Messiaen's Modes of Limited Transposition
Put simply, a mode of limited transposition is a special kind of twelve-note scale in which there are fewer than twelve unique transpositions of the scale. This occurs only with special scales. For example a Major scale is not a mode of limited transposition because it can be transposed eleven more times to D
Major, D Major, and so on. A whole tone scale is a limited mode of transposition since it has only two unique transpositions (the starting note of the scale does not matter here).
A little background information on the distribution of these scales:
- There are 2,048 (2 to the 11th power) possible pitch collections which begin on the note C.
- There are thirty-seven modes of limited transposition which can only be transposed twice, three times, four times, or six times. The chromatic scale can not be transposed and the remaining 2,010 pitch collections are transposable twelve times.
- The modes of limited transposition account for 1.8% of all possible pitch collections.
- The whole-tone scale is the only mode that is twice-transposable.
- There are three modes which are thrice-transposable.
- There are six modes that are four-times transposable.
- There are twenty-seven modes that are six-times transposable.
- For any mode that is transposable three, four, or six times, a second unique mode can be derived by taking the mode's inverse (remove the existing notes and add the absent notes).
- In the thirty seven modes, there are only three modes which do not contain the tri-tone interval, and only four which do not have a tri-tone above the root of the mode (F in our case) all of which are in the modes of four transpositions.
- In the modes of three transpositions, every note is part of a diminished seventh chord.
- In the modes of four transpositions, every note is part of an augmented fifth chord.
- In the modes of six tranposition, every note is paired with another note a tri-tone away.
The following is an enumeration of the limited modes of transposition with C as the beginning note and using sharps for accidentals:
|1. C D E F G A||Whole tone scale|
|1. C D F A||Diminished seventh chord|
|2. C C D E F G A A||Octatonic scale m2, M2, . . .|
|3. C D D F F G A B||Octatonic scale, reverse order M2, m2, . . .|
|1. C E G||Augmented fifth chord|
|2. C C E F G A||Sextonic m2, m3, . . . or two augmented fifth chords m2 apart|
|3. C C D E F F G A A||Inverse of augmented fifth chord|
|4. C D E G G B||Sextonic m3, m2, . . . or two augmented fifth chords m3 apart|
|5. C C D E F G G A B||Inverse of augmented fifth chord|
|6. C D D E F G G A B||Inverse of augmented fifth chord|
|1. C F||Tri-tone|
|2. C C F G||Two tri-tones m2 apart|
|3. C D F G||Two tri-tones M2 apart|
|4. C C D F G G||Three tri-tones stacked m2 apart|
|5. C C D F G A||Three tri-tones stacked m2, then M2 apart|
|6. C D D F G A||Three tri-tones stacked M2, then m2 apart|
|7. C C D D F G G A||Four tri-tones stacked m2 apart|
|8. C E F A||Two tri-tones M3 apart|
|9. C C E F G A||Three tri-tones stacked m2, then m3 apart|
|10. C C D E F G G A||Four tri-tones stacked m2, m2, then M2 apart|
|11. C D E F A A||Three tri-tones stacked m3, then m2 apart|
|12. C D D E F G A A||Four tri-tones stacked M2, m2, then m2 apart|
|13. C D F F G B||Three tri-tones stacked M2, then m3 apart|
|14. C C D D E F G G A A||Five tri-tones stacked m2 apart or tri-tone inverse|
|15. C F F B||Two tri-tones P4 apart|
|16. C C F F G B||Three tri-tones stacked m2, then M3 apart|
|17. C C D F F G G B||Four tri-tones stacked m2, m2, then m3 apart|
|18. C D F F A B||Three tri-tones stacked m3, then M2 apart|
|19. C C D F F G A B||Four tri-tones stacked m2, M2, then m3 apart|
|20. C C D D F F G G A B||Tri-tone inverse|
|21. C E F F A B||Three tri-tones stacked M3, the m2 apart|
|22. C C E F F G A B||Four tri-tones stacked m2, m3, then m2 apart|
|23. C D E F F G A B||Four tri-tones stacked M2, M2, then m2 apart|
|24. C C D E F F G G A B||Tri-tone inverse|
|25. C D E F F A A B||Four tri-tones stacked m3, m2, then m2 apart|
|26. C C D E F F G A A B||Tri-tone inverse|
|27. C D D E F F G A A B||Tri-tone inverse|