Acoustics & Music Theory
why it sounds like that
Concept

Random walk — where it appears

A quantity whose successive changes accumulate rather than cancel, so that its spread grows without bound. Confined between two limits it stops spreading and starts regressing, which reproduces melodic statistics with no melodic rule in it.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

Where the two accounts part company. The spread of the error between the hands, in milliseconds, against bar number, averaged over 120 seeded runs of each model at 100 bars a minute. With one timekeeper it is flat at about 9 ms after 24 bars; with two it reaches 106 ms and is still climbing, because a random walk has nothing to return to. Measured players hold 3 against 2 inside about 25 ms indefinitely.

One player is not two clocks

Two accounts of a pianist playing three against two, simulated from the same noise. With a timekeeper in each hand the hands drift apart by a hundred milliseconds inside two dozen bars. With one timekeeper they never drift at all. The measurement everybody cites as evidence for a timekeeper turns out to be the same number under both accounts.

rhythm · Polyrhythm Two players, and the correction that keeps them together. The spread of the asynchrony between two players, in milliseconds, against beat number, for 3 correction gains, averaged over 120 seeded runs each. It reaches 120 ms after 64 beats at a gain of 0, 27 ms after 64 beats at a gain of 0.1, 20 ms after 64 beats at a gain of 0.3. With no correction at all the asynchrony is a random walk and grows without bound; with any correction it settles at a fixed spread within a few beats and stays there. Two people cannot share a timekeeper, so the fact that ensembles do not come apart is itself the evidence that they are correcting.

Two players and no clock

Two people cannot share a timekeeper, and two independent ones drift a hundred and twenty milliseconds apart inside a minute. Ensembles do not, so something is correcting — and the measurement everybody reaches for recovers the pair's total responsiveness exactly and cannot tell which of the two is doing it. Four tenths from one player and two tenths each give the identical number.

instruments · Microtiming Where a twelfth comes from. The range of a walk with no walls, against how many notes it runs for, at three settings of the one parameter it has. The parameter is fitted to the post-skip reversal rate and to nothing else; the range is then read off. With no central tendency at all the walk passes two octaves by 60 notes and keeps going. At the setting that reproduces 70 per cent reversal — κ = 0.78 — the range is 11.9 semitones at thirty notes and 17.9 at a hundred and twenty. It grows logarithmically, so over the whole plausible length of a tune it sits between an octave and a fifteenth, and a twelfth is the middle of that. The three tunes carried here are marked and all three fall below the curve.

The twelfth, and where it comes from

Melodies occupy about an octave and a fifth, and an earlier essay set out to explain that by the singer's register break and found that it does not: the chest mechanism alone spans two octaves and a semitone. The answer is in a parameter the essay on leaps fitted and then put down. A walk with no walls whose central tendency reproduces the post-skip reversal rate has a range that grows logarithmically — six semitones at eight notes, twelve at thirty, eighteen at a hundred and twenty — so across every length a tune plausibly has, the span is between an octave and a fifteenth.

form · Melody The same kind of walk covers more of the keyboard on a scale with larger steps. A walk of thirty notes with the tunes' distribution of steps in scale degrees, pulled back towards its start, on the major scale. This seeded walk uses 8 different notes and spans 12 semitones. Over three thousand such walks the mean is 11.4 semitones, 6.6 degrees and 7.4 different notes.

A tune's range is a number of notes, not an interval

An earlier essay derived the width of a tune from a walk pulled back towards its middle, and read it in semitones. That reading has a scale inside it. Put the same walk on five notes, seven and twelve, and at thirty notes it covers 16, 11 and 7 semitones, while using the same seven or so different pitches on all three. The width a tune inherits from its walk is a count of notes, which an interval then measures. And on a pentatonic scale, the taught rule about leaps cannot even be stated in semitones.

scales · Melody On an unequal scale the two meanings of a melody's pull come apart at particular notes, and they do not do it evenly. The hirajōshi scale, with the walk's centre on its first note and both pulls fitted to a post-skip reversal of 60 per cent: the chance of moving inwards at each degree. Pull in semitones κ 0.715, pull in degrees κ 0.849. It takes about 337 notes of melody to tell the two at a likelihood ratio of 99.

A melody's pull can be counted only where a step is a skip

A melody drifts back towards its centre, and nobody has said in what unit: a distance on the keyboard or a distance in the scale. On even scales the two cannot be told apart. On unequal ones they can, and an exact reckoning puts a price on it: about five hundred thirty-note tunes on the major scale, a dozen on hirajōshi. The price falls with the square of how uneven the scale is, and then drops seventeenfold at the point where a step of the scale gets wide enough to be counted as a skip.

scales · Melody

Named alongside it

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

Melodic intervalEntrainmentMotor delayPost-skip reversalRegression to the meanTessituraTimekeeperMicrotimingNull modelPhrasePolyrhythmRegister

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