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

A melody tends back towards the middle of its range. That much has been the working assumption of every essay here since the leap that pays itself back showed that the familiar rule about leaps being followed by steps the other way needs nothing else behind it: a walk with a central tendency produces it, as a consequence of a leap usually landing far from the centre. The twelfth and where it comes from found the same walk produces a tune’s range. And a tune’s range is a number of notes ran the walk on scales of five, seven and twelve notes and reached an awkward place. A pull needs a distance, and nothing has ever said what the distance is measured in. It might be register, the singer’s or the keyboard’s sense of how high the note is, in semitones. It might be the scale, how many steps of the mode the note is from home, in degrees.

On the three scales considered there the question could not be answered, since it moved the range by under a semitone and each scale is roughly even within itself. On an even scale, distance in semitones and distance in degrees are one distance multiplied by a constant. The essay ended by pointing at scales that are not even: hirajōshi and its gaps of four semitones next to steps of one, the harmonic minor with its augmented second. There the two readings disagree differently at every note, and the question could perhaps be decided, if the difference were large enough to be counted.

The answer can be given exactly, in a form nobody has been able to state before, because the walk does not actually have to be sampled. It is a Markov chain on the degrees of the scale, and two such chains can be compared directly: how many notes of melody would have to be counted before one explanation became ninety-nine times as likely as the other. That number, not a difference between two averages, is what “can be told apart” means.

Set the walk up as before. It moves by degree steps in the proportions three tunes use, 31 repeats, 55 steps of one degree, and a few of two to four. At each note the next step goes inward with probability 0.5 + κ/2 × min(1, x/R), with x the distance from the centre. In one chain x is semitones and R is six, half the octave. In the other, x is degrees and R is half the scale. Those definitions make them identical on an equal division, and the calculation checks that: on five equal steps the information between the two chains is zero to within rounding, whatever κ is. Each chain’s κ is then fitted so that it gives the same post-skip reversal, 60 per cent, which is the one thing a melodic corpus reliably reports. The figure below draws the result, on the dial’s choice of scale.

The two curves of inward probability would lie on top of each other if the scale were even. On hirajōshi they cross, and they disagree by most at the notes near the centre.

Five hundred tunes on the major scale, a dozen on hirajōshi #

On the major scale it would take 13,645 notes to tell the two apart, about 450 tunes of thirty notes. On hirajōshi it would take 337, about a dozen. The major scale is the one every count so far has been made on, and it is close to the worst case: its steps are ones and twos and a pull in either unit picks almost the same chain, κ 0.497 against 0.494. A corpus of a few hundred major-key tunes exists, and the essay on the twelfth used a fraction of one. So the question is not out of reach even there, but a model simple enough to run on one has only a few parameters, and a real tune differs from it in many ways larger than the difference between these two pulls. For a difference this small, no count will be able to attribute its answer to the unit of a pull rather than to something else about the tunes.

Hirajōshi is different in kind. Three hundred and thirty-seven notes is a couple of pages of music. The fitted pulls are quite far apart there, κ 0.715 against 0.849, and on a scale with steps of one and four, a walk pulled by semitones and a walk pulled by degrees are not the same walk. So there is a scale on which the question should be decidable by counting, and any body of music in a mode like it would answer it, provided that the mode is what it really is: a question about Japanese court and koto repertoire, not about a scale in the abstract. That count has not been made here.

The order of the scales in that figure is the first sign that something other than unevenness is at work. The pentatonic’s steps are 2, 2, 3, 2, 3, quite even. The harmonic minor’s are 2, 1, 2, 2, 1, 3, 1, far less even. Yet it takes under half as many notes on the pentatonic. A smooth measure of unevenness cannot put them in that order, and the reason it comes out that way is the thing the essay is about.

The cliff where a step becomes a skip #

Take a family of five-note scales that starts even and is pulled gradually apart, with steps 2.4 + d, 2.4 − d, 2.4 + d, 2.4 − d, 2.4, and ask how the notes needed fall with d.

For small d the count goes as the inverse square of d, exactly as information does for a small difference between two chains: 942,694 notes at 0.1, 37,451 at 0.5, a factor of twenty-five for a factor of five. Then between 0.5 and 0.6 it falls seventeenfold in a single step. At d = 0.6 the wide step of the scale is three semitones, and a skip, in the definition the reversal is fitted to, is anything of three semitones or more. So at that point a simple step of the scale has started to count as a skip. A move by one degree, which a moment earlier was not in the statistic at all, is now in it, and the fitted pull jumps to compensate. The two chains now have to be fitted to very different things, and they part at once.

This is a property of how a skip is defined and of nothing in the music, and it is the part worth knowing. The published statistic is defined in semitones, for reasons of convenience, and a corpus study reporting post-skip reversal on a mode with an augmented second or a gap is counting something different in that mode from what the same number means in the major scale. The earlier essay already had the symptom of this, without the diagnosis: no pull takes a pentatonic walk to seventy per cent reversal with a three-semitone skip, because the pentatonic’s own minor-third step dilutes the statistic. It is why every number in this essay is at 60 per cent, since on the pentatonic a higher one is out of reach or close to it. Here is the same dilution operating on a question it has no business answering. A pentatonic melody and a harmonic-minor melody take such different numbers of notes partly because they have different numbers of steps that have reached three semitones, and for no more musical reason than that.

So the surprising connection is between a theoretical question, what a melody’s pull is measured in, and a definition nobody thinks of as theoretical. The question becomes easy to answer at exactly the place where the statistic it is answered with stops meaning what it says. That should make anyone cautious of the low count on hirajōshi. Part of what separates the two pulls there is that a four-semitone step of the mode is a skip by definition. A test of the pull’s unit would be sounder with a skip defined in the scale’s own terms, and then the count would rise, perhaps a good deal. The next two figures keep the semitone definition and read where the evidence is, since that does not depend on it.

Half of it comes from one note #

The evidence does not come from everywhere. On hirajōshi, more than half of what separates the two pulls comes from standing on one note, the third degree of the scale, three semitones above the centre. It is two degrees from home but only three semitones. On the degree reckoning that is four fifths of the way to the strongest pull, and on the semitone reckoning it is halfway. So the two explanations make their most different prediction there: on one account, a note on that degree goes back towards the centre 83 per cent of the time, and on the other 66. It is also a place the walk spends a good deal of time, since it is close to home. It would be enough to count what melodies in this mode do next when they are on that note.

There is no equivalent on the other side. The mirror degree, two below the centre, is five semitones away, and there the two reckonings nearly agree. The asymmetry of a scale’s steps around its centre is what produces a place like this, and it is a property of where the tune has its centre, so the same scale should give different counts according to its centre. That was the next thing to try.

Moving the centre changes the count by a factor of two or three, and no more. On the major scale it runs from 10,032 notes when centred on the fourth or the seventh degree to 30,569 on the second. The modes of the major scale do differ, in this respect, in how legible a melody’s pull in them would be. The rotations of the harmonic minor are similar. But the scales keep their order. No choice of centre takes the major scale to where hirajōshi begins. Which is also why a corpus of tunes in one mode, centred on one tonic, is a fair test of nothing wider than that mode.

A count that uses this is unusually simple to specify, which is the practical point. Take a body of melodies in a mode, name each one’s centre, find every occurrence of the note two degrees and three semitones above it, and record whether the next different note lies above or below. On a semitone reckoning about two in three go inward; on a degree reckoning, about five in six. That is a test a student could run in an afternoon on any transcribed repertoire in a suitable mode, and the difference between 66 and 83 per cent is large enough to be seen in a few dozen occurrences. Whether it would then mean what it appears to mean is the harder matter: a mode’s third degree may have a function of its own, as a goal or a passing note, and the smoothness of a line is one of several things that could produce a preference at that note with no pull behind it at all.

The number everyone actually counts is no use #

There is one more thing worth reading, because it is what a corpus study would reach for first.

Fitted to one reversal, the two pulls differ in a tune’s distance from its centre by under half a semitone on every scale, and by nothing at all on the major scale. Hirajōshi, on which the chains are most different, shows 4.01 against 3.58. That is what makes the earlier essay’s “undecidable” true of range in particular. A count of range would need a very large corpus and a very quiet model to detect a difference of 0.4 semitones in the mean. A count of what happens next from one chosen note needs a dozen tunes. The difference between the two has nothing to do with the data and everything to do with asking the question at the place where the two explanations disagree. The likelihood ratio points there automatically, and it is the general moral of the method: a test of two models should count the transition they disagree about, and a statistic that was developed for a different purpose may have averaged over it.

What the walk does not know #

It is the wrong kind of model for a real tune, on purpose. A walk has no phrases, no harmony, no cadences, no sense of the tonic other than one point, and it has first-order memory. The part of the tune that is kept is what it leaves out. Every number here is what it would take to separate two toy models if a tune were one or the other, and a tune is neither. The likelihood ratio of 99 is also the most generous form of the test, assuming the true model is known exactly apart from the one distinction. So these counts are lower bounds, and they should be read as ratios between scales, as the size of the question, rather than as a sampling plan for a corpus study.

The step distribution is borrowed from three tunes in the major scale. It is the one that a melody is a walk, not a set established, and it is held constant on every scale here. Tunes on hirajōshi certainly do not use the same step sizes in the same proportions. A scale with gaps could plausibly have fewer steps of one degree, and that would change the counts by amounts not computed.

Reversal is fitted at 60 per cent throughout, because a higher value is at or beyond the ceiling of a pull on some of these scales: on the pentatonic and on hirajōshi, even a pull of 1.4 leaves a degree-reckoned chain just above 65 per cent. A real corpus’s figure is higher than sixty, by a margin that depends on what is counted, and this has to do with the dilution above. At seventy per cent the chains are clamped on several scales, and at those limits the comparison is not meaningful.

Hirajōshi is not one scale. The label covers several tunings and several note sets in the theoretical literature, and these steps are a Western-derived twelve-tone reading of it. On a scale that is a set of tuned intervals rather than a keyboard, even the semitones here would be approximations.

Whose melodies #

Everything is a claim about single-line melody, in the way that the corpus literature on post-skip reversal considers it: folk song and hymn tunes in the European tradition for the major scale, and everything else by extension. The other scales are drawn for their step patterns, and no claim is being made here about any repertoire that actually uses them, whether Japanese koto music, Klezmer, Roma, Hindustani or Arabic melody, any of which has its own structure far beyond a scale and a centre. A melodic theory might be in semitones for a singer and in degrees for a keyboard player, too. That would make the question this essay has been answering a question about instruments, and it has not been asked.

Still open: a skip counted in the scale’s own terms #

If the cliff is an artefact of defining a skip in semitones, then the honest next count defines it in degrees: a skip is a move of two or more degrees, whatever that measures. The calculation is the same chain and the same fit with that one change. It would say whether hirajōshi’s dozen tunes survive, whether the major scale’s five hundred come down or rise, and whether the seventeenfold drop vanishes as it should. If it does, the sharp ordering of the scales here is a property of a definition. If it does not, the gap in a scale really is what makes a melody’s pull legible. The answer would also say which statistic a corpus study of a non-Western mode should count from the start.