Editing Octatonic scale (section)

==Harmonic implications==

===Petrushka chord===
{{Image frame|content=<score sound="1" override_midi="Petrushka chord Second Tableau.mid">
    \new PianoStaff <<
         \new Staff = "up" \relative c'' {
            \time 3/4
            s2.
          }
        \new Staff = "down" \relative c' {
            \time 3/4
            \voiceOne
            \repeat tremolo 12 {
                  {
                    \change Staff = "up"
                    \voiceTwo
                    <e g c>32
                  }
                {
                    \change Staff = "down"
                    \voiceOne
                    <cis fis ais>32
                }
            }
          }
    >>
</score>|width=300|caption=The [[Petrushka chord]] in the piano during the second tableau of [[Stravinsky]]'s ballet ''[[Petrushka (ballet)|Petrushka]]''{{sfn|Taruskin|1987|loc=269}}}}
The [[Petrushka chord]] is a recurring [[Polytonality|polytonal]] device used in [[Igor Stravinsky]]'s ballet ''[[Petrushka (ballet)|Petrushka]]'' and in later music. In the Petrushka chord, two [[Major chord|major triads]], C major and F{{music|sharp}} major – a [[tritone]] apart – clash, "horribly with each other", when sounded together and create a [[Consonance and dissonance|dissonant]] [[Chord (music)|chord]].{{sfn|Pogue|1997|loc=80}} The six-note chord is contained within an octatonic scale.

=== French sixth and Mystic chord ===
While [[Passing chord|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>{{Citation |title=Ears Wide Open Online {{!}} Deconstructing Rimsky-Korsakov's Scheherazade | date=15 June 2020 |url=https://www.youtube.com/watch?v=vRlnhaLEWMc |access-date=2023-08-19 |language=en}}</ref> Scriabin's ''Five Preludes, Op. 74'',<ref>{{Citation |title=Scriabin's Use Of The Octotonic Scale | date=20 March 2022 |url=https://www.youtube.com/watch?v=2zsmJ-XSlQI |access-date=2023-08-19 |language=en}}</ref> Debussy's ''Nuages'' and Ravel's ''[[Gaspard de la nuit|Scarbo]]''.<ref>{{Cite web |last=Ravel |first=Maurice |date=1908 |title=Gaspard de la nuit: III. Scarbo, piano score |url=https://imslp.hk/files/imglnks/euimg/2/22/IMSLP813748-PMLP2576-Gaspard_de_la_nuit_-_III._Scarbo.pdf }}</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.

===Bitonality===
In [[Béla Bartók]]'s piano piece, "Diminished Fifth" from ''[[Mikrokosmos (Bartók)|Mikrokosmos]]'', octatonic collections form the basis of the pitch content. In mm. 1–11, all eight pitch classes from the E{{music|b}} 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{{music|b}}, F, G{{music|b}}, and A{{music|b}} 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{{music|b}} minor scale. In mm. 5–11, the left and right hand switch—the A minor tetrachord appears in the left hand, and the E{{music|b}} minor tetrachord appears in the right hand.{{Original research inline|date=April 2015}}

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".{{sfn|Wilson|1992|loc=27}}

Bartók also utilizes the two other octatonic collections so that all three possible octatonic collections are found throughout this piece (D{{music|b}}, D, and E{{music|b}}). In mm. 12–18, all eight pitch classes from the D{{music|b}} octatonic collection are present. The E{{music|b}} octatonic collection from mm. 1–11 is related to this D{{music|b}} 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{{music|b}} octatonic collection from mm. 1–11 by the following transposition operations: T2, T5, T8, T11.{{Original research inline|date=April 2015}}

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{{music|b}}, E{{music|b}}, and G{{music|b}}, 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{{music|b}}, C, E{{music|b}} and E, G{{music|b}}, A respectively. Later on, in mm. 20, the right hand moves on to A− and the left back to E{{music|b}}−. 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{{music|b}}.{{Original research inline|date=April 2015}}

===Alpha chord===
<!--[[Alpha chord]] and [[Golden section chord]] redirect directly here-->
{{Image frame|content=<score sound="1">
{
\override Score.TimeSignature #'stencil = ##f
\relative c' {
  \clef treble \key c \major
  <cis e g bes c es fis a>1
} }
</score>

<score sound="1">
{
\override Score.TimeSignature #'stencil = ##f
\relative c' {
  \clef treble \key c \major
  \once \override NoteHead.color = #red des8 \once \override NoteHead.color = #blue es \once \override NoteHead.color = #red e! \once \override NoteHead.color = #blue fis \once \override NoteHead.color = #red g \once \override NoteHead.color = #blue a \once \override NoteHead.color = #red bes \once \override NoteHead.color = #blue c
} }
</score>|width=300|caption=Two diminished seventh chords in the octatonic scale (one red, one blue) may be rearranged as the alpha chord}}
The alpha chord (α chord) collection is, "a vertically organized statement of the octatonic scale as two [[diminished seventh chord]]s", such as: C{{music|sharp}}–E–G–B{{music|flat}}–C–E{{music|flat}}–F{{music|sharp}}–A.{{sfn|Wilson|1992|loc=7}}

One of the most important subsets of the alpha collection, the alpha chord ([[Forte number]]: 4-17, [[pitch class]] [[prime form (music)|prime form]] (0347)), such as E–G–C–E{{music|flat}}; using the theorist [[Ernő Lendvai]]'s terminology,{{sfn|Lendvai|1971}} the C alpha chord may be considered a [[mistuning|mistuned]] [[major chord]] or major/minor in first inversion (in this case, C major/minor).{{sfn|Wilson|1992|loc=9}}{{clarify|date=May 2020|reason=How is it mistuned?}} The number of semitones in the [[interval array]] of the alpha chord corresponds to the [[Fibonacci sequence]].{{sfn|Slayton|2010|loc=15}}{{Explain|date=May 2020|reason=How does the array correspond to what part of the sequence?}}

===Beta chord===
<!--[[Beta chord]] redirects directly here.-->
{{Image frame|content=<score sound="1">
{
\override Score.TimeSignature #'stencil = ##f
\relative c' {
  \clef treble \key c \major
  <cis e g bes c>1 <cis e g c> <cis g' c>
} }
</score>|width=300|caption=A beta chord on C{{music|#}}, with two reduced versions}}

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;{{sfn|Honti|2007|loc=305}} notes: C{{music|#}}, E, G, B{{music|b}}, C{{music|natural}}). The beta chord can also occur in its reduced form, that is, limited to the characteristic tones (C{{music|#}}, E, G, C{{music|natural}} and C{{music|#}}, G, C{{music|natural}}). 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{{music|#}}: C{{music|sharp}}, E, G, C{{music|natural}}),{{sfn|Anon.|1977|loc=12}} or from a diminished triad by adding the root's major 7th (called a diminished major 7th, or C#{{music|diminished}}<sup>Maj7</sup>. The diminished octave is inverted creates a minor ninth,{{clarify|date=August 2021|reason=Mangled grammar, incomprehensible.}} creating a C<sup>({{music|flat}}9)</sup> chord, a sound commonly heard in the V chord during an authentic cadence in a minor key.{{citation needed|date=August 2021}}

===Gamma chord===
<!--[[Gamma chord]] redirects directly here.-->
[[File:5-32A pentachord on C.mid|thumb|Gamma chord]]
The gamma chord (γ chord) is 0,3,6,8,11 (Forte number 5-32A){{sfn|Honti|2007|loc=305}} It is the beta chord with one interval diminished: C{{music|#}}, E, G, A, C{{music|natural}}. It may be considered a major-minor minor seventh chord on A: A, C{{music|natural}}, C{{music|#}}, E, G. See also: [[Elektra chord]]. This is also commonly known as the [[Jimi Hendrix|Hendrix]] chord,{{citation needed|date=August 2021}} or in jazz music as a Dominant 7{{music|sharp}}9 chord; the notes in this case creating an A<sup>7{{music|sharp}}9</sup>.

===Hungarian major and Romanian major===
{{further|Hungarian major scale|Romanian major scale}}
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 {{music|flat}}2 degree removed, while the Romanian major scale has the {{music|flat}}3 degree removed.