Symmetric scales

The major scale has a pattern that does not repeat: tone, tone, semitone, tone, tone, tone, semitone. The two semitones sit in different places, and that asymmetry is what gives each note of the scale its own function. The tonic is the tonic because only it has those surroundings.

Now imagine a scale whose pattern repeats, with the octave split into equal parts. In it, every note has the same surroundings. That is a symmetric scale.

Splitting the octave into equal parts

The octave has twelve semitones, and twelve divides evenly in several ways. Each division gives a symmetric object:

Parts	Repeated interval	What it makes
12	semitone	the chromatic scale
6	whole tone	the whole-tone scale
4	minor third	the diminished chord
3	major third	the augmented triad
2	tritone	a pair of notes

The first two are scales; with four and three parts the notes are spaced so widely that the result sounds like a chord:

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The diminished chord splits the octave into four minor thirds, and the augmented triad into three major thirds. That is why both are known for having no clear root: any of their notes could be the one at the bottom.

The whole-tone scale

The whole-tone scale is made of whole tones only, six of them closing the octave. Built strictly, tone by tone and letter by letter, its last note does not have the same name as its first (Med 1996, ch. XXXV):

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B sharp sounds like C but is not written as C. Since the scale has six notes and there are seven letters, one letter is left over along the way. The way out is to swap one of the notes for its enharmonic (Med 1996, ch. XXXV), and the most common spelling trades A sharp for B flat:

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Between G sharp and B flat there is a diminished third, which sounds exactly like a whole tone. It is a scale whose symmetry in sound does not fit the asymmetry of the writing.

Only two whole-tone scales

Start the whole-tone scale on D: D, E, F sharp, G sharp, A sharp, C. Those are the same notes as the scale that started on C. Start on E, on F sharp, on any of its notes, and the result is always the same set of six sounds.

To get new notes you have to start on a note that was not there, such as D flat:

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And that second scale holds all six notes the first one lacked. So, technically, there are only two whole-tone scales, one through C and one through D flat, and every other one is one of these two started on another note (Med 1996, ch. XXXV). Blatter counts the same way: two whole-tone scales, one including C and the other including C sharp (Blatter 2007, part 2, art. 10).

This is the property that defines symmetric scales. A major scale can be transposed to twelve different keys. A symmetric scale runs out sooner, because transposing it by the repeated interval gives back the same notes.

A scale without a tonic

The same symmetry takes away the sense of a centre. With no semitone, no note pulls towards another, and every note has the same surroundings. Med calls the whole-tone scale "non-functional", because in it you feel no tonic, subdominant and dominant; he also notes that its ascending and descending forms are the same (Med 1996, ch. XXXV).

In music, what decides the tonic of a symmetric scale is the composer, with the bass, repetition and rhythm. The scale on its own decides nothing, and that suspended, groundless sound is exactly what draws people to it.

The octatonic scale

Alternating semitone and tone, the octave closes in eight notes. This is the octatonic scale, which Blatter presents in two forms: one starts with the semitone, the other with the tone, and so the two make a complementary pair (Blatter 2007, part 2, art. 10).

Starting with the semitone:

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Starting with the tone:

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The semitone-tone pattern repeats four times, and each repetition spans a minor third. That is why the octatonic scale holds two interlocking diminished chords, and why jazz also calls it the diminished scale. How many different octatonic scales are there? Transposing by a minor third gives back the same notes, and there are only three minor thirds before you are back where you started: C, C sharp and D. So there are three octatonic scales, against two whole-tone scales and twelve major ones.

Rimsky-Korsakov and Stravinsky used the octatonic scale a great deal, and in jazz it is the standard scale over the diminished chord and over the dominant chord with a flat ninth.

Med's mathematical scales

Med gathers scales of this kind under a curious name: mathematical scales, combinations of tones and semitones fixed in advance and repeated (Med 1996, ch. XXXV). The octatonic is one of them: semitone and tone, repeated to form four equal groups of three notes. Another alternates tone, semitone and semitone, forming three equal groups of four notes (Med 1996, ch. XXXV):

Hear the passage

The pattern spans a major third and starts again. Three groups, three major thirds, one octave.

The modes of limited transposition

Med matches each of these scales with a mode of the French composer Olivier Messiaen: the whole-tone scale is the first mode, the octatonic the second, the one of three four-note groups the third, and there are four more, the fourth to the seventh (Med 1996, ch. XXXV).

Messiaen, who described them in a 1944 book on his compositional technique, called them modes of limited transposition, and the name states exactly the property seen above: each one runs out before twelve transpositions.

Messiaen mode	Scale	Different transpositions
1st	whole tone	2
2nd	octatonic	3
3rd	tone, semitone, semitone	4
4th to 7th	patterns that repeat every tritone	6

The chromatic scale, with a single semitone repeated, is the extreme case: it has one transposition, itself.

What comes next

Symmetric scales are where the theory of scales meets twentieth-century harmony. The same reasoning, splitting the octave evenly and seeing what the symmetry does to the tonic, returns in diminished and augmented chords and in music that leaves tonality behind.