Generator

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.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 29 of its 29 placements carry sound, built from the same numbers as the drawing, offering these buttons: A chromatic walk of thirty notes, A major-scale walk pulled in semitones, A major-scale walk, where a minor third is a skip, A nearly equal five-note scale, A pentatonic walk of thirty notes, A pentatonic walk pulled in semitones, A pentatonic walk, where a minor third is a step, A tune: the corpus melody these walks are compared against and 19 more.

Called by 4 essays

the blast radius of changing it

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

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

All figures · What can be heard