Acoustics & Music Theory
why it sounds like that
Concept

Description length — where it appears

The number of bits a stated coder needs to write a passage down, used as a measure of how repetitive it is. It makes repetition a quantity rather than an impression, and it is sensitive to which coder is chosen.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 What a contour costs to remember. A melody of n notes over 8 degrees carries 3 bits a note. Its contour carries fewer, and fewer than the number of distinct contours suggests, because the contours are not equally likely: at 6 notes there are 243 of them but the entropy is 6.59 bits, an effective alphabet of 96. Each further note adds 1.28 bits of contour against three of melody, so the shape keeps a stable 37 per cent of what is there however long the tune.

The part of the tune that is kept

Contour survives transposition, retuning, a change of instrument and a doubling of every interval, and the usual explanation is that it is what a listener retains. That can be counted rather than assumed. A six-note melody over eight degrees carries eighteen bits; its contour carries 6.59 — not the 7.92 the number of distinct shapes suggests, because the shapes are wildly unequal — and the effective alphabet is ninety-six out of two hundred and forty-three. Each further note adds 1.28 bits of shape against three of melody, and at about nine notes a contour is specific enough to pick one tune out of a thousand.

perception · Melody 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 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

Named alongside it

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

RedundancyRepetitionMemory decayNoveltyRefrainSection boundarySegmentationCadenceClosureContourDurationEnumeration

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