Acoustics & Music Theory
why it sounds like that
Concept

Refrain — where it appears

A section that returns unchanged between contrasting sections, and whose return is what a listener uses to locate the form. Its unchangedness is what makes it useful: a landmark that varied would not be one.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

thirty-two-bar AABA, every bar against every other bar. A self-similarity matrix of 32 bars of thirty-two-bar AABA. Each cell is the cosine similarity of two bars' pitch-class vectors, with the chord's own notes weighted 1 and the rest of the key 0.5. Similarity is quantised into four bands for drawing and anything under 0.35 is left as paper. Nothing in the computation knows what a section is; the blocks and stripes are what the arithmetic returns.

A piece is mostly itself again

Take a piece of music, encode each bar as the notes sounding in it, and compare every bar with every other bar. The picture that comes out has blocks and stripes in it, and those blocks and stripes are the form — arrived at by arithmetic that has never heard of an exposition, a chorus or a refrain.

form · Repetition Redundancy in bits, and why the number needs a length beside it. Left: the LZ78 cost of each scheme divided by the cost of sending the same symbols flat, against how many bars are sent, with each bar coded as its chord and key. Every scheme is above 1 at a single chorus — the coder loses — and every one falls under it as the piece runs. Right: new dictionary phrases per bar at one chorus, which is the statistic that survives at short lengths.

How much of this is new

Repetition can be counted rather than looked at. Feed a piece's bars to a compressor and the bits it needs are a measure of how much of the piece is a repeat of an earlier part of itself. The measurement works, the number is real, and it turns out to be a statement about the description rather than about the music — which is the most useful thing it has to say.

form · Repetition How long until it comes back. Mean similarity along each diagonal of the self-similarity matrix, minus the matrix's own mean off-diagonal similarity, against lag in bars, for 1 case. Lags run to half the length of each scheme, because a longer diagonal holds too few pairs to average. All rows share one vertical scale and the spread of each is printed beside it; the largest is 0.427 and the smallest 0.427. 1 of 1 cases with any spread at all put their strongest lag at the scheme's own repeat unit or a multiple of it.

How long until it comes back

A self-similarity matrix has a second reading that nobody looks for. Add up each diagonal instead of walking along one, and out falls repetition as a function of how long ago — a period, in bars, with no segmentation, no kernel width and no bar numbers anywhere in the answer. Five of the six schemes here report the length a listener would have named. The sixth reports something better.

form · Repetition Sensitivity and specificity on one dial. Aligned similarity — bar i against bar i+L, which is what a return is — for 4 eight-bar comparisons, as the key-invariance dial turns. One comparison is constructed: a literal repeat in the encoding, moved up a fifth, which is a stated manipulation because no scheme encoded here repeats a section in a new key. The shaded band is the margin between the two named comparisons, and it runs from 0.021 to 0.106.

The same thing somewhere else

A measure built on which notes are sounding calls a passage that comes back a fifth higher a stranger. There is a dial that fixes this, and turning it is supposed to be a trade — more sensitivity to a transposed return, less specificity against a coincidental one. It is not that trade. Two different statistics answer opposite ways, and the setting that would compromise between them is the worst one available.

form · Repetition thirty-two-bar AABA, as a strip of time. thirty-two-bar AABA laid out one cell per bar, coloured by section, with the roman numeral in each bar. the A section's turnaround is the ii-V every variant keeps; the bridge is a chain of applied dominants. At 108 beats a minute in 4/4 the whole of it lasts 71 seconds. Cut into 4 repeat units of 8 bars, 3 pairs of units agree on more than 50 per cent of their bars. 2 of them are not identical, and 2 of those 2 differ in a run of bars ending at the last bar of the unit; the changed bars are marked in orange.

Where a repeat is changed

Cut every scheme into its own repeat unit, compare each unit with every other, and ask where a repeat stops agreeing with what it repeats. The answer is that it stops at the end, in every case the corpus contains — and the number of cases the corpus contains depends entirely on where the threshold for "a repeat" is put. Moving it by nothing at all takes the count from two to twenty and the finding with it.

form · Repetition 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 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

Named alongside it

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

RepetitionMemory decayMusical formProportionSegmentationSelf-similarityHierarchyStrophicDescription lengthNotationOstinatoRedundancy

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