Mode of limited transposition
Modes of limited transposition are musical modes or scales that fulfill specific criteria relating to their symmetry and the repetition of their interval groups. These scales may be transposed to all twelve notes of the chromatic scale, but at least two of these transpositions must result in the same pitch classes, thus their transpositions are "limited". They were compiled by the French composer Olivier Messiaen, and published in his book La technique de mon langage musical ("The Technique of my Musical Language").
Messiaen's list[edit]
Messiaen's first mode, also called the whole-tone scale, is divided into six groups of two notes each. The intervals it contains are tone, tone, tone, tone, tone, tone – it has two transpositions and one mode.
The second mode, also called the octatonic, diminished, whole-half, or half-whole scale, is divided into four groups of three notes each. It contains the intervals semitone, tone, semitone, tone, semitone, tone, semitone, tone – it has three transpositions, like the diminished 7th chord, and two modes:
The third mode is divided into three groups of four notes each. It contains the intervals tone, semitone, semitone, tone, semitone, semitone, tone, semitone, semitone – it has four transpositions, like the augmented triad, and three modes.
The fourth mode contains the intervals semitone, semitone, minor third, semitone, semitone, semitone, minor third, semitone – it has six transpositions, like the tritone, and four modes.
The fifth mode contains the intervals semitone, major third, semitone, semitone, major third, semitone – it has six transpositions, like the tritone, and three modes.
The sixth mode has the intervals tone, tone, semitone, semitone, tone, tone, semitone, semitone – it has six transpositions, like the tritone, and four modes.
The seventh mode contains the intervals semitone, semitone, semitone, tone, semitone, semitone, semitone, semitone, tone, semitone – it has six transpositions, like the tritone, and five modes.
Expansion and alteration of the modes[edit]
Are there others?[edit]
Messiaen wrote, "Their series is closed, it is mathematically impossible to find others, at least in our tempered system of 12 semitones."[1] More modes can be found that fit the criteria, but they are truncations of the original seven modes. Indeed, all these modes can be regrouped mathematically speaking in a lattice (wherein one set of notes is connected with those containing it). If one excludes modes with 0, 1, 11 or 12 notes which can be construed as too extreme, there are 15 generalized Messiaen modes which are represented on clock diagrams. Some of the ones omitted by Messiaen are arguably of musical importance, like the hexatonic collection 0 1 4 5 8 9. The augmented scale: 0 3 4 7 8 11 (Root, minor 3rd, major 3rd, 5th, augmented 5th, major 7th, or minor 3rd, semitone, minor 3rd, semitone, minor 3rd, semitone) may appear to be an inexplicable omission on Messiaen's part. It is a symmetrical scale used frequently by modern jazz improvisers. However closer inspection reveals it to be a truncated version of his Third Mode.
Use and sound[edit]
Messiaen found ways of employing all of the modes of limited transposition harmonically, melodically, and sometimes polyphonically. The whole-tone and octatonic scales have enjoyed quite widespread use since the turn of the 20th century, particularly by Debussy (the whole-tone scale) and Stravinsky (the octatonic scale).
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 composer Tōru Takemitsu made frequent use of Messiaen's modes, particularly the third mode.[2] The composer Alexander Tcherepnin has adopted the third mode as one of his synthetic scales, and it is known as the "Tcherepnin scale".[3]