Editing Atonality (section)

==Composing atonal music==
Setting out to compose atonal music may seem complicated because of both the vagueness and generality of the term. Additionally [[George Perle]] explains that, "the 'free' atonality that preceded dodecaphony precludes by definition the possibility of self-consistent, generally applicable compositional procedures".{{sfn|Perle|1962|p=9}} However, he provides one example as a way to compose atonal pieces, a pre-twelve-tone technique piece by Anton Webern, which rigorously avoids anything that suggests tonality, to choose pitches that do not imply tonality. In other words, reverse the rules of the [[common practice period]] so that what was not allowed is required and what was required is not allowed. This is what was done by [[Charles Seeger]] in his explanation of [[dissonant counterpoint]], which is a way to write atonal counterpoint.{{sfn|Seeger|1930}}

{{Image frame|content=<score sound="1">
\layout { \context { \PianoStaff \accidentalStyle dodecaphonic } }
\midi {
  \context { \Staff \consists "Dynamic_performer" }
  \context { \Voice \remove "Dynamic_performer" }
}
\new PianoStaff <<
  \new Staff \fixed c' {
    \dynamicUp
    r4 b( gis |
    \voices "",2 <<
      { \voiceOne g4. a8 f4) | 2( e4) | \oneVoice } \\
      { r4 b,2 | r4 des2 | }
    >>
    r4 \voices "",2 << { \voiceOne e(\< g~ | 8)\> <>\! \oneVoice } \\ { c( bes, b,4~ | 8) } >>
    r r \voices "",2 << { \voiceOne e4.(\< | g)\> <>\! \oneVoice } \\ { r8 c( bes, | b,4.) } >> r8\! r4 |
  }
  \new Dynamics {
    \time 3/4
    \tempo "Mässig"
    s4 s2\p\> | s4 s2\! | s2. | s2 s4\< | s4. s4.\! | s8 s4.\< s4\> | <>\!
  }
  \new Staff {
    \clef bass
    R2. | r4 <ges, f>2 | r4 <bes, a>2 |
    r4 r << { d8( fis | a[ ais] b) } \\ { gis,4~ | 4. } >> r8 r4 |
    r8 << { d( fis a ais b) } \\ { gis,4.~ 4 } >> |
  }
>>
</score>|align=right|caption=Opening of Schoenberg's [[Drei Klavierstücke (Schoenberg)|''Klavierstück'', Op. 11, No. 1]], exemplifying his four procedures as listed by {{harvnb|Kostka & Payne|1995}}}}

Kostka and Payne list four procedures as operational in the atonal music of Schoenberg, all of which may be taken as negative rules. Avoidance of melodic or harmonic octaves, avoidance of traditional pitch collections such as major or minor triads, avoidance of more than three successive pitches from the same diatonic scale, and use of disjunct melodies (avoidance of conjunct melodies).{{sfn|Kostka & Payne|1995|p=513}}

Further, Perle agrees with Oster{{sfn|Oster|1960}} and Katz{{sfn|Katz|1945}} that, "the abandonment of the concept of a [[root (chord)|root]]-generator of the individual chord is a radical development that renders futile any attempt at a systematic formulation of chord structure and progression in atonal music along the lines of traditional harmonic theory".{{sfn|Perle|1962|p=31}} Atonal compositional techniques and results "are not reducible to a set of foundational assumptions in terms of which the compositions that are collectively designated by the expression 'atonal music' can be said to represent 'a system' of composition".{{sfn|Perle|1962|p=1}} Equal-interval chords are often of indeterminate root, [[mixed-interval chord]]s are often best characterized by their interval content, while both lend themselves to atonal contexts.{{sfn|DeLone and Wittlich|1975|pp=362–372}}

Perle also points out that structural coherence is most often achieved through operations on intervallic cells. A cell "may operate as a kind of microcosmic set of fixed intervallic content, statable either as a chord or as a melodic figure or as a combination of both. Its components may be fixed with regard to order, in which event it may be employed, like the twelve-tone set, in its literal transformations. … Individual tones may function as pivotal elements, to permit overlapping statements of a basic cell or the linking of two or more basic cells".{{sfn|Perle|1962|pp=9–10}}

Regarding the post-tonal music of Perle, one theorist wrote: "While&nbsp;... montages of discrete-seeming elements tend to accumulate global rhythms other than those of tonal progressions and their rhythms, there is a similarity between the two sorts of accumulates spatial and temporal relationships: a similarity consisting of generalized arching tone-centers linked together by shared background referential materials".{{sfn|Swift|1982–83|p=272}}

Another approach of composition techniques for atonal music is given by Allen Forte who developed the theory behind atonal music.{{sfn|Forte|1977}} Forte describes two main operations: [[Transposition (music)|transposition]] and [[Melodic inversion|inversion]]. Transposition can be seen as a rotation of ''t'' either clockwise or anti-clockwise on a circle, where each note of the chord is rotated equally. For example, if ''t'' = 2 and the chord is [0 3 6], transposition (clockwise) will be [2 5 8]. Inversion can be seen as a symmetry with respect to the axis formed by 0 and 6. If we carry on with our example [0 3 6] becomes [0 9 6].

An important characteristic are the invariants, which are the notes which stay identical after a transformation. No difference is made between the octave in which the note is played so that, for example, all C{{Music|sharp}}s are equivalent, no matter the octave in which they actually occur. This is why the 12-note scale is represented by a circle. This leads us to the definition of the similarity between two chords which considers the subsets and the interval content of each chord.{{sfn|Forte|1977}}