Modes of limited transposition

Modes of limited transposition are musical modes or scales that fulfill specific criteria relating to their symmetry and the repetition of their interval groups. They were compiled by the French composer Olivier Messiaen, and published in his book La technique de mon langage musical ("The Technique of my Musical Language").

Based on our present chromatic system, a tempered system of 12 sounds, these modes are formed of several symmetrical groups, the last note of each group always being common with the first of the following group. At the end of a certain number of chromatic transpositions which varies with each mode, they are no longer transposable, giving exactly the same notes as the first.

There are two complementary ways to view the modes: considering their possible transpositions, and considering the different modes contained within them.

Transposing the diatonic major scale up in semitones results in a different set of notes being used each time. C major consists of C, D, E, F, G, A, B, and the scale a semitone higher (D♭ major) consists of D♭, E♭, F, G♭, A♭, B♭, C. When transposing a mode of limited transposition this is not the case. For example, the mode of limited transposition that Messiaen labelled "Mode 1", which is the whole tone scale, contains the notes C, D, E, F♯, G♯, A♯; transposing this mode up a semitone produces C♯, D♯, F, G, A, B. Transposing this up another semitone produces D, E, F♯, G♯, A♯, C. Since transposing the mode up a whole tone produces the same set of notes, mode 1 has only 2 transpositions.

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