Octatonic scale

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Octatonic scales on C About this sound Play .

An octatonic scale is any eight-note musical scale. The scale most often meant by this term is one 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 (or octatonic collection), although there are forty-two other non-enharmonically equivalent, non-transpositionally equivalent eight-tone sets possible. In St. Petersburg at the turn of the 20th century, this scale had become so familiar in the circle of composers around Nikolai Rimsky-Korsakov that it was referred to there as the Korsakovian scale (Корсаковская гамма) (Taruskin 1985, 132). As early as 1911 the Russian theorist Boleslav Yavorsky described this collection of pitches as the diminished mode (уменьшенный лад), because of the stable way the diminished fifth functions in it (Taruskin 1985, 111–13, citing Yavorsky 1911), and in jazz theory it is called the diminished scale (Campbell 2001, 126), or symmetric diminished scale (Hatfield 2005, 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. In more recent Russian theory the term "octatonic" is not used. Instead this scale is placed among other symmetrical modes (total 11) under its historical name Rimsky-Korsakov scale, or Rimsky-Korsakov mode (Kholopov 1982, 30; Kholopov 2003, 227). Because it was associated in the early 20th century with the Dutch composer Willem Pijper, in the Netherlands it is called the Pijper scale (Taruskin 1985, 73).

The earliest systematic treatment of the octatonic scale was Edmond de Polignac's unpublished treatise, "Etude sur les successions alternantes de tons et demi-tons (Et sur la gamme dite majeure-mineure)" from c. 1879 (Kahan 2009,[page needed]), which preceded Vito Frazzi's Scale alternate per pianoforte of 1930 by a full half-century (Sanguinetti 1993,[page needed]). The term octatonic pitch collection was first introduced into English by Arthur Berger in 1963 (Van den Toorn 1983,[not in citation given]).

Construction and enumeration[edit]

The twelve tones of the chromatic scale are covered by three disjoint diminished seventh chords. The notes from exactly two such seventh-chords combination form an octatonic collection. Because there are exactly three ways to select two from three, there are exactly three octatonic scales in the twelve-tone system.

Each octatonic scale has exactly two modes: the first begins its ascent with a whole step between its first two notes, while the second begins its ascent with a half step (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, 109).

The three octatonic collections are transpositionally and inversionally symmetric—they are related by a variety of transposition and inversion operations:

They are each closed under transpositions by 3, 6, or 9 semitones. A transposition by 1, 4, 7, or 10 semitones will transform the Eb scale into the Db scale, the Db scale into the D scale, and the D scale into the Eb scale. Conversely, transpositions by 2, 5, 8, or 11 semitones acts in the reverse way; the Eb scale goes to the D scale, D to Db and Db to Eb. Thus, the set of transpositions acts on the set of diminished collections as the integers modulo 3. If the transposition is congruent to 0 mod 3 the pitch collection is unchanged and the transpositions by 1 semitone or by 2 semitones are inverse to one another.[original research?]

The Eb and Db collections can be swapped by inversions around Eb, F#, A or C (the tones common to both scales). Similarly, the Db and D collections can be swapped by inversions around E, G, Bb, Db/C# and the D and Eb collections by inversions around D, F, Ab, or B. All other transformations do not change the classes (e.g. reflecting the Eb collection around E gives the Eb collection once again). This unfortunately means that the inversions do not act as a simple cyclic group on the set of diminished scales.[original research?]

Properties[edit]

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 1991,[page needed]). 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 the first four notes of four minor scales separated by minor thirds. For Example: C, D, E, F and (enharmonically) F, G, A, B. Also E, F, G, A, and A, B, C, D.

The scale "allows familiar harmonic and linear configurations such as triads and modal tetrachords to be juxtaposed unusually but within a rational framework" though the relation of the diatonic scale to the melodic and harmonic surface is thus generally oblique (Pople 1991, 2).

History[edit]

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 (Schillinger n.d.,[page needed])

Octatonic scales first occurred in Western music as byproducts of a series of minor-third transpositions. Agmon (1990,[page needed]) 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 (Langlé 1797, 72, ex. 25.2).

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 (1983) catalogues many octatonic moments in Stravinsky's music. 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, Fourth Quartet, Cantata Profana, and Improvisations, the octatonic is used with the diatonic, whole tone, and other "abstract pitch formations" all "entwined...in a very complex mixture" (Antokoletz 1984,[page needed]). 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, 26–27).

Twentieth-century composers who used octatonic collections include Samuel Barber, Béla Bartók, Ernest Bloch, Benjamin Britten, Julian Cochran, George Crumb, Claude Debussy, Irving Fine, Ross Lee Finney, Alberto Ginastera, John Harbison, Aram Khatchaturian, Witold Lutosławski, Olivier Messiaen, Darius Milhaud, Henri Dutilleux, Robert Morris, Carl Orff, Jean Papineau-Couture, Krzysztof Penderecki, Francis Poulenc, Sergei Prokofiev, Maurice Ravel, Alexander Scriabin, Dmitri Shostakovich, Igor Stravinsky, Toru Takemitsu, Joan Tower (Alegant 2010, 109), Robert Xavier Rodriguez, and Frank Zappa (Clement 2009, 214). Other composers include Willem Pijper (Chan 2005, 52), who may have inferred the collection from Stravinsky's Rite of Spring, which he greatly admired, and composed at least one piece—his Piano Sonatina No. 2—entirely in the octatonic system (van den Toorn 1983, 464n11).

In the 1920s Heinrich Schenker criticized the use of the octatonic scale, specifically Stravinsky's Concerto for Piano and Wind Instruments, for the oblique relation between the diatonic scale and the harmonic and melodic surface (Pople 1991, 2).

Harmonic implications[edit]

Jazz[edit]

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 "G79" chord).

Petrushka chord[edit]

Petrushka chord in piano during the second tableau of Petrushka (Taruskin 1987, 269) (About this sound Play )

Bitonality[edit]

In Bartók’s piano piece "Diminished Fifth," from Mikrokosmos, octatonic collections form the basis of the pitch content. In mm. 1–11, all eight pitch classes from the E diminished scale (see Construction and Enumeration section) appear. In mm 1–4, the pitch classes A, B, C, and D appear in the right hand, and the pitch classes E, F, G, and A appear in the left hand. The collection in the right hand outlines the first four notes of an A minor scale, and the collection in the left hand outlines the first four notes of an E minor scale. In mm. 5–11, the left and right hand switch—the A minor tetrachord appears in the left hand, and the E minor tetrachord appears in the right hand.[original research?]

From this, one can see that Bartók has partitioned the octatonic collection into two (symmetrical) four-note segments 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" (Wilson 1992, 27).

Bartók also utilizes the two other octatonic collections so that all three possible octatonic collections are found throughout this piece (D, D, and E). In mm. 12–18 all eight pitch classes from the D octatonic collection are present. The E octatonic collection from mm. 1–11 is related to this D octatonic collection by the following transposition operations: T, T4, T7, T10. In mm. 26–29, all eight pitch classes from the D octatonic collection appear. This collection is related to the E octatonic collection from mm. 1–11 by the following transposition operations: T2, T5, T8, T11.[original research?]

Other relevant features of the piece include the groups of three notes taken from the whole-half diminished scale in mm. 12–18. In these measures, the right hand features D, E, and G, the tetrachord without the 3rd (F). The left hand has the same tetrachord transposed down a tritone (G, A, C). In mm. 16, both hands transpose down three semitones to B, C, E and E, G, A respectively. Later on, in mm. 20, the right hand moves on to A- and the left back to E-. After repeating the structure of mm. 12–19 in mm. 29–34 the piece ends with the treble part returning to A- and the bass part returning to E.[original research?]

Triads[edit]

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 D whole/half (and the C half whole) diminished scale supports two diminished triads, C, E, G, and D, F, Adouble flat(or G), both of which can be inverted three times to produce four diminished triads as well.

  • C dim (C E G)
  • E dim (E G Bdouble flat)
  • G dim (G Bdouble flat C)
  • Bdouble flat dim (Bdouble flat C E)

and

  • D dim (D F Adouble flat)
  • F dim (F Adouble flat(G) B)
  • G dim (Adouble flat(G) B D)
  • B dim (B D F)

This is of particular interest to jazz musicians as it facilitates the creation of chord voicings, especially polychords (e.g. C/G or A/E) as expressing a dominant function on B7, F7, A7 and C7, 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[edit]

Two diminished seventh chords in the octatonic scale (one red, one blue) may be rearranged as the alpha chord About this sound Play .

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, 7).

One of the most important subsets of the alpha collection, the alpha chord (such as E–G–C–E About this sound Play ; 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). (Wilson 1992, p. 9) The number of semitones in the interval array of the alpha chord corresponds to the Fibonacci sequence (Slayton 2010, 15).

See also[edit]

Sources[edit]

  • Agmon, Eytan (1990). "Equal Divisions of the Octave in a Scarlatti Sonata." In Theory Only 11, no. 5:[page needed].
  • Alegant, Brian (2010). The Twelve-Tone Music of Luigi Dallapiccola.[full citation needed] ISBN 978-1-58046-325-6.
  • 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.[where?] 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, no. 1:[page needed].
  • Berger, Arthur (1963). "Problems of Pitch Organization in Stravinsky". Perspectives of New Music 2, no. 1 (Fall–Winter):[page needed].
  • Campbell, Gary (2001). Triad Pairs for Jazz: Practice and Application for the Jazz Improvisor.[full citation needed] ISBN 0-7579-0357-6.
  • Chan, Hing-yan (2005). New Music in China and the C.C. Liu Collection at the University of Hong Kong.[full citation needed] ISBN 9789622097728.
  • Cohn, Richard (1991). "Bartók's Octatonic Strategies: A Motivic Approach." Journal of the American Musicological Society 44[full citation needed].
  • Frazzi, Vito (1930). Scale alternate per pianoforte con diteggiature di Ernesto Consolo. Florence: A. Forlivesi.
  • Hatfield, Ken (2005). Mel Bay Jazz and the Classical Guitar Theory and Applications.[full citation needed] ISBN 0-7866-7236-6.
  • 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 
  • Keeling, Andrew (2007). King Crimson: Red: An Analysis by Andrew Keeling.[unreliable source]
  • Kholopov, Yuri (2003). Harmony. Theoretical Course. Orig. title: Гармония. Теоретический курс. Moscow: Lan'. [full citation needed]
  • Kholopov, Yuri (1982). "Modal harmony. Modality as a type of harmonic structure". Art of Music. General Questions of Music Theory and Aesthetics: 16–31; Orig. title: Модальная гармония: Модальность как тип гармонической структуры // Музыкальное искусство. Общие вопросы теории и эстетики музыки. Ташкент: Издательство литературы и искусства им. Г. Гуляма[full citation needed]
  • Langlé, Honoré François Marie (1797). Traité d'harmonie et de modulation. Paris: Boyer.
  • Lendvai, Ernő (1971). Béla Bartók: An Analysis of his Music. introd. by Alan Bush. London: Kahn & Averill. ISBN 0-900707-04-6. OCLC 240301.  Cited in Wilson (1992).
  • Pople, Anthony (1991). Berg: Violin Concerto. ISBN 0-521-39976-9.
  • Sanguinetti, Giorgio (1993). "Il primo studio teorico sulle scale octatoniche: Le 'scale alternate' di Vito Frazzi." Studi Musicali 22, no. 2:[page needed]
  • Schillinger, Joseph (n.d.) The Schillinger System of Musical Composition, Vol. 1.[full citation needed]
  • Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts.[full citation needed] ISBN 978-1-58046-270-9.
  • Slayton, Michael K. (2010). Women of Influence in Contemporary Music: Nine American Composers.[full citation needed]ISBN 9780810877481.
  • Taruskin, Richard (1985). "Chernomor to Kashchei: Harmonic Sorcery; or, Stravinsky's 'Angle'". Journal of the American Musicological Society 38, no. 1 (Spring): 72–142.
  • Taruskin, Richard (1987). "Chez Pétrouchka- Harmony and Tonality "chez" Stravinsky". 19th-Century Music 10, no. 3 (Spring, Special Issue: Resolutions I): 265–86.
  • Tymoczko, Dmitri (2002). "Stravinsky and the Octatonic: A Reconsideration". Music Theory Spectrum 24, no. 1 (Spring): 68–102.
  • Van den Toorn, Pieter (1983). The Music of Igor Stravinsky. New Haven and London: Yale University Press.
  • Wollner, Fritz (1924) "7 mysteries of Stravinsky in Progression" 1924 German international school of music study.[full citation needed]
  • Wilson, Paul (1992). The Music of Béla Bartók.[full citation needed]ISBN 0-300-05111-5.
  • Yavorsky, Boleslav Leopoldovich (1911). "Neskol'ko myslei v sviazi s iubileem Frantsa Lista". Muzyka no. 45 (8 October): 961. Cited in Taruskin (1985, 113).