Here is my final Word on the Modes of Limited Transposition in the work of Messaien.

This link gives an exhaustive list of the possibilities for "Modes of Limited Transposition":

http://williamandrewburnson.com/media/M ... ition.htmlThe author mathematically derives the 37 possibilities as he defines them. Analyzing each one by its own sequence of intervals, we see that many of them are modally equivalent... eg 1-1-4-1-1-4 is modally equivalent to 1-4-1-1-4-1 and to 4-1-1-4-1-1 since they follow the same sequence but starting at a different place in the sequence.

Because the repeating unit (1-1-4) repeats twice, we say this gives two "Axes of Symmetry".

Because the repeating unit has three members, we say this gives three "Modes".

Because the total of these numbers 1+1+4=6, we say this gives six possible Transpositions.

When we reduce the 37 possibilities by their modal equivalents, we get a remaining group of 15 items. One of them, sequence 3-2-1-3-2-1, is the reverse of another 1-2-3-1-2-3. So let us discount that as not being meaningfully different, leaving us with a group of 14 items. These are the SAME as the 14 items shown in the Lattice discussed previously.

As to which ones are qualified to be considered Modes by Messaien, we exclude those with only two or four members as being too simple or too far reduced. This leaves nine remaining items. We can exclude another, sequence 1-2-3-1-2-3 (as well as its reverse) because there are three different kinds of step. Messaien clearly prefers to use only those Modes with one or two kinds of step.

The remaining eight items contain the seven Modes of Messaien, plus the sequence 1-3-1-3-1-3. On what basis did Messaien choose to exclude this potential Mode from any further consideration?

I believe the answer lies in the Tritone (Augmented Fourth or Diminished Fifth). Every Mode of Messaien does contain the Tritone, and more specifically if every Mode is written as beginning on C, there is an F# also included in that Mode.

The sequence 1-3-1-3-1-3 can be written as beginning on C, but then does NOT contain an F#. In fact, there is NO Tritone included in that sequence on ANY of its notes.

Looking around on-line, it has been suggested that Messaien uses the Tritone as a kind of Dominant. If this is so, then he surely would not wish to include a Mode in which this centrally-important Interval simply does not exist.

In further discussion on-line, it has been stated that Messaien did not actually use all of his seven Modes. When he does us them, it is primarily for a "Coloristic" effect. But for three of his own Modes, he never even discussed what color effects he believed they could produce.

It was stated that Messaien's most commonly used Mode was his Mode III, which stands apart from the other Modes on the Lattice because it is the only one containing exactly three Axes of Symmetry. Also, it is the only one containing exactly four unique Transpositions. And as it happens, it is the only one containing nine different notes.

The other Modes which Messaien used regularly were his Modes II, IV, and VI. We note that all four of these preferred modes have either eight or nine members. In effect excluding his Mode VII (with ten members) and his Modes I and V (with only six members each).

For further discussion, see:

http://www.geocities.ws/jeharris56/chapter05.pdf