Octatonic Scales
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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 Jazz harmony is the music theory, theory and practice of how chord (music), chords are used in jazz music. Jazz bears certain similarities to other practices in the tradition of Western harmony, such as many chord progressions, and the incorpora ...
), this symmetrical scale is commonly called the ''octatonic scale'' (or the ''octatonic collection''), although there are a total of 42 enharmonically non-equivalent, transpositionally non-equivalent eight-note sets. The earliest systematic treatment of the octatonic scale was in
Edmond de Polignac Prince Edmond Melchior Jean Marie de Polignac (19 April 18348 August 1901) was a French aristocrat and composer. Ancestry Edmond was a member of the Polignac family, one of the more illustrious families of France. His grandmother, the duchesse ...
's unpublished treatise "Étude sur les successions alternantes de tons et demi-tons (Et sur la gamme dite majeure-mineure)" (''Study of the Succession of Alternating Whole Tones and Semitones (and of the so-called Major-Minor Scale)'') from c. 1879, which preceded
Vito Frazzi Vito Frazzi (1 August 1888 – 7 July 1975) was an Italian neo-romantic composer. He was born in San Secondo Parmense, and studied at the Parma Conservatory, where he learnt composition from Guido Alberto Fano. From 1912 to 1958 he taught p ...
's ''Scale alternate per pianoforte'' of 1930 by a full half-century.


Nomenclature

In Saint Petersburg at the turn of the 20th century, this scale had become so familiar in the circle of composers around
Nikolai Rimsky-Korsakov Nikolai Andreyevich Rimsky-Korsakov . At the time, his name was spelled Николай Андреевичъ Римскій-Корсаковъ. la, Nicolaus Andreae filius Rimskij-Korsakov. The composer romanized his name as ''Nicolas Rimsk ...
that it was referred to as the ''Korsakovian scale'' (Корсаковская гамма). As early as 1911, the Russian theorist
Boleslav Yavorsky Boleslav Leopoldovich Yavorsky (russian: Болеслав Леопольдович Яворский; June 22, 1877, Kharkiv – November 26, 1942) was a Russian musicologist, music teacher, administrator and pianist. Through his teachings and e ...
described this collection of pitches as the ''diminished mode'' (уменьшённый лад), because of the stable way the diminished fifth functions in it. 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''.) In jazz theory, it is called the ''diminished scale'' or ''symmetric diminished scale'' because it can be conceived as a combination of two interlocking diminished seventh chords, just as the
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 ...
can be conceived as a combination of two interlocking augmented triads. The two modes are sometimes referred to as the ''half-step/whole step diminished scale'' and the ''whole step/half-step diminished scale''. Because it was associated in the early 20th century with the Dutch composer
Willem Pijper Willem Frederik Johannes Pijper (; 8 September 189418 March 1947) was a Dutch composer, music critic and music teacher. Pijper is considered to be among the most important Dutch composers of the first half of the 20th century. Life Pijper was b ...
, in the Netherlands, it is called the ''Pijper scale''.


Construction

The twelve tones of the chromatic scale are covered by three disjoint diminished seventh chords. The notes from two such seventh-chords combination form an octatonic collection. Because there are three ways to select two from three, there are three octatonic scales in the twelve-tone system. Each octatonic scale has exactly two
modes Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
: the first begins its ascent with a
whole step In Western culture, Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a interval (music), musical interval encompassing two adjacent staff positions ( ...
, while the second begins its ascent with a half step ( semitone). These modes are sometimes referred to as the ''whole step/half-step diminished scale'' and the ''half-step/whole step diminished scale'', respectively. 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 below in square brackets, and return to tonic in parentheses, the three are, ascending by semitones: * C D E F G A A B (C) * D E F G G A B C (D) * E F G A A B C D (E) It may also be represented as semitones, either starting with a whole tone (as above): , or starting with a semitone: , or labeled as set class 8‑28. With one more scale tone than present in the western
diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) 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, ...
, it is not possible to notate music in the octatonic scale in any conventional occidental key signature, without the use of accidentals. In any conventional key signature, at least one of the semitone steps must be written as two notes with the same letter/on the same line or space on the staff. (That is, there must be at least one note that regularly appears with two different accidentals.) There are usually several equally succinct combinations of key signature and accidentals, and different composers have chosen to notate their music differently, sometimes ignoring the niceties of notation conventions designed to facilitate
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 ...
tonality Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is call ...
.


Properties


Symmetry

The three octatonic collections are transpositionally and inversionally symmetric—that is, 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 E scale into the D scale, the C scale into the D scale, and the D scale into the E scale. Conversely, transpositions by 2, 5, 8, or 11 semitones acts in the reverse way; the E scale goes to the D scale, D to C and C to E. 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. The E and C collections can be swapped by inversions around E, F, A or C (the tones common to both scales). Similarly, the C and D collections can be swapped by inversions around E, G, B/A, D/C and the D and E collections by inversions around D, F, A, or B. All other transformations do not change the classes (e.g. reflecting the E collection around E gives the E collection once again). This unfortunately means that the inversions do not act as a simple cyclic group on the set of diminished scales.


Subsets

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. 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 different 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
tetrachord In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency propo ...
s to be juxtaposed unusually but within a rational framework" though the relation of the
diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) 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, ...
to the melodic and harmonic surface is thus generally oblique.


History


Early examples

Joseph Schillinger suggests that the scale was formulated already by
Persian traditional music Persian traditional music or Iranian traditional music, also known as Persian classical music or Iranian classical music, refers to the classical music of Iran (also known as ''Persia''). It consists of characteristics developed through the coun ...
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. Octatonic scales first occurred in Western music as byproducts of a series of minor-third transpositions. While
Nikolai Rimsky-Korsakov Nikolai Andreyevich Rimsky-Korsakov . At the time, his name was spelled Николай Андреевичъ Римскій-Корсаковъ. la, Nicolaus Andreae filius Rimskij-Korsakov. The composer romanized his name as ''Nicolas Rimsk ...
claimed he was conscious of the octatonic collection "as a cohesive frame of reference" in his autobiography ''My Musical Life'', instances can be found in music of previous centuries. Eytan Agmon locates one in
Domenico Scarlatti Giuseppe Domenico Scarlatti, also known as Domingo or Doménico Scarlatti (26 October 1685-23 July 1757), was an Italian composer. He is classified primarily as a Baroque composer chronologically, although his music was influential in the deve ...
's Sonata K. 319. In the following passage, according to Richard Taruskin, "its descending whole-step/half-step bass progression is complete and continuous". Taruskin also cites the following bars from
J. S. Bach Johann Sebastian Bach (28 July 1750) was a German composer and musician of the late Baroque period. He is known for his orchestral music such as the '' Brandenburg Concertos''; instrumental compositions such as the Cello Suites; keyboard w ...
's English Suite No. 3 as octatonic:
Honoré Langlé Honoré François Marie Langlé (1741–1807) was a French theorist of music of Monegasque origin, composer, and author of a ''Traité d'harmonie et de modulation'' (Paris: Boyer, 1795). Napoleon named him to the newly founded Paris Conservatoire ...
'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.


19th century

In 1800, Beethoven composed his Piano Sonata No. 11 in B , Op. 22. The slow movement of this work contains a passage of what was, for its time, highly dissonant harmony. In a lecture (2005), pianist
András Schiff Sir András Schiff (; born 21 December 1953) is a Hungarian-born British classical pianist and conductor, who has received numerous major awards and honours, including the Grammy Award, Gramophone Award, Mozart Medal, and Royal Academy of Musi ...
describes the harmony of this passage as "really extraordinary". The chord progressions at the beginning of the second and third bars of this passage are octatonic: Later in the 19th century, the notes in the chords of the coronation bells from the opening scene of Modest Mussorgsky's opera ''
Boris Godunov Borís Fyodorovich Godunóv (; russian: Борис Фёдорович Годунов; 1552 ) ruled the Tsardom of Russia as ''de facto'' regent from c. 1585 to 1598 and then as the first non-Rurikid tsar from 1598 to 1605. After the end of his ...
'', which consist of "two dominant seventh chords with roots a tritone apart" according to Taruskin, are entirely derived from an octatonic scale. Taruskin continues: "Thanks to the reinforcement the lesson has received in some equally famous pieces like ''
Scheherazade Scheherazade () is a major female character and the storyteller in the frame narrative of the Middle Eastern collection of tales known as the ''One Thousand and One Nights''. Name According to modern scholarship, the name ''Scheherazade'' deri ...
'', the progression is often thought of as being peculiarly Russian." Tchaikovsky was also influenced by the harmonic and coloristic potential of octatonicism. As Mark DeVoto points out, the cascading arpeggios played on the celesta in the "Sugar Plum Fairy" from ''
The Nutcracker ''The Nutcracker'' ( rus, Щелкунчик, Shchelkunchik, links=no ) is an 1892 two-act ballet (""; russian: балет-феерия, link=no, ), originally choreographed by Marius Petipa and Lev Ivanov with a score by Pyotr Ilyich Tchaiko ...
'' ballet are made up of dominant seventh chords a minor third apart. "Hagens Watch", one of the darkest and most sinister scenes in
Richard Wagner Wilhelm Richard Wagner ( ; ; 22 May 181313 February 1883) was a German composer, theatre director, polemicist, and conductor who is chiefly known for his operas (or, as some of his mature works were later known, "music dramas"). Unlike most op ...
's opera '' Götterdämmerung'' features chromatic harmonies using eleven of the twelve chromatic notes, within which the eight notes of the octatonic scale may be found in bars 9–10 below:


Late 19th and 20th century

The scale is also found in the music of
Claude 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 ...
and
Maurice Ravel Joseph Maurice Ravel (7 March 1875 – 28 December 1937) was a French composer, pianist and conductor. He is often associated with Impressionism along with his elder contemporary Claude Debussy, although both composers rejected the term. In ...
. Melodic phrases that move by alternating tones and semitones frequently appear in the works of both these composers. Allen Forte identifies a five-note segment in the
cor anglais The cor anglais (, or original ; plural: ''cors anglais''), or English horn in North America, is a double-reed woodwind instrument in the oboe family. It is approximately one and a half times the length of an oboe, making it essentially an alto ...
melody heard near the start of Debussy's "Nuages" from his orchestral suite ''
Nocturnes A nocturne is a musical composition that is inspired by, or evocative of, the night. History The term ''nocturne'' (from French ''nocturne'' 'of the night') was first applied to musical pieces in the 18th century, when it indicated an ensemble ...
'' as octatonic. Mark DeVoto describes "Nuages" as "arguably ebussy'sboldest single leap into the musical unknown. 'Nuages' defines a kind of tonality never heard before, based on the centricity of a diminished tonic triad (B-D-F natural)." According to Stephen Walsh, the
cor anglais The cor anglais (, or original ; plural: ''cors anglais''), or English horn in North America, is a double-reed woodwind instrument in the oboe family. It is approximately one and a half times the length of an oboe, making it essentially an alto ...
theme "hangs in the texture like some motionless object, always the same and always at the same pitch". There is a particularly striking and effective use of the octatonic scale in the opening bars of Liszt's late piece '' Bagatelle sans tonalité'' from 1885. The scale was extensively used by Rimsky-Korsakov's student
Igor 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 composers of the ...
, particularly in his Russian-period works such as '' Petrushka'' (1911), ''
The Rite of Spring ''The Rite of Spring''. Full name: ''The Rite of Spring: Pictures from Pagan Russia in Two Parts'' (french: Le Sacre du printemps: tableaux de la Russie païenne en deux parties) (french: Le Sacre du printemps, link=no) is a ballet and orchestral ...
'' (1913), up to the ''
Symphonies of Wind Instruments The ''Symphonies of Wind Instruments'' (French title: ''Symphonies d'instruments à vent'') is a concert work written by Igor Stravinsky in 1920, for an ensemble of woodwind and brass instruments. The piece is in one movement, lasting about 9 min ...
'' (1920). Passages using this scale are unmistakable as early as the '' Scherzo fantastique'', ''
Fireworks Fireworks are a class of Explosive, low explosive Pyrotechnics, pyrotechnic devices used for aesthetic and entertainment purposes. They are most commonly used in fireworks displays (also called a fireworks show or pyrotechnics), combining a l ...
'' (both from 1908), and '' The Firebird'' (1910). It also appears in later works by Stravinsky, such as the '' Symphony of Psalms'' (1930), the ''
Symphony in Three Movements The Symphony in Three Movements is a work by Russian expatriate composer Igor Stravinsky. Stravinsky wrote the symphony from 1942–45 on commission by the Philharmonic Symphony Society of New York. It was premièred by the New York Philharmoni ...
'' (1945), most of the neoclassical works from the
Octet Octet may refer to: Music * Octet (music), ensemble consisting of eight instruments or voices, or composition written for such an ensemble ** String octet, a piece of music written for eight string instruments *** Octet (Mendelssohn), 1825 compos ...
(1923) to ''
Agon Agon (Greek ) is a Greek term for a conflict, struggle or contest. This could be a contest in athletics, in chariot or horse racing, or in music or literature at a public festival in ancient Greece. Agon is the word-forming element in 'agony', ...
'' (1957), and even in some of the later serial compositions such as the '' Canticum Sacrum'' (1955) and '' Threni'' (1958). In fact, "few if any composers have been known to employ relations available to the collection as extensively or in as varied a manner as Stravinsky". The second movement of Stravinsky's Octet for wind instruments opens with what Stephen Walsh calls "a broad melody completely in the octatonic scale". Jonathan Cross describes a highly rhythmic passage in the first movement of the ''Symphony in Three Movements'' as "gloriously octatonic, not an unfamiliar situation in jazz, where this mode is known as the 'diminished scale', but Stravinsky of course knew it from Rimsky. The ' rumba' passage... alternates chords of E-flat7 and C7, over and over, distantly recalling the coronation scene from Mussorgsky's ''Boris Godunov''. In celebrating America, the émigré looked back once again to Russia." Van den Toorn catalogues many other octatonic moments in Stravinsky's music. The scale also may be found in music of
Alexander Scriabin Alexander Nikolayevich Scriabin (; russian: Александр Николаевич Скрябин ; – ) was a Russian composer and virtuoso pianist. Before 1903, Scriabin was greatly influenced by the music of Frédéric Chopin and composed ...
and
Béla Bartók Béla Viktor János Bartók (; ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt are regarded as H ...
. In Bartók's ''Bagatelles'', Fourth Quartet, ''
Cantata Profana ''Cantata Profana'' (subtitled ''A kilenc csodaszarvas'' 'The Nine Enchanted Stags'' Sz 94) is a work for tenor, baritone, double mixed chorus and orchestra by the Hungarian composer Béla Bartók. Completed on 8 September 1930, it received its ...
'', and '' Improvisations'', the octatonic is used with the diatonic, whole tone, and other "abstract pitch formations" all "entwined... in a very complex mixture". '' Mikrokosmos'' Nos. 99, 101, and 109 are octatonic pieces, as is No. 33 of the ''
44 Duos for Two Violins ''44 Duos for Two Violins'' (sometimes also entitled ''44 Duets''), Sz. 98, BB 104 is a series of duets composed in 1931 by Hungarian composer Béla Bartók. Composition Béla Bartók did not intend this work to be played in performances, but ...
''. "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' ".
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 ...
made frequent use of the octatonic scale throughout his career as a composer, and indeed in his seven modes of limited transposition, the octatonic scale is Mode 2. Peter Hill writes in detail about "La Colombe" (The Dove), the first of a set of '' Preludes'' for piano that Messiaen completed in 1929, at the age of 20. Hill speaks of a characteristic "merging of tonality (E major) with the octatonic mode" in this short piece. Other twentieth-century composers who used octatonic collections include
Samuel Barber Samuel Osmond Barber II (March 9, 1910 – January 23, 1981) was an American composer, pianist, conductor, baritone, and music educator, and one of the most celebrated composers of the 20th century. The music critic Donal Henahan said, "Proba ...
,
Ernest Bloch Ernest Bloch (July 24, 1880 – July 15, 1959) was a Swiss-born American composer. Bloch was a preeminent artist in his day, and left a lasting legacy. He is recognized as one of the greatest Swiss composers in history. As well as producing music ...
,
Benjamin Britten Edward Benjamin Britten, Baron Britten (22 November 1913 – 4 December 1976, aged 63) was an English composer, conductor, and pianist. He was a central figure of 20th-century British music, with a range of works including opera, other ...
,
Julian Cochran 200px, Julian Cochran in 1998 Julian Cochran (born 1974) is an English-born Australian composer. Cochran's earlier works show stylistic influences from Impressionist music and his later works are more noticeably influenced by Classical music an ...
,
George Crumb George Henry Crumb Jr. (24 October 1929 – 6 February 2022) was an American composer of avant-garde contemporary classical music. Early in his life he rejected the widespread modernist usage of serialism, developing a highly personal musical ...
, Irving Fine,
Ross Lee Finney Ross Lee Finney Junior (December 23, 1906–February 4, 1997) was an American composer who taught for many years at the University of Michigan. Life and career Born in Wells, Minnesota, Finney received his early training at Carleton College a ...
,
Alberto Ginastera Alberto Evaristo Ginastera (; April 11, 1916June 25, 1983) was an Argentinian composer of classical music. He is considered to be one of the most important 20th-century classical composers of the Americas. Biography Ginastera was born in Buen ...
, John Harbison,
Jacques Hétu Jacques Hétu (August 8, 1938 – February 9, 2010) was a Canadian composer and music educator. Biography Jacques Hétu was born in Trois-Rivières, Quebec; he began his professional training at the University of Ottawa where he was a pupil ...
, Aram Khachaturian, Witold Lutosławski, Darius Milhaud,
Henri Dutilleux Henri Paul Julien Dutilleux (; 22 January 1916 – 22 May 2013) was a French composer active mainly in the second half of the 20th century. His small body of published work, which garnered international acclaim, followed in the tradition of ...
, Robert Morris, Carl Orff, Jean Papineau-Couture, Krzysztof Penderecki,
Francis Poulenc Francis Jean Marcel Poulenc (; 7 January 189930 January 1963) was a French composer and pianist. His compositions include songs, solo piano works, chamber music, choral pieces, operas, ballets, and orchestral concert music. Among the best-kno ...
,
Sergei Prokofiev Sergei Sergeyevich Prokofiev; alternative transliterations of his name include ''Sergey'' or ''Serge'', and ''Prokofief'', ''Prokofieff'', or ''Prokofyev''., group=n (27 April .S. 15 April1891 – 5 March 1953) was a Russian composer, p ...
,
Alexander Scriabin Alexander Nikolayevich Scriabin (; russian: Александр Николаевич Скрябин ; – ) was a Russian composer and virtuoso pianist. Before 1903, Scriabin was greatly influenced by the music of Frédéric Chopin and composed ...
,
Dmitri Shostakovich Dmitri Dmitriyevich Shostakovich, , group=n (9 August 1975) was a Soviet-era Russian composer and pianist who became internationally known after the premiere of his Symphony No. 1 (Shostakovich), First Symphony in 1926 and was regarded throug ...
, Toru Takemitsu,
Joan Tower Joan Tower (born September 6, 1938)http://www.schirmer.com/default.aspx?TabId=2419&State_2872=2&ComposerId_2872=1605 Biography on Schirmer is a Grammy-winning contemporary American composer, concert pianist and conductor. Lauded by ''The New York ...
,
Robert Xavier Rodriguez Robert Xavier Rodríguez (born June 28, 1946) is an American classical composer, best known for his eight operas and his works for children. Life and career Rodríguez received his early musical education in his native San Antonio and in Aus ...
,
John Williams John Towner Williams (born February 8, 1932)Nylund, Rob (15 November 2022)Classic Connection review ''WBOI'' ("For the second time this year, the Fort Wayne Philharmonic honored American composer, conductor, and arranger John Williams, who wa ...
and
Frank Zappa Frank Vincent Zappa (December 21, 1940 – December 4, 1993) was an American musician, composer, and bandleader. His work is characterized by wikt:nonconformity, nonconformity, Free improvisation, free-form improvisation, sound experimen ...
. Other composers include
Willem Pijper Willem Frederik Johannes Pijper (; 8 September 189418 March 1947) was a Dutch composer, music critic and music teacher. Pijper is considered to be among the most important Dutch composers of the first half of the 20th century. Life Pijper was b ...
, who may have inferred the collection from Stravinsky's ''The Rite of Spring'', which he greatly admired, and composed at least one piece—his Piano Sonatina No. 2—entirely in the octatonic system. In the 1920s,
Heinrich Schenker Heinrich Schenker (19 June 1868 – 14 January 1935) was a Galician-born Austrian music theorist whose writings have had a profound influence on subsequent musical analysis. His approach, now termed Schenkerian analysis, was most fully exp ...
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.
Radiohead Radiohead are an English rock band formed in Abingdon, Oxfordshire, in 1985. The band consists of Thom Yorke (vocals, guitar, piano, keyboards); brothers Jonny Greenwood (lead guitar, keyboards, other instruments) and Colin Greenwood (bass) ...
used the ascending octatonic scale in the introduction and the chorus of their 1995 single "
Just Just or JUST may refer to: __NOTOC__ People * Just (surname) * Just (given name) Arts and entertainment * ''Just'', a 1998 album by Dave Lindholm * "Just" (song), a song by Radiohead * "Just", a song from the album ''Lost and Found'' by Mudvayne ...
". This song was included on their second album,
The Bends "The bends" is a colloquialism for decompression sickness. The Bends may also refer to: * ''The Bends'' (album), a 1995 studio album by Radiohead * "The Bends" (song), a 1995 song by Radiohead * "The Bends", a song by Mr. Bungle from the 1995 alb ...
. The introduction of Gospel musician Israel Houghton's "Alive", title track of album Alive in South Africa (track 3), is built around a descending octatonic scale.


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 Edim7 chord) while the half-whole scale is used in dominant harmony (e.g., with an F9 chord). Examples of octatonic jazz include Jaco Pastorius' composition "Opus Pocus" from the album '' Pastorius'' and
Herbie Hancock Herbert Jeffrey Hancock (born April 12, 1940) is an American jazz pianist, keyboardist, bandleader, and composer. Hancock started his career with trumpeter Donald Byrd's group. He shortly thereafter joined the Miles Davis Quintet, where he help ...
's piano solo on "Freedom Jazz Dance" Tymoczko, D. (2017, bars 18-21, right hand part) Transcription of Piano solo from “Freedom Jazz dance” https://dmitri.mycpanel.princeton.edu/transcriptions.html accessed 24/11/2021. from the album ''
Miles Smiles ''Miles Smiles'' is an album by jazz musician Miles Davis, released on February 16, 1967 on Columbia Records. It was recorded by Davis and his second quintet at Columbia 30th Street Studio in New York City on October 24 and October 25, 1966. It i ...
'' (1967).


Petrushka chord

The Petrushka chord is a recurring
polytonal Polytonality (also polyharmony) is the musical use of more than one key simultaneously. Bitonality is the use of only two different keys at the same time. Polyvalence or polyvalency is the use of more than one harmonic function, from the same key, a ...
device used in
Igor 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 composers of the ...
's ballet '' Petrushka'' and in later music. In the Petrushka chord, two major triads, C major and F major – a tritone apart – clash, "horribly with each other", when sounded together and create a
dissonant In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...
chord. The six-note chord is contained within an octatonic scale.


Bitonality

In
Béla Bartók Béla Viktor János Bartók (; ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt are regarded as H ...
'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 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 are 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. 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". 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 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. 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.


Alpha chord

The alpha chord (α 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. One of the most important subsets of the alpha collection, the alpha chord ( Forte number: 4-17, pitch class
prime form In algebraic geometry, the Schottky–Klein prime form ''E''(''x'',''y'') of a compact Riemann surface ''X'' depends on two elements ''x'' and ''y'' of ''X'', and vanishes if and only if ''x'' = ''y''. The prime form ''E'' is not quite ...
(0347)), such as E–G–C–E; using the theorist
Ernő Lendvai __NOTOC__ Ernő Lendvai (6 February 1925 – 31 January 1993) was one of the first music theorists to write on the appearance of the golden section and Fibonacci series and how these are implemented in Bartók's music. He also formulated the a ...
'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). The number of semitones in the interval array of the alpha chord corresponds to the Fibonacci sequence.


Beta chord

The beta chord (β chord) is a five-note chord, formed from the first five notes of the alpha chord (integers: 0,3,6,9,11; notes: C, E, G, B, C). The beta chord can also occur in its reduced form, that is, limited to the characteristic tones (C, E, G, C and C, G, C). Forte number: 5-31B. The beta chord may be created from a diminished seventh chord by adding a diminished octave. It may be created from a major chord by adding the sharpened root (solfege: in C, di is C: C, E, G, C), or from a diminished triad by adding the root's major 7th (called a diminished major 7th, or C#Maj7. The diminished octave is inverted creates a minor ninth, creating a C(9) chord, a sound commonly heard in the V chord during an authentic cadence in a minor key.


Gamma chord

The gamma chord (γ chord) is 0,3,6,8,11 (Forte number 5-32A) It is the beta chord with one interval diminished: C, E, G, A, C. It may be considered a major-minor minor seventh chord on A: A, C, C, E, G. See also: Elektra chord. This is also commonly known as the
Hendrix James Marshall "Jimi" Hendrix (born Johnny Allen Hendrix; November 27, 1942September 18, 1970) was an American guitarist, singer and songwriter. Although his mainstream career spanned only four years, he is widely regarded as one of the most ...
chord, or in jazz music as a Dominant 79 chord; the notes in this case creating an A79.


See also

*
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 ...
has a ten-note analogue * Complexe sonore * Alpha scale * Beta scale * Delta scale * Gamma scale *
List of pieces which use the octatonic scale Compositions using the octatonic scale: * Radiohead ::" Just" (1995). Jonny Greenwood plays a series of OCT02 scales on the guitar during the intro (0:06-0:16) and each chorus (0:55-1:05, 1:44-1:55, 2:47-3:09)Victoria Malawey, “Ear Training wit ...


References

Sources * * * * * * * * * * * . * * * * * * * * * * * Cited in . * * * * * * * * * * * * * * * * * * * Cited in .


Further reading

* Baur, Steven (1999). "Ravel's 'Russian' Period: Octatonicism in His Early Works, 1893–1908." '' Journal of the American Musicological Society'' 52, no. 1:. * Berger, Arthur (1963). "Problems of Pitch Organization in Stravinsky". ''
Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief). ''Perspectives'' was first ...
'' 2, no. 1 (Fall–Winter): 11–42. * Gillespie, Robert (2015).
Herbie Hancock: Freedom Jazz Dance Transcription
. (Accessed 1 October 2015). * * Tymoczko, Dmitri (2002). "Stravinsky and the Octatonic: A Reconsideration". '' Music Theory Spectrum'' 24, no. 1 (Spring): 68–102. * Wollner, Fritz (1924) "7 mysteries of Stravinsky in Progression" 1924 German international school of music study. {{Scales Musical symmetry Post-tonal music theory Persian music