Symmetry

Form

Symmetry has been used as a formal constraint by many composers, such as the arch form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney (or swell).

Pitch structures

Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord. Symmetrical scales or chords, such as the whole tone scale, augmented chord, or diminished seventh chord (diminished-diminised seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the 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 cycles in an analogous way to keys or non-tonal tonal centers.

Perle (1992) explains "C-E, D-F#, [and] Eb-G, are different instances of the same 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:"

 

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).

 

+	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 progressions in the works of 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 Varese, and the Vienna school. At the same time, these progressions signal the end of tonality.

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)

 

Equivalency

Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm.

 

Symmetry in telecommunications

Some telecommunications services (specifically data products) may be referred to as symmetrical or asymmetrical. This refers to the bandwidth allocated for data sent and received. Most internet services used by residential customers are asymmetrical: the data sent to the server normally is far less than that returned by the server.