Acoustics & Music Theory
why it sounds like that
Concept

Hierarchy — where it appears

An arrangement in which units of one size are made of units of the next size down, so that a description applies at several scales at once. Metre is one, tonal function is another, and both let a listener attend at whichever level is useful.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

Boundaries found by a local operator, at three kernel widths. Foote's checkerboard novelty computed on the self-similarity matrix of thirty-two-bar AABA, at kernel widths of 2, 4, 8 bars. The dashed verticals are where the encoding's sections actually change; nothing about them enters the computation. A peak is a place where the bars before resemble each other, the bars after resemble each other, and the two groups do not resemble each other.

The boundary is where the neighbourhood changes

A section boundary can be found by an operator that never sees a section. It walks the diagonal of a similarity matrix asking one local question — do the bars behind me resemble each other, do the bars ahead resemble each other, and do the two groups resemble each other — and where the answer is yes, yes, no, there is an edge. What it cannot find turns out to say more than what it can.

form · Repetition 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 Twinkle, twinkle, phrased at the level each tempo selects. The number of boundaries the model finds when its smoothing scale is set by the psychological present rather than chosen, against the tempo the tune is taken at. The scale in notes is the present's 3.5 seconds divided by the mean note length, so a fast tempo puts more notes inside the present and smooths harder. The page's own phrasing has 5 boundaries, drawn as the flat line; the model matches it best at 160 beats per minute, where the present holds 8.2 notes. The same tune at two tempos is read at two levels, which is the prediction and is not a free parameter.

The level the tempo chooses

The boundary detector has a scale parameter and an earlier essay left it free, ending with the sentence that names this one: the scale is in notes and the psychological present is in seconds. The psychological present is two to eight seconds, a tempo converts one to the other, and the level a listener reads then stops being a parameter at all — which is a prediction with teeth, because the same tune at two tempos should be phrased differently at levels the arithmetic names in advance.

form · Phrase The same tune read at six widths of the psychological present. A later essay made the detector's smoothing width a function of position, which removed its last free parameter but one — and the one it cannot remove is the width of the psychological present, because that is a fact about listeners rather than a choice. So the honest object is not a reading but a family of them, one per width. A short present finds 6 boundaries and a long one finds 2, and the family agrees on 0 of them. The fixed-width control, at its own best width, scores 0.67 against the adaptive readings' 0.67, 0.75, 0.33, 0.33, 0.40, 0.40 — so the adaptation does not win, which is what that essay reported too. What the family adds is the ordering: a boundary in every row is a different claim from one in a single row, and a single reading has no way to say so.

A family of readings

Removing the detector's free parameter, and then its constant tempo, cost persistence both times — the property that made its boundaries ordered rather than merely found. Recovering it means a family of adaptive readings rather than one, indexed by the width of the psychological present, which is the one parameter that cannot be removed, because it is a fact about listeners.

rhythm · 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 A count at every level turns a form's vague top into its sharpest level. 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, counted three ways: in 2.0 s units alone, with one checkpoint count every 16 s, and with a count at every level of 2.0 s, 16 s, 64 s, 128 s, each mending the one below, every count failing independently. 480 s: 2.2, 3.9 and 76.4; 240 s: 2.5, 7.1 and 54.0; 120 s: 3.1, 13.2 and 38.1; 60 s: 4.1, 19.4 and 19.4; 30 s: 5.4, 18.1 and 18.1; 15 s: 12.2, 12.2 and 12.2; 7.5 s: 10.3, 10.3 and 10.3. The whole piece goes from 2.2 with one count and 3.9 with one checkpoint to 76.4 with a count at every level.

Only a lapse of attention keeps a form's top vague

A listener who keeps a count at every level of a form — bars, phrases, periods, sections — each mending the one below when it lapses, does not have a vague top at all. The whole eight-minute piece becomes the sharpest level, at seventy-six distinguishable proportions against two. Failures shared between neighbouring counts barely dent it, because a shared failure has to climb every level to reach the top. What puts the vagueness back is a lapse that takes every count at once, and one lapse in a hundred of that kind is enough.

form · Proportion A lapse of attention that ends costs a form nothing unless a section is heard twice in a row. How many proportions between 1 : 1 and 3 : 1 a listener who knows the form tells apart across a whole 480-second piece, counted at every level of 2.0 s, 16 s, 64 s, 128 s, with 10 per cent of lapses taking every count at once and lasting 10 seconds on average, after which the listener re-anchors at the section the music says they are in. through-composed, ABCDEFGH: 77.1, 0.0 per cent of returns misplaced; rondo, ABACABAD: 76.4, 0.0 per cent of returns misplaced; sonata, exposition repeated, ABABCDAB: 76.6, 0.0 per cent of returns misplaced; minuet and trio, halves repeated, AABBCCDDAB: 50.2, 5.7 per cent of returns misplaced; thirty-two-bar song, twice, AABAAABA: 47.7, 4.3 per cent of returns misplaced; strophic, AAAAAAAA: 35.4, 10.1 per cent of returns misplaced. With lapses that never end every form reads 4.5, and with every returning listener placed exactly 77.2. The forms that lose most are the ones that play a section twice in a row, not the ones with the most repetition.

A repeat sign loses a listener that a refrain does not

A lapse of attention loses every count a listener is keeping, and when it ends the listener has only the music to say where they are. A listener who knows the form comes back to a rondo's refrain and is almost never misplaced, because the episodes between refrains make only one of them reachable. They come back near the join of a section repeated immediately and are misplaced nearly half the time, and a form built of such repeats loses between a third and a half of the precision a count at every level had restored. One of the most familiar signs in notation does more damage than the most repetitive form.

form · Proportion A first hearing costs the through-composed piece most and the rondo almost nothing. How many proportions between 1 : 1 and 3 : 1 a listener tells apart across a whole 480-second piece of eight one-minute sections, counted at every level, with a tenth of lapses taking every count at once and lasting 20 seconds on average, for a listener who knows the piece, one who knows only its plan of sections, one who knows the style's five forms, and one who knows nothing of the style. through-composed, ABCDEFGH: knows the piece 75.7, knows only the plan 62.7, knows the style 61.4, knows nothing 59.3; rondo, ABACABAD: knows the piece 56.2, knows only the plan 56.0, knows the style 54.6, knows nothing 51.6; sonata, ABABCDAB: knows the piece 61.0, knows only the plan 59.7, knows the style 57.8, knows nothing 53.9; song form, AABAAABA: knows the piece 35.3, knows only the plan 35.3, knows the style 34.5, knows nothing 32.8; strophic, AAAAAAAA: knows the piece 25.9, knows only the plan 25.9, knows the style 25.5, knows nothing 25.0. What a first hearing lacks is not the plan but the ability to recognise a section never heard: new music says only that it is new.

New music says only that it is new

A listener who knows a piece comes back from a lapse of attention and is placed at once by whatever they hear, and a through-composed piece, which never repeats anything, is the easiest of all forms to come back to. A listener hearing it for the first time is not so lucky. What they hear on returning is either something they have heard before, which names itself, or something they have not, which says only that it is new. For a rondo that costs almost nothing. For the through-composed piece it costs its whole lead: at lapses of half a minute or more, a first-time listener is placed no better in it than in a rondo — and being told the piece's plan in advance gives back almost none of the difference.

form · Proportion A function label gives back a first hearing's place only if it does not recur. A through-composed piece of eight one-minute sections, heard for the first time by a listener told its plan, with a tenth of lapses taking every count and lasting 40 seconds on average: proportions told apart across the whole piece, against the chance that on returning into music not yet heard the listener recognises the section's function — opening (O), continuing (C), intensifying (I) or closing (Z) — for four arrangements of the functions. OIOIOIOI, two functions alternating: 0% 29.8, 25% 31.5, 50% 33.6, 75% 36.2, 100% 39.5. OCCCIIIZ, an arc in long runs: 0% 29.8, 25% 31.4, 50% 33.3, 75% 35.7, 100% 38.6. OOCCIIZZ, four in blocks of two: 0% 29.8, 25% 32.1, 50% 35.1, 75% 39.1, 100% 45.0. OCIZOCIZ, four, cycled: 0% 29.8, 25% 33.1, 50% 37.8, 75% 45.4, 100% 60.8. With no label the listener reads 29.8; knowing the piece, 66.1.

A section that says what it is for places a first hearing

A first-time listener who loses count in a through-composed piece comes back into music they have never heard, and new music says only that it is new. Suppose instead each section announces what it is doing — opening, continuing, intensifying, closing — and the listener recognises it. How much of their place that gives back depends hardly at all on how many functions there are and almost entirely on how soon the same function comes back. Four functions cycled restore 85 per cent of what a first hearing costs at forty-second lapses; the same four in the familiar arc, two sections each, restore 42; two functions alternating restore 85 per cent at twenty seconds and 14 at a minute.

form · Proportion Four functions cycled lose their advantage the moment neighbours are confused. A through-composed piece of eight one-minute sections, a first hearing, a listener told the plan of functions, lapses of forty seconds on average: proportions told apart across the whole piece against the chance that a recognised function is heard as its neighbour in the arc opening, continuing, intensifying, closing, for five arrangements. OCIZOCIZ, four, cycled: 0% 60.8, 5% 47.7, 10% 44.1, 15% 41.3, 20% 39.0, 30% 36.2, 40% 35.1, 50% 34.9. OOCCIIZZ, four in blocks of two: 0% 45.0, 5% 40.0, 10% 35.3, 15% 34.4, 20% 33.0, 30% 32.3, 40% 31.3, 50% 31.5. OIZOIZOI, three, cycled: 0% 51.4, 5% 43.2, 10% 41.9, 15% 40.8, 20% 39.8, 30% 37.9, 40% 37.1, 50% 36.7. OZOZOZOZ, opening and closing alternated: 0% 39.5, 5% 39.5, 10% 39.5, 15% 39.5, 20% 39.5, 30% 39.5, 40% 39.5, 50% 39.5. OCOCOCOC, opening and continuing alternated: 0% 39.5, 5% 37.3, 10% 35.3, 15% 33.7, 20% 31.8, 30% 31.0, 40% 30.4, 50% 30.6. Unlabelled, 29.8; knowing the piece, 66.1.

Two functions far apart outlast four that blur

A first-time listener who loses their place in a through-composed piece gets most of it back if each section announces what it is doing and the same function does not come back too soon: four functions cycled restore 85 per cent of what a first hearing costs. That rests on hearing each function exactly. Let a listener mistake a function for its neighbour in the arc — a continuation for an intensification — one time in ten, and the cycle keeps less than half of its gain. Two functions at the opposite ends of the arc, opening and closing alternated, lose nothing to such confusion, and from a little over one confusion in five no arrangement does better.

form · Proportion A listener who has to learn the plan loses most where the plan said most. Eight one-minute sections, a first hearing, lapses of forty seconds: proportions told apart across the whole piece, against the chance a recognised function is heard as its neighbour in the arc, for a listener told the plan of functions (dashed) and one who must infer it from the functions heard before each lapse, among five plans it thinks equally likely (solid). four, cycled: told 60.8, 44.1, 39.0, 34.9; learning 44.2, 38.5, 35.4, 33.3; four in blocks of two: told 45.0, 35.3, 33.0, 31.5; learning 39.1, 33.2, 32.4, 30.7; opening and closing alternated: told 39.5, 39.5, 39.5, 39.5; learning 37.1, 37.1, 37.1, 37.1 at confusions of 0, 0.1, 0.2 and 0.5. A listener who knows the piece reaches 66.1; one with no functions, 29.8.

A listener who learns the plan keeps only the far pair's immunity

A first-time listener who recognises what each section of a through-composed piece is doing gets most of their place back after a lapse — if they know the plan of functions. A listener who has to infer the plan from the sections heard so far loses most where the plan said most: four functions cycled give up more than half of what their labels were worth, nearly all of it to lapses in the first two minutes, before the second function has settled which plan the piece has. Opening and closing alternated lose a little and stay immune to confusing neighbouring functions. The far pair's immunity was never knowledge of the plan; it was the arc.

form · Proportion

Named alongside it

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

Memory decayMusical formRepetitionProportionHypermetrePhraseRefrainSegmentationGroupingScale-spaceWeber fractionAntecedent consequent

All concepts