Mode of limited transposition

At the end of a certain number of chromatic transpositions which varies with each mode, they are no longer transposable, giving exactly the same notes as the first.

Transposing the diatonic major scale up in semitones results in a different set of notes being used each time.

Messiaen's first mode, also called the whole-tone scale, is divided into six groups of two notes each.

Playⓘ The second mode, also called the octatonic, diminished, whole-half, or half-whole scale, is divided into four groups of three notes each.

Playⓘ Messiaen wrote, "Their series is closed, it is mathematically impossible to find others, at least in our tempered system of 12 semitones.

Indeed, all these modes can be regrouped mathematically speaking in a lattice (wherein one set of notes is connected with those containing it).

Removing alternate notes creates a new truncated mode of limited transposition.

Removing two notes for every one kept creates a new truncated mode of limited transposition.

Keeping two notes for every one removed creates another truncated mode of limited transposition.

Messiaen found ways of employing all of the modes of limited transposition harmonically, melodically, and sometimes polyphonically.

The symmetry inherent in these modes (which means no note can be perceived as the tonic), together with certain rhythmic devices, Messiaen described as containing "the charm of impossibilities".

The 12-note chromatic scale can itself be considered such a mode when viewed as a subset of a larger system that contains it, such as quarter tones or 72 equal temperament.