In my opinion, Olivier Messiaen has established a quite ingenious way of building his chords. The charm of impossibilities is what encouraged him to come up with these, even mathematically, fascinating structures or groups of chords. “Their impossibility of transposition makes their strange charm. […] Their series is closed. It is mathematically impossible to find others of them.”♪
His modes of limited transposition are formed based on the current chromatic system of Western music: the tempered system of the twelve sounds that we get if we divide the octave in semitones. The last note of each group is aways the first of the following. He divides the octave (from C to C) in all possible ways: meaning semitone-tone-semitone, tone-semitone, and so on. This is what he calls limited modes of transposition since these patterns can only be transposed chromatically (starting from D flat, or D and so on) only for a finite number of times (the exact number depends on the mode): the pitch collection (in Western music theory) is symmetrically systemized and then transposed to exhaust all possibilities. By organizing the pitches and chords in such a way, Messiaen provides a new insight into the harmonic analysis of pieces that were composed even before he introduced this system.