Octatonic scale

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An octatonic scale is any eight-note musical scale. Among the most famous of these is a scale in which the notes ascend in alternating intervals of a whole step and a half step, creating a symmetric scale. In classical theory, in contradistinction to jazz theory, this scale is commonly simply called the octatonic scale, although there are forty-two other non-enharmonically equivalent, non-transpositionally equivalent eight-tone sets possible. In jazz theory this scale is more particularly called the diminished scale (Campbell 2001, p.126), or symmetric diminished scale (Hatfield 2005, p.125), because it can be conceived as a combination of two interlocking diminished seventh chords, just as the augmented scale can be conceived as a combination of two interlocking augmented triads. The first systematic treatment of the octatonic scale was provided by Vito Frazzi (1930), in Scale alternate per pianoforte (Sanguinetti 1993). The term octatonic pitch collection was first introduced into English by Arthur Berger in 1963 (Van den Toorn 1983).

The twelve tones of the chromatic scale partition into three non-overlapping diminished seventh chords. A combination of any two, omitting the third, forms an octatonic collection. As there are three diminished-seventh chords that can be omitted, it follows that there are only three distinct (non-transpositionally equivalent) diminished scales in the 12-tone system. Thus Olivier Messiaen considered it one of the modes of limited transposition. A given diminished scale has only two modes (one beginning its ascent with a whole step between its first two notes, while the other begins its ascent with a half step or semitone). T.

Each of the three distinct scales can form differently named scales with the same sequence of tones by starting at a different point in the scale. With alternative starting points listed in parentheses, the three are:

E♭ diminished (F♯/G♭, A, C diminished): E♭, F, F♯, G♯, A, B, C, D, E♭
D diminished (F, A♭, B diminished): D, E, F, G, A♭, B♭, B, C♯, D
D♭ diminished (E, G, B♭ diminished): D♭, E♭, E, F♯, G, A, B♭, C, D♭

It may also be represented as 0134679t or labeled as set 8-28.

Among the collection's remarkable features is that it is the only collection that can be disassembled into four transpositionally-related pitch pairs in six different ways, each of which features a different interval class (Cohn). For example:
semitone: (C, C#), (D#, E) (F#, G), (A, Bb)
whole step: (C#,D#), (E, F#), (G, A), (Bb, C)
minor third:(C, Eb), (F#, A), (C#, E), (G, Bb)
major third:(C, E), (F#, Bb), (Eb, G), (A, C#)
perfect fourth: (C#, F#), (Bb,Eb), (G,C), (E,A)
tritone: (C, F#), (Eb,A), (C#,G), E, Bb)

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