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

The twelfth, and where it comes from tried to explain why melodies are narrow, and its answer was unusually economical. A free walk with a tune’s distribution of steps wanders far too wide. Add a pull back towards the centre, fit that one parameter to a statistic which says nothing about width, the seven in ten leaps that are followed by a turn back, and the range comes out by itself: an octave at thirty notes and something near a twelfth for the longest tunes.

That argument says it is about melodies, and it took all its numbers from three tunes in the major scale. Its steps were in semitones: 43 per cent whole tones, 12 per cent semitones, then a few larger ones. So it had quietly built the major scale’s spacing into its walk, and it read its answer in the same unit. What it calls the width of a tune could therefore be a fact about walking, or a fact about walking on that scale, and nothing in the essay can tell them apart.

The parameter nobody varied is the scale. Vary it, keeping the walk the same in every way a walk can be the same, and the result takes a different form from the one the essay stated. The interval moves by a factor of more than two. The number of different notes hardly moves at all.

The same walk, in degrees #

To put one walk on several scales, its steps have to be in the scale’s own units. So the walk here counts degrees: of the tunes’ 101 steps, 31 repeat a note, 55 move one degree, 6 move two, 5 three and 4 four. A step on the pentatonic scale is therefore usually a whole tone or a minor third, a step on the major scale a whole tone or a semitone, and a step on the chromatic scale always a semitone. The pull is the earlier essay’s: the chance of stepping inward rises with the distance from the first note over a stated reach, and is fitted on each scale to give the same seventy per cent reversal after a skip.

On the major scale this agrees with the earlier calculation, though not exactly, because a walk in degrees is a slightly different object from a walk in semitones that happens to contain the major scale’s sizes. Its pull comes out at 0.83 against the earlier 0.78, and at thirty notes its range is 11.4 semitones against 11.9. That is close enough for the comparison to be made between scales in one model.

At thirty notes the same walk spans 15.9 semitones on five notes, 11.4 on seven and 7.1 on twelve. That is about a major tenth, an octave and a fifth, and it is not a small correction. The major scale’s result is not a general fact about melody. A tune on a five-note scale, with the same walk and the same pull fitted to the same reversal, has a range nearly half as wide again. And the point at which the walk reaches a twelfth, 19 semitones, moves from past two hundred notes to about fifty.

That is the first consequence and it is an easy one to state. The figure an octave and a fifth, if it comes from the walk, is a figure about seven-note scales. It predicts something measurable about five-note repertoires, namely that they should span appreciably more than diatonic tunes of the same length, and it predicts the opposite for any music moving by semitones alone. No corpus is counted here. These are three tunes and a model, and that prediction is what a count over a real pentatonic repertoire would test.

What stays the same #

If the walk’s range is not conserved across scales, something else ought to be, since nothing about the walk has changed.

What the walk keeps is the number of different notes it uses: 7.3, 7.4 and 7.8 at thirty notes, whatever scale it stands on. Its range measured in degrees is 6.6, 6.6 and 7.1. So the quantity a walk produces is a count, and the semitones are that count times the scale’s own step size. The major scale’s average degree is 12/7 of a semitone and the pentatonic’s 12/5, and the ratio of the two ranges is 1.4, which is almost exactly the ratio of the two steps.

That suggests a different way of stating the original fact. A tune of thirty notes on the pull and step statistics here will visit about seven different pitches. On the major scale, seven adjacent pitches make up about an octave. On the pentatonic scale they make up a tenth or more. It is the seven that the walk is responsible for. The octave is what the scale then does with it.

There is a tempting connection here that should be stated and left. A scale of seven and a tune using seven distinct notes look related, and the part of the tune that is kept has shown how much of a tune’s information is compressible. But the count here is not a capacity. It rises steadily with the tune’s length, from four at eight notes to eleven at a hundred and twenty, with no ceiling, and it would be seven at thirty notes on a scale with fifty tones. It is what a walk of that length covers, and nothing here says a listener needs it to be that.

Where the taught rule stops being stateable #

The fit of the pull exposed something that the major-scale version could not have shown, because of how the statistic it fits is defined. A skip, as it has been counted here, is defined in semitones. The leap that pays itself back uses three semitones or more, so a minor third counts as a leap.

On the major scale that is reasonable: the only steps are one and two semitones, so a minor third is the next interval up from a step. On a pentatonic scale, a minor third is a step. Two of the scale’s five adjacent pairs are that far apart.

Count a minor third as a skip and no pull takes a pentatonic walk to seventy per cent. The curve reaches 67 per cent at the strongest pull drawn, 1.4. The statistic has been diluted by scale steps, and it is reporting something else. Define a skip as a fourth or more, which is the next size up on that scale, and the same walk reaches seventy with a pull of 0.71 and passes 85.

Two things follow, one about the model and one about the literature it borrows from. For the model, the walk on the five-note scale is only comparable to the others if the skip is defined in scale steps; that is why every other pentatonic figure in this essay uses a fourth. For the literature, a rule stated in semitones is a rule with a scale in it. The reversal rate measured in semitones across a repertoire that mixes five-note and seven-note melodies is measuring partly the rule and partly how often the scale’s own steps fall on the wrong side of the definition, and the seventy per cent the leap essay took from the corpus literature was quoted as a single number. Whether any particular count is affected depends on how that study defined its interval and which tunes it held, which has not been checked here. It is something to check before the seventy per cent is treated as universal.

Five notes, and no semitones gives the reason the pentatonic scale is built as it is: five is the longest a chain of fifths can run before it produces a semitone. The same fact makes it awkward for this statistic. A scale with no semitones has larger steps, and what is a leap on one scale is a step on the next.

The arithmetic extends to any five-note scale, not just the anhemitonic one. Slendro, the gamelan scale whose five steps are close to 240 cents each, is one tuned by the pair rather than by a table, and it would give the walk the same 6.6 degrees at thirty notes and about the same width as this pentatonic. In its case the statistic has no sensible semitone reading, since none of its steps is a whole number of semitones, and the only form the rule can take is in degrees.

And the operation used throughout this essay has a name elsewhere. Reading one path of degrees on a different scale is a transformation that changes every interval and leaves the ups and downs alone. The shape that survives everything else showed that a tune’s contour is what comes through such a change. The same thing can now be said from the other side: contour is what the walk conserves, along with its count of notes, and range is the thing that gives way. A tune carried from seven notes to five comes out recognisably itself and about four tenths wider, and a listener following its shape would have no reason to notice that the second part of that sentence is about the instrument or the singer rather than about the tune.

In which unit does the pull act? #

One choice inside the model might have decided all this. The pull could act in register, a distance in semitones from the first note, or it could act in the scale, a distance in degrees. They are the same on the major scale, up to the definition of the reach, and they might have come apart on the others.

It makes no measurable difference. At thirty notes the two versions agree to within a quarter of a semitone on all three scales. That is a little surprising. The reason is the fit: each version of the pull is tuned to the same reversal rate, and whichever units it acts in, a walk with that reversal has about the same tendency to return. So a question which sounds as though it should decide what kind of thing a central tendency is, register against scale, cannot be answered from the range. It would need a measurement that separated them, such as a tune transposed across a singer’s break or a melody on a scale with very unequal steps.

Three real tunes against the walk #

Any claim about walks should be set beside what the three actual tunes do. Since the count of different notes is the quantity under discussion now, that is what to compare.

All three tunes use fewer different notes than a walk of their length. Ode to Joy has five in thirty, Twinkle six in forty-two and Frère Jacques seven in thirty-two, against 7.4, 8.2 and 7.6 for the walk. Frère Jacques comes closest and also has by far the widest range for its length, 14 semitones against 11.7, because it drops a fourth below its tonic at the end. Ode to Joy, with seven semitones, is at about half of what the walk predicts.

Two of the three lie well short of the walk in both measures, and all three in the count, so the walk’s economy has a limit that the earlier essay already acknowledged. What keeps them shorter is not hard to name: a real tune repeats itself, and a walk does not. All three of these are built by repeating a phrase, and a phrase repeated covers no new notes. A piece is mostly itself again has measured how much music does that. A walk that sometimes returned to a segment it had already taken would use fewer notes, and it would come nearer these tunes on both counts.

Which assumptions carry the figures #

The model. A walk on degrees, with a step distribution counted from three major-scale tunes and applied unchanged elsewhere. That is the strongest assumption here. A real pentatonic tune may take fewer one-degree steps, since those are larger, and that would narrow the gap between the scales. A singer’s preference for small intervals has been reported as a preference in pitch distance rather than in degrees, and if the step distribution in semitones is what is really conserved then a pentatonic tune has no reason to span more than a diatonic one. The figures here are the other extreme: the scale changes and the walk in degrees does not.

Which seed is shown. Each figure’s means come from three thousand seeded walks. Only the trace is one walk, chosen because it sits near the mean on all three scales. Seeds are spread widely, and a pentatonic walk of thirty notes spans anywhere from about seven to nearly thirty semitones.

No singer or instrument is present. The argument is about statistics on paper. The register where the series becomes a scale and the essays on singing give the constraints that could confine a tune as well, and on a wide-stepped scale those might start to bind at a much smaller note count. At fifty notes a pentatonic walk is at a twelfth, and at a hundred it covers nearly two octaves, which is near what most untrained singers have.

What a count of notes does not show #

It does not show melody. A walk and a tune can have the same range and the same number of notes and be nothing alike. Every figure above is a marginal statistic of a walk whose notes have no key, no harmony and no phrase, and a melody is a walk, not a set said only that a tune’s steps have this shape, not that a walk is a tune.

It does not show which pitches. The trace in the first figure and the means in the others say how many, and they do not say where in a scale those pitches fall. Seven different notes on a pentatonic walk spread over more than one octave repeat some pitch classes: the count of different notes is not the count of pitch classes, and a tune using all five of a pentatonic scale’s pitch classes may use them in two octaves. The count treats a note and its octave as different, as a singer has to.

Whose scales #

The three scales here are one major scale, its pentatonic subset and the equal-tempered chromatic. They are tools for the argument and not a survey. A chromatic walk with a semitone for every step does not correspond to much of any repertoire. The pentatonic tradition in question could be anything from Scottish song to East Asian court music, and each has its own range conventions, its own treatment of leaps and its own instruments, all of which the walk ignores. Nothing here says what those repertoires are like. It says what they would have to be like if this walk were all there was.

Still open: a scale with very unequal steps #

The pull’s unit came out undecidable because all three scales here are roughly even within themselves. A scale with some very wide and some very narrow steps, the hirajoshi pentatonic or the harmonic minor’s augmented second, would pull those two readings apart: a distance in semitones and a distance in degrees would disagree by a different amount at every position. The computation is this walk on such a scale, with both pulls fitted to the same reversal, and it would say whether a melody’s return towards its centre is more likely a matter of the voice’s register or of the scale’s own degrees, and how large a distortion of a scale’s steps it takes before the two predictions can be told apart by any count that could actually be made.