The diatonic set — the series · Acoustics & Music Theory

The major scale as a cycle. The twelve semitones drawn as a cycle, with the notes of the scale filled in. The gaps between filled positions are the step pattern, and reading them round the circle is what makes the scale's asymmetry obvious.

Seven of the twelve, chosen unevenly

A major scale is a selection of seven positions out of twelve, and the selection is lopsided on purpose. The two semitones sit where they do because an even choice would destroy the thing that makes a scale usable.

part 1 · scales

Stopping the chain of fifths at every length. The chain of fifths taken two notes at a time, three, four and so on, with each collection folded into a single octave. The smallest gap in the collection is measured at every stage: it is two semitones until the sixth note arrives, and then it is one. Five is as far as the chain goes without producing a semitone.

Five notes, and no semitones

The five black keys sound acceptable in any combination, in any order, over almost any bass note. That is not mysticism and not luck — five is the longest a chain of fifths can run before it produces a semitone, and the number is derived rather than chosen.

part 2 · scales

How many sizes each interval comes in. Each generic interval of the diatonic scale and of a seven-note set that is not a mode of it, with the specific sizes it takes as the starting degree moves round. The first gives exactly two sizes for every one of them; the second gives 3, 4, 4, 4, 4, 3. Of the 462 seven-note selections from the twelve that contain the tonic, 14 have two sizes for every generic interval — 3.0% of them, and they are the rotations of just 2 step patterns: 1·1·1·1·1·1·6 and 1·2·2·1·2·2·2.

Two sizes of every step, which is why the names work

A third is three semitones or four, and every musician learns to call both of them thirds without being told why that is allowed. It is allowed because of a property the diatonic scale has and 448 of the other 461 seven-note selections from the twelve do not.

part 3 · scales

The 7-note sets in which every interval occurs a different number of times. Each set of 7 notes containing C whose six interval counts are all different, with the counts printed. Every one of them is a rotation of one of two shapes, and only one of the two has steps a scale could use — the other is a run of semitones with the gap at the end.

Every interval a different number of times

Count the intervals inside a major scale and the six answers are 2, 5, 4, 3, 6 and 1 — six different numbers, no two alike. That is not decoration. It means the number of notes a key shares with a transposition of itself identifies the distance uniquely, so a listener who can only count common tones can still tell exactly how far a modulation went.

part 4 · scales

One construction, a scale on one side and a rhythm on the other. Both rings are the same computation: k things placed as evenly as possible in n positions. On the left it is 7 notes in 12 semitones, which is the major scale; on the right 5 strikes in 8 steps, which is the cinquillo. The filled positions come from the J-function and the open marks from Bjorklund's algorithm, and they are the same set of positions turned by 9 and 6.

The same algorithm made a Cuban rhythm

Ask Bjorklund's algorithm for seven onsets in twelve steps and it returns 101101011010, which read as pitch classes is the natural minor scale. That function has been producing the tresillo and the cinquillo here from the beginning, and nobody has said that the scale field's central object comes out of it unchanged.

part 5 · scales

Four properties, ten sizes, one survivor. Every subset of the twelve pitch classes, counted by shape, at each size from two notes to eleven, and how many shapes of each size have each property. Exactly one shape in the whole table has all four at once: the 7-note set with steps 1221222, which is the diatonic scale.

Why seven

Four properties, each shared with some other set. Run all four over every subset of the twelve, at every size from two notes to eleven — three hundred and forty-nine shapes in all — and exactly one has all four at once. It is the diatonic set, and it is the only survivor in the whole twelve-note universe.

part 6 · scales

The sizes a chain of a pure fifth will make. Chains of a pure fifth of two to 12 notes, each folded into one octave and drawn on a 1200-cent line. A chain has exactly two step sizes at 2, 3, 5, 7, 12 notes and three or more at every other size in this range. The five-note and seven-note cases are the chain's pentatonic and its diatonic set, and they are neighbours in a series rather than two separate facts.

The only sizes a fifth will make

Stack pure fifths and fold them into an octave, and at almost every number of notes the result has three or more different step sizes. At 2, 3, 5, 7, 12, 17 and 29 it has exactly two — and at no other size below thirty. The pentatonic and the diatonic set are not two discoveries. They are neighbours in one series, and it is the comma's series.

part 7 · scales

The same census, in every universe from four to thirty. How many sets have all four properties, in a universe of n equal steps. None at all when n is two more than a multiple of four — 6, 10, 14, 18, 22, 26, 30 — exactly one when n is a multiple of four, and exactly two when n is odd. The multiples of four each have their survivor at n/2 + 1 notes generated by n/2 − 1 steps, so twelve's seven notes generated by the fifth is the general answer with n put at twelve rather than a fact about twelve.

Every universe has one, or none

Run the census that found the diatonic set in a universe of nineteen equal steps, or twenty-four, or fifty-three. The answer is completely regular and nobody appears to have written it down — none at all when the universe is two more than a multiple of four, exactly one when it is a multiple of four, and exactly two when it is odd.

part 8 · tuning

The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time.

Nothing in the census knows which note is home

All four properties six earlier essays are about are invariant under rotation and transposition — one property tuple over all eighty-four rotations and transpositions of the diatonic set. So the census that separates 349 shapes cannot separate a major scale from its own Aeolian mode, and everything that makes one note a tonic is outside it.

part 9 · scales