... of limited transposition. This is part 2 of the nested series, Theory in Abstraction where we talk about cool theoretical concepts without focusing on how the concepts are applied.

The modes of limited transposition are a set of scales or modes that were devised by Olivier Messiaen (mess-yah(n) ah with a nasal n). As the name suggests, these are scales that have a limited number of transpositions, which in context means fewer than 12. The diatonic scale for example is not a mode of limited transposition because for every semitone transposition, of which there are obviously 12, a unique set of pitches is obtained. Hence why there are twelve keys. Messiaen's modes are different in that they can only be transposed a limited number of times before the pitches are all the same. To demonstrate let's take Mode 1:

C D E F# G# A# C

If we transpose it up a semitone we get:

C# D# F G A B C#

This is a unique set of pitches, but if we transpose it up again see what happens:

D E F# G# A# C D

These are all the same pitches as in the first transposition, just starting on a different note. Mode 1 has exactly two transpositions because any other transpositions would just be repeats of one of these two.

Mode 1 is also known as the whole tone scale. Mode 2 is also know as the octatonic scale or half-whole or whole-half or diminished scale. There are a total of seven modes and the rest don't have names.

Mode 1 (whole tone scale) has 2 transpositions.

Mode 2 (octatonic scale) has 3 transpositions.

Mode 3 has 4 transpositions.

Mode 4 and all the remaining modes have 6 transpositions.

Mode 5

Mode 6

Mode 7

These are all the modes that Messiaen defined, but in reality there are more. For example the tritone, C-Gb is a mode of limited transposition because it only has six transpositions (in this way it's related to modes 4-7), the augmented triad, C-E-G#, has only four transpositions (related to mode 3), the diminished seventh chord, C-E-Gb-Bbb, has only three transpositions (related to mode 2), etc.

There is a fundamental element of symmetry or sameness in all MOLT. The tritone is a symmetrical interval because it isn't invertible, the augmented triad and diminished seventh chords are stacks of the same interval (major third and minor third respectively) the whole tone scale is just a stack of major seconds, the octatonic scale is an alternation of whole steps and half steps and so on. If we look at the interval series of each mode we'll see some symmetry or repetition in all of them:

Mode 1: TTTTTT
Mode 4: SSm3S-SSm3S
Mode 5: SM3S-SM3S

Two related things arise from this symmetry. The first is that the MOLT don't have tonics. They can't have tonics because there is always some point where the interval series repeats itself and so there is always at least two potential tonics in any mode. Look at Mode 6 for example. If we start on C we get the interval series TTSSTTSS and if we start on F# we get the interval series TTSSTTSS. The exact same one. So is our tonic C or F#? You can't tell.

The second thing is an extension of this idea. All modes of limited transposition also have limited modes. If we look at the diatonic scale there are unlimited modes because the scale from any of the notes results in a unique interval series (the diatonic modes). If we look at Mode 1 though, starting on any note results in the exact same interval series (TTTTTT). With Mode 2 we get more options because by starting on different modes we can get two unique interval series out of it: TSTSTSTS or STSTSTST.

Mode 3 has 3 modes
Mode 4 has 4 modes
Mode 5 has 3 modes
Mode 6 has 4 modes
Mode 7 has 5 modes

The last thing I'll say is that although Messiaen labels 7 modes and there are in fact many more that fit his definition of a mode of limited transposition, there are only in reality two MOLT: Mode 3 and Mode 7. The rest can all be derived by "truncating" these two modes.

Get some Messiaen all up in ya

Oh. I'm okay with this. You should point out which ones are the symmetric diminished scale (both modes) and the augmented scale for clarity/pop crossover.
"There are two styles of music. Good music and bad music." -Duke Ellington

"If you really think about it, the guitar is a pointless instrument." - Robert Fripp
What a boss indeed.
"There are two styles of music. Good music and bad music." -Duke Ellington

"If you really think about it, the guitar is a pointless instrument." - Robert Fripp