whatever the tempo, which is 2.0 seconds at 200 ms, 5.5 seconds at 550 ms, 20.0 seconds at 2000 ms. The shaded band is the psychological present, 2 to 8 seconds, measured by a completely different literature and used in this collection to bound a phrase. The two answers overlap: a silence under about 2 seconds is a rest inside something and one over about 5.5 is after it.

A silence long enough to be an ending

Four of the five closure components are present or absent. Silence is the one with a continuous scale, so it is the one that can be given a threshold — and the threshold arrives from two literatures that were not chosen to agree, landing between three and a half and five and a half seconds. In a large hall, the room's own decay uses up most of it.

form · Closure 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 Two passages, one note apart, and opposite cadences. Two passages that share eight bars drawn from the six pitch classes C major and G major have in common, and differ only in their last three bars: one cadences in C and one in G, and the single note that separates them is an F against an F sharp. The cadence evidence names C for the first and G for the second, by 9 to 3 and 6 to 4. The profile model names E minor for both, with correlations differing in the third decimal.

The cadence as evidence

Every model so far infers a key from a bag of notes, and the thing that most obviously declares a key is a cadence — an ordered pair of chords in which the order is the whole content. The essays on closure built a five-component cadence vector long before this and nobody has used it for this. Two passages differing in one note, one cadencing in C and one in G, get opposite answers from the ordered pairs and the same answer from the profile.

harmony · Progression 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 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 ritardando does not spend the diminuendo. The arrival reading under a deceleration into the ending, from no ritardando at all to a final tempo 30 per cent of the starting one — which stretches the closing bars from 12.0 seconds to 20.2. The expectation was that it would matter: a ritardando lengthens exactly the bars the gesture is happening in, so a diminuendo that would have been absorbed at a steady tempo gets more of the smoother's own time to be absorbed in. It moves the reading by 0.00 per cent. Every line here is flat to within the thickness of the line, which is the second time a tempo parameter has been swept here and found to do nothing.

The reading was a step response

Sweeping the tempo found it did not decide the answer. This one sweeps the deceleration across a factor of three and finds a null to five figures, and then sweeps the length of the closing gesture across a factor of forty-eight and finds it moves the reading by eight per cent — but not as a function of seconds. Sorted by seconds the twelve runs scatter; sorted by how many bars the instruction covers they fall into three tight groups. One sentence explains the null and the not-null together.

form · Closure 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 A general pause of a bar is worth 4.9 decibels in a hall and 8.1 in silence. What a rest is worth as a diminuendo, in rooms with different reverberation times. The upper curve is the earlier figure, which assumed the sound stops when the players do; each curve below it lets the hall go on sounding, falling sixty decibels in its own reverberation time until it reaches the background. At the bar of silence a general pause usually is — about 1.2 seconds — the dry value is 8.1 decibels, shoebox concert hall keeps 60 per cent of it and gothic cathedral keeps 22. At 3.5 seconds, where a gap stops being a pause and becomes an ending, the same hall keeps 86 per cent. A long silence outlives any hall's tail and a short one does not, which is why the room costs the device most at exactly the length a composer writes it.

A rest needs a dry room

The twenty-four decibels a general pause is worth assume the sound stops when the players do. Put a hall under it and a bar of silence keeps 60 per cent of its value in a shoebox concert hall and 22 per cent in a cathedral, while a three-and-a-half-second one keeps 86 and 45 — because a hall's tail has a length and a rest either outlives it or does not. A written bar of silence is worth half its dry value at 2.76 seconds of reverberation, which falls between the concert hall and the stone church.

form · Closure 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 A final chord stands above the impression for a fraction of a second. How far a final chord at the tutti's own level stands above the listener's running impression at the instant it is released, against how long it lasts, for four ways of arriving at it. Straight out of the tutti it stands above nothing at any length; after 1.2 s of silence the impression is 8.1 dB down, and the chord stands highest, 5.14 dB, when it lasts 54 ms; after 3.5 s of silence the impression is 23.7 dB down, and the chord stands highest, 14.42 dB, when it lasts 28 ms; after a 6 dB diminuendo the impression is 5.0 dB down, and the chord stands highest, 3.18 dB, when it lasts 42 ms. Every curve is level again by half a second, because the impression's attack of 99 ms catches the note's attack of 22 ms, so a held chord is released at the impression's level whatever preceded it.

A final chord stands out for a twentieth of a second

A general pause drives a listener's running impression down, and the final chord that follows is supposed to cash the fall in. It cashes in at most two thirds of it. The note's own loudness rises with a 22-millisecond constant and the impression with a 99-millisecond one, so after a bar of silence the chord stands furthest above the impression 54 milliseconds in, by 5.1 of the 8.1 decibels the silence bought, and after 206 milliseconds the two are within a phon of each other. A short stamp spends most of its life standing out; a chord held a second and a half spends a seventh of it.

form · Closure A louder final chord stands higher and still stands for a fraction of a second. How far a final chord stands above the listener's running impression at the instant it is released, against how long it lasts. The chord is a struck six-note tonic; "a step louder" is the hammer velocity doubled, which raises its loudness 7.12 phons above the tutti's. At the tutti's level, after 1.2 s of silence, the impression is 8.09 dB below the chord and the chord stands highest, 5.14 dB, at 54 ms; a step louder, straight out of the tutti, the impression is 7.12 dB below the chord and the chord stands highest, 4.51 dB, at 36 ms; a step louder, after 1.2 s of silence, the impression is 15.21 dB below the chord and the chord stands highest, 9.46 dB, at 38 ms. Every curve returns to zero by half a second: the impression climbs to whatever level the chord is played at, so a louder mark is not a stand that lasts but a deeper fall to climb out of, and a silence and a mark add as depths.

A louder final chord is a deeper silence and a brighter sound

A final chord marked a step louder than the passage was supposed to stand above a listener's running impression for as long as it sounded, since the impression can climb no higher than the chord. It climbs exactly that high, and the stand closes in half a second as it always did. What a louder mark actually buys is depth — about seven phons, the same depth a second of silence buys — and a spectrum whose balance point sits most of a whole tone higher, which, unlike the stand, lasts for the whole chord.

form · Closure

Named alongside it

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

CadenceDynamicsLoudnessExpectationIntegration windowSilenceTempoNotationOrchestrationTextureEntrainmentPerceptual present

All concepts