Editing Symmetry (section)

====Pitch structures====
Symmetry is also an important consideration in the formation of [[scale (music)|scale]]s and [[chord (music)|chords]], traditional or [[tonality|tonal]] music being made up of non-symmetrical groups of [[pitch (music)|pitches]], such as the [[diatonic scale]] or the [[major chord]]. [[Symmetrical scale]]s or chords, such as the [[whole tone scale]], [[augmented chord]], or diminished [[seventh chord]] (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are [[ambiguous]] as to the [[Key (music)|key]] or tonal center, and have a less specific [[diatonic functionality]]. However, composers such as [[Alban Berg]], [[Béla Bartók]], and [[George Perle]] have used axes of symmetry and/or [[interval cycle]]s in an analogous way to [[musical key|keys]] or non-[[tonality|tonal]] tonal [[Tonic (music)|center]]s.<ref name=Perle1992>{{Cite journal |title=Symmetry, the twelve-tone scale, and tonality |first=George |last=Perle |author-link=George Perle |journal=Contemporary Music Review |volume=6 |issue=2 |year=1992 |pages=81–96 |doi=10.1080/07494469200640151}}</ref> George Perle explains that "C–E, D–F♯, [and] Eb–G, are different instances of the same [[interval (music)|interval]] … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:"<ref name=Perle1992/>

{|
|-
|D
| 
|D♯
| 
|'''E'''
| 
|F
| 
|F♯
| 
|G
| 
|G♯
|-
|D
| 
|C♯
| 
|'''C'''
| 
|B
| 
|A♯
| 
|A
| 
|G♯
|}

Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0).<ref name=Perle1992/>

{|
|rowspan=3|+
|2
|
|3
|
|'''4'''
|
|5
|
|6
|
|7
|
|8
|-
|2
|
|1
| 
|'''0'''
|
|11
|
|10
|
|9
|
|8
|-
|4
|
|4
|
|4
|
|4
|
|4
|
|4
|
|4
|}

Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are [[enharmonic]] with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal [[chord progression|progressions]] in the works of [[Romantic music|Romantic]] composers such as [[Gustav Mahler]] and [[Richard Wagner]] form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, [[Alexander Scriabin]], [[Edgard Varèse]], and the Vienna school. At the same time, these progressions signal the end of tonality.<ref name=Perle1992/><ref name="Perle1990">{{cite book |author-link=George Perle |author=Perle, George |year=1990 |title=The Listening Composer |url=https://archive.org/details/listeningcompose00perl |url-access=limited |page=[https://archive.org/details/listeningcompose00perl/page/n31 21] |publisher=University of California Press |isbn=978-0-520-06991-6}}</ref>

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's [[String Quartet (Berg)|''Quartet'', Op. 3]] (1910).<ref name="Perle1990"/>