Mode of limited transposition

Last updated

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").


Technical criteria

Based on our present chromatic system, a tempered system of 12 sounds, these modes are formed of several symmetrical groups, the last note of each group always being common with the first of the following group. 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. [1]

There are two complementary ways to view the modes: considering their possible transpositions, and considering the different modes contained within them.

Definition by chromatic transposition

Transposing the diatonic major scale up in semitones results in a different set of notes being used each time. For example, C major consists of C, D, E, F, G, A, B, and the scale a semitone higher (D major) consists of D, E, F, G, A, B, C. By transposing D major up another semitone, another new set of notes (D major) is produced, and so on, giving 12 different diatonic scales in total. When transposing a mode of limited transposition this is not the case. For example, the mode of limited transposition that Messiaen labelled "Mode 1", which is the whole tone scale, contains the notes C, D, E, F, G, A; transposing this mode up a semitone produces C, D, F, G, A, B. Transposing this up another semitone produces D, E, F, G, A, C, which is the same set of notes as the original scale. Since transposing the mode up a whole tone produces the same set of notes, mode 1 has only 2 transpositions.

Any scale having 12 different transpositions is not a mode of limited transposition.

Definition by shifting modal degrees

Consider the intervals of the major scale: tone, tone, semitone, tone, tone, tone, semitone. Starting the scale on a different degree will always create a new mode with individual interval layouts—for example starting on the second degree of a major scale gives the "Dorian mode"—tone, semitone, tone, tone, tone, semitone, tone. This is not so of the modes of limited transposition, which can be modally shifted only a limited number of times. For example, mode 1, the whole tone scale, contains the intervals tone, tone, tone, tone, tone, tone. Starting on any degree of the mode gives the same sequence of intervals, and therefore the whole tone scale has only 1 mode. Messiaen's mode 2, or the diminished scale, consists of semitone, tone, semitone, tone, semitone, tone, semitone, tone, which can only be arranged 2 ways, starting with either a tone or a semitone. Therefore mode 2 has two modes.

Any scale having the same number of modes as notes is not a mode of limited transposition.

Messiaen's list

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.

MOLT 1.png Loudspeaker.svg Play  

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:

MOLT 2.png Loudspeaker.svg Play  

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.

MOLT 3.png

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.

MOLT 4.png

The fifth mode contains the intervals semitone, major third, semitone, semitone, major third, semitone – it has six transpositions, like the tritone, and three modes.

MOLT 5.png

The sixth mode has the intervals tone, tone, semitone, semitone, tone, tone, semitone, semitone – it has six transpositions, like the tritone, and four modes.

MOLT 6.png

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.

MOLT 7.png Loudspeaker.svg Play  

Expansion and alteration of the modes

Are there others?

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.


Truncation involves the removal of notes from one of the modes to leave a new truncated mode. Both the notes removed and the notes remaining must preserve the symmetry of the parent mode, and must therefore fulfill the conditions for limited transposition. For example, consider mode 1.

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.

Only Messiaen's mode 7 and mode 3 are not truncated modes: the other modes may be constructed from them or from one or more of their modes. Mode 7 contains modes 1, 2, 4, 5, and 6. Mode 6 contains modes 1 and 5. Mode 4 contains mode 5. Mode 3 contains mode 1.

Pure intervallic truncations

  • Tritones, truncation of modes 1, 2, 3, 4, 5, 6 and 7: augmented fourth, augmented fourth – 1 mode and 6 transpositions
  • Major thirds, truncation of modes 1, 3, 6 and 7: major third, major third, major third – 1 mode and 4 transpositions. See Augmented triad
  • Minor thirds, truncation of modes 2, 4, 6 and 7: minor third, minor third, minor third, minor third – 1 mode and 3 transpositions. See Diminished seventh chord
  • Whole tones (mode 1), truncation of modes 3, 6 and 7: tone, tone, tone, tone, tone, tone – 1 mode and 2 transpositions

Other truncations

  • Truncation of modes 2, 4, 6 and 7: semitone, tone, minor third, semitone, tone, minor third – 3 modes, 6 transpositions. (Modes are "mirror" inversions of Petrushka Chord modes.)
  • Truncation of modes 1, 2, 3, 4, 5, 6 and 7: major third, tone, major third, tone – 2 modes, 6 transpositions. See French Sixth and Dominant seventh flat five chord
  • Truncation of modes 2, 3, 4, 5, 6 and 7: perfect fourth, semitone, perfect fourth, semitone – 2 modes, 6 transpositions. See 1:5 Distance model
  • Truncation of mode 3: minor third, semitone, minor third, semitone, minor third, semitone – 2 modes, 4 transpositions. See augmented scale
  • Truncation of modes 2, 4, 6 and 7: minor third, tone, semitone, minor third, tone, semitone – 3 modes, 6 transpositions. See Petrushka Chord

Use and sound

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]

In other temperaments

There are no modes of limited transposition in any prime equal division of the octave, such as 19 equal temperament or 31 equal temperament.

Composite divisions, such as 15 equal temperament or 22 equal temperament, have them. 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.

Scale step01234567
NoteCD Llpd- 1/2 .svg /C Llpd+1 1/2 .svg E/DF Llpd- 1/2 .svg /E Arabic music notation half sharp.svg G/FA Three quarter flat.svg /G Arabic music notation half sharp.svg AB Three quarter flat.svg /A Arabic music notation half sharp.svg

For example, eight equal temperament, the lowest non-prime equal temperament not completely included in 12-tet (due to a scale step in 24-tet), would have modes of limited transposition. The first would be 0, 2, 4, 6 (steps: 2222), which has only two transpositions and one mode. Another would be 0, 1, 4, 5 (steps: 1313 and 3131), which has 4 transpositions and 2 modes (the other is 0, 3, 4, 7).


  1. 1 2 Messiaen, O. The Technique of my Musical Language, trans. John Satterfield, p. 58. Alphonse Leduc, Paris, 1956.
    • Burt, Peter (2001). The Music of Toru Takemitsu. Cambridge University Press. p. 34. ISBN   0-521-78220-1.

Further reading

Related Research Articles

In music theory, a diatonic scale is any heptatonic scale that includes five whole steps and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.

In music theory, the term minor scale refers to three scale patterns – the natural minor scale, the harmonic minor scale, and the melodic minor scale – rather than just one as with the major scale.

In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. A tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half as 6 of 12 semitones or 600 of 1200 cents.

In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or hexatonic scales. A single whole tone scale can also be thought of as a "six-tone equal temperament".

An octatonic scale is any eight-note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory, this scale is commonly called the octatonic scale, although there are a total of 42 enharmonically non-equivalent, transpositionally non-equivalent eight-note sets.

A jazz scale is any musical scale used in jazz. Many "jazz scales" are common scales drawn from Western European classical music, including the diatonic, whole-tone, octatonic, and the modes of the ascending melodic minor. All of these scales were commonly used by late nineteenth and early twentieth-century composers such as Rimsky-Korsakov, Debussy, Ravel and Stravinsky, often in ways that directly anticipate jazz practice. Some jazz scales, such as the bebop scales, add additional chromatic passing tones to the familiar diatonic scales.

In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, was further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.

Chromatic circle Clock diagram for displaying relationships among pitch classes

The chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle.

Harmonic major scale

In music theory, the harmonic major scale is a musical scale found in some music from the common practice era and now used occasionally, most often in jazz. In George Russell's Lydian Chromatic Concept it is the fifth mode (V) of the Lydian Diminished scale. It corresponds to the Raga Sarasangi in Indian Carnatic music.

Quarter-comma meantone, or 14-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80), with respect to its just intonation used in Pythagorean tuning ; the result is 3/2 × 14 = 45 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

Interval vector

In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. Other names include: ic vector, PIC vector and APIC vector

Rothenberg propriety

In diatonic set theory, Rothenberg propriety is an important concept, lack of contradiction and ambiguity, in the general theory of musical scales which was introduced by David Rothenberg in a seminal series of papers in 1978. The concept was independently discovered in a more restricted context by Gerald Balzano, who termed it coherence.

Diatonic and chromatic Terms in music theory to characterize scales

Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

In music, a symmetric scale is a music scale which equally divides the octave. The concept and term appears to have been introduced by Joseph Schillinger and further developed by Nicolas Slonimsky as part of his famous Thesaurus of Scales and Melodic Patterns. In twelve-tone equal temperament, the octave can only be equally divided into two, three, four, six, or twelve parts, which consequently may be filled in by adding the same exact interval or sequence of intervals to each resulting note.

Regular diatonic tuning

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

Post-tonal music theory is the set of theories put forward to describe music written outside of, or 'after', the tonal system of the common practice period. It revolves around the idea of 'emancipating dissonance', that is, freeing the structure of music from the familiar harmonic patterns that are derived from natural overtones. As music becomes more complex, dissonance becomes indistinguishable from consonance.

Anhemitonic scale

Musicology commonly classifies scales as either hemitonic or anhemitonic. Hemitonic scales contain one or more semitones, while anhemitonic scales do not contain semitones. For example, in traditional Japanese music, the anhemitonic yo scale is contrasted with the hemitonic in scale. The simplest and most commonly used scale in the world is the atritonic anhemitonic "major" pentatonic scale. The whole tone scale is also anhemitonic.

A decatonic scale is a ten note musical scale. If the notes are ordered, a decatonic set has 3,628,800 permutations, however, in twelve tone equal temperament only six unordered ten note sets exist, 10-1—10-6: