Messiaen's Modes of Limited Transposition
Put simply, a mode of limited transposition is a special kind of twelvenote 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 thirtyseven 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 wholetone scale is the only mode that is twicetransposable.
 There are three modes which are thricetransposable.
 There are six modes that are fourtimes transposable.
 There are twentyseven modes that are sixtimes 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 tritone interval, and only four which do not have a tritone 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 tritone away.
The following is an enumeration of the limited modes of transposition with C as the beginning note and using sharps for accidentals:
Two Transpositions 
Scale  Intervallic Significance 
1. C D E F G A  Whole tone scale 
Three Transpositions 
Scale  Intervallic Significance 
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, . . . 
Four Transpositions 
Scale  Intervallic Significance 
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 
Six Transpositions

Scale  Intervallic Significance 
1. C F  Tritone 
2. C C F G  Two tritones m2 apart 
3. C D F G  Two tritones M2 apart 
4. C C D F G G  Three tritones stacked m2 apart 
5. C C D F G A  Three tritones stacked m2, then M2 apart 
6. C D D F G A  Three tritones stacked M2, then m2 apart 
7. C C D D F G G A  Four tritones stacked m2 apart 
8. C E F A  Two tritones M3 apart 
9. C C E F G A  Three tritones stacked m2, then m3 apart 
10. C C D E F G G A  Four tritones stacked m2, m2, then M2 apart 
11. C D E F A A  Three tritones stacked m3, then m2 apart 
12. C D D E F G A A  Four tritones stacked M2, m2, then m2 apart 
13. C D F F G B  Three tritones stacked M2, then m3 apart 
14. C C D D E F G G A A  Five tritones stacked m2 apart or tritone inverse 
15. C F F B  Two tritones P4 apart 
16. C C F F G B  Three tritones stacked m2, then M3 apart 
17. C C D F F G G B  Four tritones stacked m2, m2, then m3 apart 
18. C D F F A B  Three tritones stacked m3, then M2 apart 
19. C C D F F G A B  Four tritones stacked m2, M2, then m3 apart 
20. C C D D F F G G A B  Tritone inverse 
21. C E F F A B  Three tritones stacked M3, the m2 apart 
22. C C E F F G A B  Four tritones stacked m2, m3, then m2 apart 
23. C D E F F G A B  Four tritones stacked M2, M2, then m2 apart 
24. C C D E F F G G A B  Tritone inverse 
25. C D E F F A A B  Four tritones stacked m3, m2, then m2 apart 
26. C C D E F F G A A B  Tritone inverse 
27. C D D E F F G A A B  Tritone inverse 
