Incubator escapee wiki

Octatonic scale

Article Discussion

Template:Short description Template:For Template:Redirect Template:Image frame

An octatonic scale is any eight-note musical scale. However, the term most often refers to the ancohemitonic symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), this symmetrical scale is commonly called the octatonic scale (or the octatonic collection), although there are a total of 43 enharmonically inequivalent, transpositionally inequivalent eight-note sets.

The earliest systematic treatment of the octatonic scale was in Edmond de Polignac's unpublished treatise "Étude sur les successions alternantes de tons et demi-tons (Et sur la gamme dite majeure-mineure)" (Study of the Succession of Alternating Whole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,Template:Sfn which preceded Vito Frazzi's Scale alternate per pianoforte of 1930Template:Sfn by 50 years.Template:Sfn

NomenclatureEdit

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 that it was referred to as the Korsakovian scale (Корсаковская гамма).Template:Sfn As early as 1911, the Russian theorist Boleslav Yavorsky described this collection of pitches as the diminished mode (уменьшённый лад), because of the stable way the diminished fifth functions in it.<ref>Template:Harvnb, citing Template:Harvnb</ref> 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.Template:SfnTemplate:Sfn)

In jazz theory, it is called the diminished scaleTemplate:Sfn or symmetric diminished scaleTemplate:Sfn 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 two modes are sometimes referred to as the half-step/whole step diminished scale and the whole step/half-step diminished scale.Template:Sfn

Because it was associated in the early 20th century with the Dutch composer Willem Pijper, in the Netherlands it is called the Pijper scale.Template:Sfn

ConstructionEdit

Template:Image frameThe 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: the first begins its ascent with a whole step, 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.Template:Sfn

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:

It may also be represented as semitones, either starting with a whole tone (as above): Template:Nowrap, or starting with a semitone: Template:Nowrap, or labeled as set class 8‑28.Template:Sfn

With one more scale tone than described by the western diatonic scale, it is not possible to perfectly notate music of the octatonic scale using any conventional key signature without the use of accidentals. Across all conventional key signatures, at least two of the octatonic notes must share similar horizontal alignment on the staves, although the precise combination of accidentals and naturals varies. 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 tonality.

PropertiesEdit

SymmetryEdit

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 ETemplate:Music scale into the DTemplate:Music scale, the CTemplate:Music scale into the D scale, and the D scale into the ETemplate:Music scale. Conversely, transpositions by 2, 5, 8, or 11 semitones acts in the reverse way; the ETemplate:Music scale goes to the D scale, D to CTemplate:Music and CTemplate:Music to ETemplate:Music. 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.Template:Original research inline

The ETemplate:Music and CTemplate:Music collections can be swapped by inversions around ETemplate:Music, FTemplate:Music, A or C (the tones common to both scales). Similarly, the CTemplate:Music and D collections can be swapped by inversions around E, G, BTemplate:Music/ATemplate:Music, DTemplate:Music/CTemplate:Music and the D and ETemplate:Music collections by inversions around D, F, ATemplate:Music, or B. All other transformations do not change the classes (e.g. reflecting the ETemplate:Music collection around E gives the ETemplate:Music collection once again). This unfortunately means that the inversions do not act as a simple cyclic group on the set of diminished scales.Template:Original research inline

SubsetsEdit

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.Template:Sfn For example:

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, ETemplate:Music, F and (enharmonically) FTemplate:Music, GTemplate:Music, A, B. Also ETemplate:Music, F, GTemplate:Music, ATemplate:Music, 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.Template:Sfn

HistoryEdit

Early examplesEdit

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.Template:Sfn

Octatonic scales first occurred in Western music as byproducts of a series of minor-third transpositions. While Nikolai Rimsky-Korsakov claimed he was conscious of the octatonic collection "as a cohesive frame of reference" in his autobiography My Musical Life,Template:SfnTemplate:Sfn instances can be found in music of previous centuries. Eytan AgmonTemplate:Sfn locates one in Domenico Scarlatti's Sonata K. 319. In the following passage, according to Richard Taruskin,Template:Sfn "its descending whole-step/half-step bass progression is complete and continuous".

File:Scarlatti Sonasta K319, bars 62-80.wav
Scarlatti Sonata K319, bars 62–80
File:Scarlatti Sonata K319, bars 62-80.png
Scarlatti's Sonata K. 319, bars 62–80

TaruskinTemplate:Sfn also cites the following bars from J. S. Bach's English Suite No. 3 as octatonic:

File:Octatonic bars from Sarabande from English Suite No 3.wav
Octatonic bars from Sarabande from English Suite No 3
File:Sarabande from J.S.Bach's English Suite No.3, bars 17-19.png
Sarabande from J. S. Bach's English Suite No. 3, bars 17–19

Honoré 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.Template:Sfn

19th centuryEdit

In 1800, Beethoven composed his Piano Sonata No. 11 in BTemplate:Music, Op. 22. The slow movement of this work contains a passage of what was, for its time, highly dissonant harmony. In a lecture (2005),Template:Sfn pianist András Schiff 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:

File:Beethoven Piano Sonata Op 22, 2nd movement, bars 30-32.wav
Adagio (2nd movement) from Beethoven's Piano Sonata No. 11, bars 31–33.
File:Adagio (2nd movement) from Beethoven's Piano Sonata Op.22, bars 31-33.png
Adagio (2nd movement) from Beethoven's Piano Sonata No. 11, bars 31–33.

Octatonic scales can be found in Chopin's Mazurka, Op. 50, No. 3 and in several Liszt piano works: the closing measures of the third Étude de Concert, "Un Sospiro," for example, where (mm. 66–70) the bass contains a complete falling octatonic scale from D-flat to D-flat, in the opening piano cadenzas of Totentanz, in the lower notes between the alternating hands, and in the First Mephisto Waltz, in which a short cadenza (m. 525) makes use of it by harmonizing it with a B-flat Diminished Seventh chord. 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, which consist of "two dominant seventh chords with roots a tritone apart" according to Taruskin,Template:Sfn are entirely derived from an octatonic scale.

File:Coronation scene from Boris Godunov.wav
Coronation scene from Boris Godunov
File:Coronation scene from Boris Godunov.png
Coronation scene from Boris Godunov. Template:YouTube

Taruskin continues: "Thanks to the reinforcement the lesson has received in some equally famous pieces like Scheherazade, the progression is often thought of as being peculiarly Russian."Template:Sfn

Tchaikovsky was also influenced by the harmonic and coloristic potential of octatonicism. As Mark DeVotoTemplate:Sfn points out, the cascading arpeggios played on the celesta in the "Sugar Plum Fairy" from The Nutcracker ballet are made up of dominant seventh chords a minor third apart.

File:Cascading arpeggios on celesta from Sugar Plum Fairy.wav
Cascading arpeggios on celesta from Sugar Plum Fairy
File:Cascading arpeggios on celesta from the Dance of the Sugar Plum Fairy.png
Cascading arpeggios on celesta from the Dance of the Sugar Plum Fairy.

"Hagens Watch", one of the darkest and most sinister scenes in Richard Wagner'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:

File:Wagner, "Hagen's Watch" from Gotterdamerung.png
Wagner, "Hagen's Watch" from Götterdämmerung, act 1<ref>"Hagen's Watch"</ref>

Late 19th and 20th centuryEdit

Template:Image frame

File:Istrian scale Schubert Symphony No. 8 in B minor (1922), 1st mvt., bars 13-20.png
Istrian scale in Schubert's Symphony No. 8 in B minor (1822), 1st mvt., bars 13–20; flat fifth marked with asteriskTemplate:SfnFile:Istrian scale Schubert Symphony No. 8 in B minor (1922), 1st mvt., bars 13-20.mid

The scale is also found in the music of Claude Debussy and Maurice Ravel. Melodic phrases that move by alternating tones and semitones frequently appear in the works of both these composers. Allen ForteTemplate:Sfn identifies a five-note segment in the cor anglais melody heard near the start of Debussy's "Nuages" from his orchestral suite Nocturnes as octatonic. Mark DeVotoTemplate:Sfn describes "Nuages" as "arguably [Debussy's] boldest 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 theme "hangs in the texture like some motionless object, always the same and always at the same pitch".Template:Sfn 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.Template:Citation needed

The scale was extensively used by Rimsky-Korsakov's student Igor Stravinsky, particularly in his Russian-period works such as Petrushka (1911), The Rite of Spring (1913), up to the Symphonies of Wind Instruments (1920). Passages using this scale are unmistakable as early as the Scherzo fantastique, Fireworks (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 (1945), most of the neoclassical works from the Octet (1923) to Agon (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".Template:Sfn

The second movement of Stravinsky's Octet<ref>Stravinsky's Octet</ref> for wind instruments opens with what Stephen WalshTemplate:Sfn calls "a broad melody completely in the octatonic scale". Jonathan CrossTemplate:Sfn describes a highly rhythmic passage<ref>Archived at GhostarchiveTemplate:Cbignore and the Wayback MachineTemplate:Cbignore: {{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref> 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 ToornTemplate:Sfn catalogues many other octatonic moments in Stravinsky's music.

The scale also may be found in music of 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".Template:Sfn Mikrokosmos Nos. 99, 101, and 109 are octatonic pieces, as is No. 33 of the 44 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 musicTemplate:'".Template:Sfn

Olivier Messiaen 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 HillTemplate:Sfn writes in detail about "La Colombe" (The Dove),<ref>"La Colombe" (The Dove)</ref> 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, Ernest Bloch, Benjamin Britten, Julian Cochran, George Crumb, Irving Fine, Ross Lee Finney, Alberto Ginastera, John Harbison, Jacques Hétu, Aram Khachaturian, Witold Lutosławski, Darius Milhaud, Henri Dutilleux, Robert Morris, Carl Orff, Jean Papineau-Couture, Krzysztof Penderecki, Francis Poulenc, Sergei Prokofiev, Alexander Scriabin, Dmitri Shostakovich, Toru Takemitsu, Joan Tower,Template:Sfn Robert Xavier Rodriguez, John WilliamsTemplate:Sfn and Frank Zappa.Template:Sfn Other composers include Willem Pijper,Template:Sfn 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.Template:Sfn

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.Template:Sfn

JazzEdit

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 FTemplate:MusicTemplate:Music9 chord).

Examples of octatonic jazz include Jaco Pastorius' composition "Opus Pocus"<ref>"Opus Pocus"</ref> from the album PastoriusTemplate:SfnTemplate:Failed verification and Herbie Hancock's piano solo on "Freedom Jazz Dance"<ref>Piano solo on "Freedom Jazz Dance"</ref><ref>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.</ref> from the album Miles Smiles (1967). The John McLaughlin (musician) composition "The Dance of Maya" is structured around modes of the octatonic scale <ref>[https://www.youtube.com/watch?v=RV64F4jJsE4 Octatonic Basis of The Dance of Maya</ref>

Rock and popEdit

Jonny Greenwood of the English rock group Radiohead uses the octatonic scale extensively, such as in the song "Just" and his soundtrack for the film The Power of the Dog.<ref name="Ross-2021">Template:Cite magazine</ref><ref>Template:Cite magazine</ref> He said "It's a slightly more grownup version of the pentatonic scales that we're all taught to do with xylophones and glockenspiels when you're a kid. It's not a major scale or a minor scale; it's something else. But all the notes work together and make a certain color that is its own thing."<ref name="Ross-2021" />

The scale is used in progressive heavy metal music such as that by Dream Theater and Opeth, both of which strive for a dissonant and tonally ambiguous sound in their music. Examples include the instrumental break in Dream Theater's Octavarium and Opeth's Deliverance. Earlier examples of the scale's use in progressive rock include King Crimson's Red and Emerson Lake & Palmer's The Barbarian.

Progressive keyboardist Derek Sherinian is also closely associated with the octatonic scale, which can be found in most of his works, both solo and as part of a band. Examples include Planet X's Desert Girl and Sons of Apollo's King of Delusion. The dissonances associated with the scale when used in conjunction with conventional tonality form an integral part of his signature sound which has influenced hundreds of keyboardists of the 21st century.

Harmonic implicationsEdit

Petrushka chordEdit

Template:Image frame The Petrushka chord is a recurring polytonal device used in Igor Stravinsky's ballet Petrushka and in later music. In the Petrushka chord, two major triads, C major and FTemplate:Music major – a tritone apart – clash, "horribly with each other", when sounded together and create a dissonant chord.Template:Sfn The six-note chord is contained within an octatonic scale.

French sixth and Mystic chordEdit

While used functionally as a pre-dominant chord in the classical period, late romantic composers saw the French sixth used as a dissonant and unstable chord. The chord can be built from the first, fourth, sixth and eighth degrees of the half-step/whole-step octatonic scale, and is transpositionally invariant about a tritone, a property somewhat contributing to its popularity. The octatonic collection contains two distinct French sixth chords a minor third apart, and since they share no notes, the scale can be thought of as the union of those two chords. For example, two French sixths based on G and E contain all the notes of an octatonic scale between them.

The octatonic scale is used very frequently for melodic material above a French sixth chord throughout the 19th and 20th centuries, particularly in Russia, in the music of Rimsky-Korsakov, Mussorgsky, Scriabin and Stravinsky, but also outside Russia in the works of Debussy and Ravel. Examples include Rimsky's Scheherezade,<ref>Template:Citation</ref> Scriabin's Five Preludes, Op. 74,<ref>Template:Citation</ref> Debussy's Nuages and Ravel's Scarbo.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> All works are full of non-functional French sixths, and the octatonic scale is almost always the mode of choice.

By adding a major sixth above the root, from within the scale, and a major second, from outside the scale, the new chord is the Mystic chord found in some of Scriabin's late works. While no longer transpositionally invariant, Scriabin teases the tritone symmetry of the French sixth in his music by alternating transpositions of the Mystic chord a tritone apart, implying the notes of an octatonic scale.

BitonalityEdit

In Béla 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 ETemplate:Music diminished scale appear. In mm. 1–4, the pitch classes A, B, C, and D appear in the right hand, and the pitch classes ETemplate:Music, F, GTemplate:Music, and ATemplate:Music 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 ETemplate:Music minor scale. In mm. 5–11, the left and right hand switch—the A minor tetrachord appears in the left hand, and the ETemplate:Music minor tetrachord appears in the right hand.Template:Original research inline

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".Template:Sfn

Bartók also utilizes the two other octatonic collections so that all three possible octatonic collections are found throughout this piece (DTemplate:Music, D, and ETemplate:Music). In mm. 12–18, all eight pitch classes from the DTemplate:Music octatonic collection are present. The ETemplate:Music octatonic collection from mm. 1–11 is related to this DTemplate:Music 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 ETemplate:Music octatonic collection from mm. 1–11 by the following transposition operations: T2, T5, T8, T11.Template:Original research inline

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 DTemplate:Music, ETemplate:Music, and GTemplate:Music, 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 BTemplate:Music, C, ETemplate:Music and E, GTemplate:Music, A respectively. Later on, in mm. 20, the right hand moves on to A− and the left back to ETemplate:Music−. 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 ETemplate:Music.Template:Original research inline

Alpha chordEdit

Template:Image frame The alpha chord (α chord) collection is, "a vertically organized statement of the octatonic scale as two diminished seventh chords", such as: CTemplate:Music–E–G–BTemplate:Music–C–ETemplate:Music–FTemplate:Music–A.Template:Sfn

One of the most important subsets of the alpha collection, the alpha chord (Forte number: 4-17, pitch class prime form (0347)), such as E–G–C–ETemplate:Music; using the theorist Ernő Lendvai's terminology,Template:Sfn the C alpha chord may be considered a mistuned major chord or major/minor in first inversion (in this case, C major/minor).Template:SfnTemplate:Clarify The number of semitones in the interval array of the alpha chord corresponds to the Fibonacci sequence.Template:SfnTemplate:Explain

Beta chordEdit

Template:Image frame

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;Template:Sfn notes: CTemplate:Music, E, G, BTemplate:Music, CTemplate:Music). The beta chord can also occur in its reduced form, that is, limited to the characteristic tones (CTemplate:Music, E, G, CTemplate:Music and CTemplate:Music, G, CTemplate:Music). 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 CTemplate:Music: CTemplate:Music, E, G, CTemplate:Music),Template:Sfn or from a diminished triad by adding the root's major 7th (called a diminished major 7th, or C#Template:MusicMaj7. The diminished octave is inverted creates a minor ninth,Template:Clarify creating a C(Template:Music9) chord, a sound commonly heard in the V chord during an authentic cadence in a minor key.Template:Citation needed

Gamma chordEdit

File:5-32A pentachord on C.mid
Gamma chord

The gamma chord (γ chord) is 0,3,6,8,11 (Forte number 5-32A)Template:Sfn It is the beta chord with one interval diminished: CTemplate:Music, E, G, A, CTemplate:Music. It may be considered a major-minor minor seventh chord on A: A, CTemplate:Music, CTemplate:Music, E, G. See also: Elektra chord. This is also commonly known as the Hendrix chord,Template:Citation needed or in jazz music as a Dominant 7Template:Music9 chord; the notes in this case creating an A7Template:Music9.

Hungarian major and Romanian majorEdit

Template:Further The Hungarian major scale and Romanian major scale are both heptatonic subsets of the octatonic scale with one scale degree removed. The Hungarian major scale has the Template:Music2 degree removed, while the Romanian major scale has the Template:Music3 degree removed.

See alsoEdit

ReferencesEdit

Template:Reflist

Sources Template:Div col

Template:Div col end

Further readingEdit

Template:Div col

Template:Div col end

Template:Atonality Template:Scales

Retrieved from "https://zalansite.site/mediawiki/index.php?title=Octatonic_scale&oldid=125779"