Octatonic scale

Octatonic scales on C  Play (help·info).

An octatonic scale is any eight-note musical scale. Among the most famous of these is a scale in which the notes ascend in alternating intervals of a whole step and a half step, creating a symmetric scale. In classical theory, in contradistinction to jazz theory, this scale is commonly simply called the octatonic scale, although there are forty-two other non-enharmonically equivalent, non-transpositionally equivalent eight-tone sets possible. In jazz theory this scale is more particularly called the diminished scale (Campbell 2001, p. 126), or symmetric diminished scale (Hatfield 2005, p. 125), because it can be conceived as a combination of two interlocking diminished seventh chords, just as the augmented scale can be conceived as a combination of two interlocking augmented triads. The earliest systematic treatment of the octatonic scale was Edmond de Polignac's unpublished treatise, "Etude sur les successions alternantes de tons et demi-tons (Et sur la gamme dite majeure-mineure)" from c. 1879 (Kahan 2009), which preceded Vito Frazzi's Scale alternate per pianoforte of 1930 by a full half-century (Sanguinetti 1993). The term octatonic pitch collection was first introduced into English by Arthur Berger in 1963 (Van den Toorn 1983).

Properties

The twelve tones of the chromatic scale partition into three non-overlapping diminished seventh chords. A combination of any two, omitting the third, forms an octatonic collection. As there are three diminished-seventh chords that can be omitted, it follows that there are only three distinct (non-transpositionally equivalent) diminished scales in the 12-tone system. Thus Olivier Messiaen considered it one of the modes of limited transposition, specifically, the second mode. 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.

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 alternative 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♭

Diminished scales on E♭, D, and D♭, ascending.

It may also be represented as 0134679t or labeled as set class 8-28 (Schuijer 2008, p.109).

Among the collection's remarkable features is that it is the only collection that can be disassembled into four transpositionally-related pitch pairs in six different ways, each of which features a different interval class (Cohn). For example:

• semitone: (C, C♯), (D♯, E) (F♯, G), (A, B♭)
• whole step: (C♯, D♯), (E, F♯), (G, A), (B♭, C)
• minor third:(C, E♭), (F♯, A), (C♯, E), (G, B♭)
• major third:(C, E), (F♯, B♭), (E♭, G), (A, C♯)
• perfect fourth: (C♯, F♯), (B♭, E♭), (G, C), (E, A)
• tritone: (C, F♯), (E♭, A), (C♯,G), (E, B♭)

Another remarkable feature of the diminished scale is that it contains first four notes of four minor scales separated by minor thirds. For Example: C, D, Eb, F and (enharmonically) F#, G#, A, B. Also Eb, F, Gb, Ab, and A, B, C, D.

History

Joseph Schillinger suggests that the scale was formulated already by Persian traditional music in the 7th century AD, where it was called "Zar 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).

Octatonic scales first occurred in Western music as byproducts of a series of minor-third transpositions. Agmon locates one in the music of Scarlatti, from the 1730s. Langlé's 1797 harmony treatise contains a sequential progression with a descending octatonic bass, supporting harmonies that use all and only the notes of an octatonic scale (p. 72, ex. 25.2).

The question of when Western composers first began to select octatonic scales as primary compositional material is difficult to determine. One strong candidate is a recurring theme in Franz Liszt's Feux Follets, the fifth of his first book of Études d'exécution transcendente (composed 1826, and twice revised). See 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 to become an idol of the Russian school, and starting with Glinka's opera Ruslan and Lyudmila (first performed 1842) the diminished scale was often used by Russian composers to evoke scenes of magic and exotic mystery. Still, Nikolai Rimsky-Korsakov claimed the diminished scale as "his discovery" in his My Musical Life (van den Toorn 1983). He certainly used the scale extensively in his opera Kashchey the Immortal, which premiered in 1902. Following that, the scale was extensively used by his student Igor Stravinsky, particularly in his Russian period ballets Petrushka and The Rite of Spring dating from 1911 and 1913 respectively. Van den Toorn catalogues many octatonic moments in Stravinsky's music, although Tymoczko argues that many of them are byproducts of other syntactic concerns. The scale also may be found with some frequency in music of Claude Debussy, Maurice Ravel, Alexander Scriabin and 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." Mikrokosmos 99, 101, and 109 are octatonic pieces, as is no. 33 of the Forty-Four Duos for two violins. "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).

Post-war composers who make significant use of the scale include Luigi Dallapiccola, Olivier Messiaen, Toru Takemitsu, and George Crumb. The Progressive Metal band Dream Theater also made use of the Octatonic scale in their 2005 song Octavarium.

Harmonic implications

Jazz

Both the half-whole diminished and its partner mode, the whole-half diminished (with a tone rather than a semitone beginning the pattern) are commonly used in jazz improvisation, frequently under different names. The whole-half diminished scale is commonly used in conjunction with diminished harmony (e.g., the "C dim7" chord) while the half-whole scale is used in dominant harmony (e.g., with a "G7♭9" chord).

Petrushka chord

Igor Stravinsky's ballet Petrushka is characterized by the so-called Petrushka chord, which combines major triads transpositionally related by a tritone. Taruskin's ascription of explanatory power to the chord's status as an octatonic subset has been challenged by Tymoczko.

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)

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 a unique aspect of the diminished scale when compared to all other octotonic scales is the presence of four major and four minor triads. 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)

The Db whole/half (and the C half whole) diminished scale supports two diminished triads, C, Eb, Gb, and Db, Fb, Abb(or G), both of which can be inverted three times to produce four diminished triads as well.

• C dim (C Eb Gb)
• Eb dim (Eb Gb Bbb)
• Gb dim (Gb Bbb C)
• Bbb dim (Bbb C Eb)

and

• Db dim (Db Fb Abb)
• Fb dim (Fb Abb(G) Bb)
• G dim (Abb(G) Bb Db)
• Bb dim (Bb Db Fb)

This is of particular interest to jazz musicians as it facilitates the creation of chord voicings, especially polychords (eg C/Gb or A/Eb) as expressing a dominant function on B7, F7, Ab7 and Cb7, and upper structure voicings on dominant seventh chords. The sequence of triads given above is an instance of the neo-Riemannian PR cycle, alternating parallels and relatives.

Alpha chord

Two diminished seventh chords in the octatonic scale (one red, one blue) may be rearranged as the alpha chord  Play (help·info).

The alpha-chord collection is, "a vertically organized statement of the octatonic scale as two diminished seventh chords," such as: C♯–E–G–B♭–C–E♭–F♯–A. (Wilson 1992, p. 7)

One of the most important subsets of the alpha collection, the alpha chord (such as E–G–C–E♭; using the theorist Ernő Lendvai's terminology, the C alpha chord) may be considered a mistuned major chord or major/minor in first inversion (in this case, C major/minor). (ibid, p. 9)

See also

• Alpha scale
• Beta scale
• Delta scale
• Gamma scale

Sources

• Agmon, Eytan (1990). "Equal Divisions of the Octave in a Scarlatti Sonata." In Theory Only 11/5.
• 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.
• Baur, Steven (1999). "Ravel's 'Russian' Period: Octatonicism in His Early Works, 1893–1908." Journal of the American Musicological Society 52.1
• Berger, Arthur (1963). "Problems of Pitch Organization in Stravinsky". Perspectives of New Music II/I (Autumn/Winter).
• Campbell, Gary (2001). Triad Pairs for Jazz: Practice and Application for the Jazz Improvisor. ISBN 0757903576.
• Cohn, Richard (1991). "Bartók's Octatonic Strategies.: A Motivic Approach." "Journal of the American Musicological Society 44.
• Frazzi, Vito (1930). Scale alternate per pianoforte con diteggiature di Ernesto Consolo (Forlivesi).
• Hatfield, Ken (2005). Mel Bay Jazz and the Classical Guitar Theory and Applications. ISBN 0786672366.
• Kahan, Sylvia (2009). In Search of New Scales: Prince Edmond de Polignac, Octatonic Explorer. Rochester, NY: University of Rochester Press. ISBN 978-1-58046-305-8.
• Langlé, Honoré François Marie (1797). Traité d'harmonie et de modulation. Paris: Boyer.
• Sanguinetti, Giorgio (1993). "Il primo studio teorico sulle scale octatoniche: Le 'scale alternate' di Vito Frazzi." Studi Musicali 22.2
• Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. ISBN 978-1-58046-270-9.
• Tymoczko, Dmitri (2002). "Stravinsky and the Octatonic: A Reconsideration" Music Theory Spectrum 24.1
• Van den Toorn, Pieter (1983). The Music of Igor Stravinsky. New Haven: Yale University Press.
• Wollner, Fritz (1924) "7 mysteries of Stravinsky in Progression" 1924 German international school of music study.
• Wilson, Paul (1992). The Music of Béla Bartók. ISBN 0-300-05111-5.
• Lendvai, Ernő (1971). Béla Bartók: An Analysis of his Music. introd. by Alan Bush. London: Kahn & Averill. ISBN 0900707046. OCLC 240301.  Cited in Wilson (1992).

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.