Acoustics & Music Theory
why it sounds like that
Concept

Self-similarity — where it appears

The degree to which one stretch of a piece resembles another, computed as a distance between descriptions of the two. Displayed as a matrix it makes a form visible without anybody deciding in advance what the sections are.

Named by 9 essays across 3 fields — 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 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 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 The period, as the piece goes by. The strongest lag of thirty-two-bar AABA computed on only the bars heard so far, against how many bars that is. The final answer is 4 bars; it is revised 6 times on the way, and is not reached for the last time until bar 29 of 32, which is 91 per cent of the way through and 64 seconds at 108 beats a minute. Nothing about the boundary operator is involved: this is the global statistic, and it is the half of the form that a first hearing cannot have.

The form a first hearing cannot have

Every figure so far was computed with the whole piece in hand. Run the same methods over only the bars already heard and one of the two methods survives intact — the boundary operator turns out to be causal at a fixed delay of a few bars — while the other collapses. The period of a piece is not knowable until the piece is nearly over, and in two of the six schemes here not until its last bar.

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 The ranking is settled either side of one narrow band. Remembered repetition — each bar's best match to an earlier bar, discounted by exp(−Δt/τ) with Δt in seconds — for 6 schemes at 108 beats a minute, against the decay constant τ on a logarithmic axis. The order of the schemes changes only between 8 and 13 seconds; outside that band it is fixed, so an estimate of τ wrong by any amount that stays outside it leaves the ranking alone.

A return has to be remembered

A stripe four bars off the diagonal and a stripe twenty-four bars off it are the same ink and are not the same experience. Convert the lag axis to seconds, discount every comparison by how long ago it was, and the ranking of these six schemes by how repetitive they are changes — and the decay constant and the tempo turn out to enter the arithmetic as one number rather than two.

perception · 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 What survives a change of encoding: verse and chorus. The same 32 bars of verse and chorus under five encodings, scored on the three things measured here measures. Mean off-diagonal similarity says how alike the piece looks to the arithmetic. Recall and precision are the novelty operator's boundaries against the 3 the section plan has, at a kernel of four bars. The period is the strongest peak of the lag profile, in bars. Under the bag of pitch classes every other figure uses, the piece is 90 per cent self-similar and the operator finds 0 per cent of the boundaries; under how far the root moved it finds 100 per cent. The period is the quantity that does not move.

The repeat that is not in the notes

Eight earlier essays compare bars by writing each one as a bag of pitch classes and taking a cosine. Nothing chose that encoding — the first used it and the other seven inherited it. Encode the same six schemes four other ways and one of the three findings survives untouched, one survives with different numbers, and one turns out to have been a statement about the encoding all along: the boundary operator finds none of the section edges in three schemes as a bag of pitch classes and every one of them as tonic, subdominant and dominant.

form · Repetition A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players.

A cycle that says where it is

Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke.

rhythm · Cyclic rhythm

Named alongside it

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

RepetitionSegmentationRefrainExpectationNoveltyPeriodicitySection boundaryModulationOstinatoPerceptual presentRelative minorStrophic

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