Octatonic scale
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In music, a diminished scale (set 8-28, 0235689e) is a scale in which the notes of the scale ascend in alternating intervals of a whole step and a half step. Because a scale constructed in such a way has eight tones (versus seven for diatonic scales), it is sometimes called an eight-tone or octatonic scale (there are other possible eight-tone scales, but the diminished is by far the most common). The latter term ("octatonic pitch collection") was first introduced by Arthur Berger in 1963 (van den Toorn 1983). The name diminished comes from the fact that the first, third, fifth, and seventh notes of a diminished scale for a key form the diminished chord for that key. The octatonic scale is, vertically, the complete alpha chord.
Because of the half-whole symmetry, there are only three distinct diminished scales, and a given diminished scale has only two modes (one beginning its ascent with a whole step between its first two notes, while the other begins its ascent with a half step or semitone). Thus Olivier Messiaen considered it one of the modes of limited transposition.
Each of the three distinct scales can form differently-named scales with the same sequence of tones by starting at a different point in the scale. With alternate starting points listed in parentheses, the three are:
- E♭ diminished (F♯/G♭, A, C diminished): E♭, F, F♯, G♯, A, B, C, D, E♭
- D diminished (F, A♭, B diminished): D, E, F, G, A♭, B♭, B, C♯, D
- D♭ diminished (E, G, B♭ diminished): D♭, E♭, E, F♯, G, A, B♭, C, D♭
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[edit] History
Formulated already by Arab musicians in the 7th century A.D., the scale was called "Zer ef Kend," meaning "string of pearls," the idea being that the two different sizes of intervals were like two different sizes of pearls (see Joseph Schillinger, The Schillinger System of Musical Composition, Vol 1).
The diminished scales may first have been used in Western music by Franz Liszt in No. 5, "Feux Follets" of his Etudes d'execution transcendante (composed 1826, and twice revised) as a recurring theme found in the descending arpeggiated figures of bars 7 and 8, 10 and 11, 43, 45 through 48, 122, and 124 through 126. In turn, all three distinct octatonic scales are used, respectively containing all, and only, the notes of each of these scales. Liszt was boldly innovative in his use of fresh scales and harmonies.
Still, Nikolai Rimsky-Korsakov, claimed the diminished scale as "his discovery" in his My Musical Life (van den Toorn 1983). Following that, it was extensively used by his student Igor Stravinsky, particularly in his major ballets Petrushka and The Rite of Spring. Other composers that experimented with the scale are Alexander Scriabin and, most often as a source set with other source sets, Béla Bartók. In Bartók's Bagatelles, Improvisations, Fourth Quartet, Cantata Profana, and Improvisations the octatonic is used with the diatonic, whole tone, and other "abstract pitch formations" (Antokoletz 1984) all "entwined...in a very complex mixture." Bartók makes use of the notes of one particular octatonic scale (E♭ diminished) exclusively in "Crossed Hands" (no. 99, vol. 4, Mikrokosmos); incidentally, this piece uses unusual, non-standard key signatures, which are different in each hand. Bartók also uses the entire octatonic collection to the exclusion of other scales in his "Diminished Fifth" (no. 101, vol. 4, Mikrokosmos) and "Harvest Song" (no. 33 of the Forty-Four Duos for two violins) and "in each piece, changes of motive and phrase correspond to changes from one of the three octatonic scales to another, and one can easily select a single central and referential form of 8-28 in the context of each complete piece." However, even his larger pieces also feature "sections that are intelligible as 'octatonic music'" (Wilson 1992, p.26-27).
[edit] Harmonic implications
[edit] In Jazz
Both the true diminished and its partner mode (with a semitone rather than a tone beginning the pattern) are commonly used in Jazz improvisation, frequently under different names. The true diminished is often called the whole-half diminished scale, while its partner is known as the half-whole diminished scale, so named for the first two intervals in their construction. The whole-half diminished scale is commonly used in conjunction with diminished harmony (e.g. the "C dim" harmony) while the half-whole scale is used in dominant harmony (e.g. with a "G7♭9" harmony.) In more advanced improvisation, the scale may be used in other circumstances, for example with a minor-major chord.
[edit] The Petrushka chord
Igor Stravinsky's ballet Petrushka is characterized by the so-called Petrushka chord. This is likely another application of one of Stravinsky's favorite devices, the diminished or octatonic scale, as both the C major and F♯ major triads chosen are obtainable from a single permutation of that scale.
[edit] Bitonality
In both of the short works by Bartók mentioned above ("Diminished Fifth" and "Harvest Song") the octatonic collection is partitioned into two (symmetrical) four-note segments (4-10 or 0235) of the natural minor scales a tritone apart. Paul Wilson argues against viewing this as bitonality since "the larger octatonic collection embraces and supports both supposed tonalities." (ibid, p.27)
[edit] Triads
As mentioned above in the context of Stravinsky's Petrushka chord, both the C major and F♯ major triads are obtainable from a single permutation of the diminished scale. In fact eight major and minor triads can be obtained from each permutation of the scale. If one takes the D♭ diminished scale as outlined above, one can produce the following triads:
- C Major (C E G)
- C Minor (C E♭ G)
- E♭ Major (E♭ G B♭)
- E♭ Minor (E♭ G♭ B♭)
- F♯ Major (F♯ A♯ C♯)
- F♯ Minor (F♯ A C♯)
- A Major (A C♯ E)
- A Minor (A C E)
This is of particular interest to jazz musicians as it facilitates the creation of chord voicings, especially polychord and upper structure voicings, and triad-based melodic improvisation.
[edit] Sources
- Berger, Arthur (1963). "Problems of Pitch Organization in Stravinsky". Perspectives of New Music II/I (Fall-Winter)
- Van den Toorn, Pieter (1983). The Music of Igor Stravinsky. New Haven: Yale University Press.
- Wilson, Paul (1992). The Music of Béla Bartók. ISBN 0-300-05111-5.
- Antokoletz, Elliott (1984). The Music of Béla Bartók: A Study of Tonality and Progression in Twentieth-Century Music. Berkeley and Los Angeles: University of California Press. Cited in Wilson directly above. ISBN 0-520-06747-9
[edit] Further reading
- Taruskin, Richard (Spring 1985). "Chernomor to Kashchei: Harmonic Sorcery; or Stravinsky's 'Angle'", Journal of the American Musicological Society 38:1, p. 74–142.
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