Modes of limited transposition
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The modes of limited transposition are musical modes, which were first compiled by the French composer Olivier Messiaen.
Based on the chromatic scale of twelve notes, these modes are made up of several symmetrical groups, the last note of each group being the first note of the next. After a certain number of chromatic transpositions (i.e. up or down a semitone) each mode cannot be transposed further - the next transposition resulting in exactly the same notes as the first.
For example, the first mode contains notes C, D, E, F#, G#, A#, C; transposing this mode up a semitone gives C#, D#, F, G, A, B, C#. Transposing this up another semitone would give D, E, F#, G#, A#, C, D which is exactly what we started with.
Messiaen found ways of employing all of these modes both harmonically and melodically.
The first mode is divided into six groups of two notes each. It has a total of two transpositions. This is the whole-tone scale, quite widespread since Debussy.
The second mode, also called octatonic scale, is divided into four groups of three notes each. It has a total of three transpositions, like the diminished-seventh chord. Here it is, expressed melodically:
The third mode is divided into three groups of four notes each. It has a total of four transpositions, like the augmented triad. Here it is, expressed melodically:
Here are the fourth, fifth, sixth and seventh modes. They each have a total of six transpositions, like the tritone.
The symmetry inherent in these modes (meaning no note can be perceived as the tonic), together with certain rhythmic devices, Messiaen described as containing "the charm of impossibilities."
