Modes of limited transposition are musical modes or

scales Scale or scales may refer to: Mathematics * Scale (descriptive set theory), an object defined on a set of points * Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original * Scale factor, a number w ...

that fulfill specific criteria relating to their

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

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 Olivier Eugène Prosper Charles Messiaen (, ; ; 10 December 1908 – 27 April 1992) was a French composer, organist, and ornithologist who was one of the major composers of the 20th century. His music is rhythmically complex; harmonically ...

, and published in his book ''La technique de mon langage musical'' ("The Technique of my Musical Language").

Technical criteria

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 Diatonic and chromatic are terms in music theory that are most often used to characterize Scale (music), scales, and are also applied to musical instruments, Interval (music), intervals, Chord (music), chords, Musical note, notes, musical sty ...

major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double i ...

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 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'' sc ...

, 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 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 (in contrast to jazz theory), this symmetrical ...

, 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. The ''second mode'', also called the

octatonic An octatonic scale is any eight-Musical note, note musical scale. However, the term most often refers to the symmetric scale composed of alternating major second, whole and semitone, half steps, as shown at right. In classical theory (in contras ...

, 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

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."Messiaen, O. ''The Technique of my Musical Language'', trans. John Satterfield, p. 58. Alphonse Leduc, Paris, 1956. More modes can be found that fit the criteria, but they are truncations of the original seven modes.

Truncation

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. *C D E F G A Removing alternate notes creates a new truncated mode of limited transposition. *C E G Removing two notes for every one kept creates a new truncated mode of limited transposition. *C F Keeping two notes for every one removed creates another truncated mode of limited transposition. *C E F A 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 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 musi ...

and

Dominant seventh flat five chord In music theory, the dominant seventh flat five chord is a seventh chord composed of a Root (chord), root note, together with a major third, a diminished fifth, and a minor seventh above the root (1, 3, 5 and 7). For example, the dominant seventh f ...

* 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 In music and music theory, a hexatonic scale is a scale with six pitches or notes per octave. Famous examples include the whole-tone scale, C D E F G A C; the augmented scale, C D E G A B C; the Prometheus scale, C D E F A B C; and the blues sca ...

* 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 (Achille) Claude Debussy (; 22 August 1862 – 25 March 1918) was a French composer. He is sometimes seen as the first Impressionist composer, although he vigorously rejected the term. He was among the most influential composers of the ...

(the whole-tone scale) and

Stravinsky Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential 20th-century clas ...

(the octatonic scale). The symmetry inherent in these modes (which means no note can be perceived as the tonic), together with certain

rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular recu ...

ic devices, Messiaen described as containing "the charm of impossibilities". The composer

Tōru Takemitsu was a Japanese composer and writer on aesthetics and music theory. Largely self-taught, Takemitsu was admired for the subtle manipulation of instrumental and orchestral timbre. He is known for combining elements of oriental and occidental phil ...

made frequent use of Messiaen's modes, particularly the third mode.*

In other temperaments

There are no modes of limited transposition in any

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

equal division of the octave, such as

19 equal temperament In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...

31 equal temperament In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31- EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equ ...

. Composite divisions, such as

15 equal temperament In music, 15 equal temperament, called 15-TET, 15- EDO, or 15-ET, is a tempered scale derived by dividing the octave into 15 equal steps (equal frequency ratios). Each step represents a frequency ratio of (=2), or 80 cents (). Because 15 fa ...

22 equal temperament In music, 22 equal temperament, called 22-TET, 22- EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps (equal frequency ratios). Each step represents a frequency ratio of , or 54.55 cents (). When composing wi ...

, 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 tone A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, a ...

s or 72 equal temperament. 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).

References

External links

My Messiaen Modes – A visual representation of the modes of limited transposition
{{DEFAULTSORT:Modes of limited transposition Modes (music) Post-tonal music theory Musical symmetry