Concept

Post-skip reversal — where it appears

The tendency of a melodic leap to be followed by a step in the opposite direction, reported at around seventy per cent across many repertoires. A range-bounded random walk with no melodic rule in it reproduces most of that figure.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

How much of the rule a walk with no rule reproduces. Post-skip reversal in 20,000-note random walks with no melodic knowledge of any kind. An unbounded walk reverses after 50.0 per cent of leaps, which is the chance rate and is the check that the measurement is right. Confining it to 12 semitones raises that to 61.3 per cent. Reaching the 70 per cent that corpus studies report needs a central tendency of 0.95 — an almost deterministic pull back toward the middle at the edges of the range. A wall is not enough; there has to be a spring.

The leap that pays itself back

Every melody textbook teaches that a leap should be followed by a step in the opposite direction, and every corpus that has been counted agrees — around seven leaps in ten are answered that way. A random walk with two walls, no memory of the leap and no rule of any kind reverses after 61 per cent of them, and the residue is not a rule either. What is left when the walls are accounted for is a prediction the rule does not make, and it is the prediction that decides between them.

form · Melody

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

Named alongside it

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

Melodic interval Regression to the mean Tessitura Null model Phrase Random walk Contour Register Scale degree

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