Symmetric scale
Encyclopedia
In music
, a symmetric scale is a music scale which equally divides the octave. The concept and term appears to have been introduced by Joseph Schillinger
and further developed by Nicolas Slonimsky
as part of his famous "Thesaurus of Scales and Melodic Patterns". In twelve-tone equal temperament
, the octave can only be equally divided into two, three, four, six, or twelve parts, which consequently may be filled in by adding the same exact interval or sequence of intervals to each resulting note (called "interpolation of notes").
Examples include the octatonic scale
(also known as the symmetric diminished scale; its mirror image is known as the inverse symmetric diminished scale) and the two-semitone tritone scale:
As explained above, both are composed of repeating sub-units within an octave. This property allows these scales to be transposed
to other notes, yet retain exactly the same notes as the original scale (Translational symmetry
).
This may be seen quite readily with the whole tone scale on C:
If transposed up a whole tone to D, contains exactly the same notes in a different permutation:
In the case of inversionally symmetrical scales, the inversion of the scale is identical. Thus the intervals
between scale degrees are symmetrical
if read from the "top" (end) or "bottom" (beginning) of the scale (mirror symmetry
). Examples include the Javanese slendro
, the chromatic scale
, whole-tone scale, Dorian
scale, the Mixolydian 13 scale (fifth mode of the melodic minor), and the double harmonic scale.
Asymmetric scales are "far more common" than symmetric scales and this may be accounted for by the inability of symmetric scales to possess the property of uniqueness (containing each interval class a unique number of times) which assists with determining the location of notes in relation to the first note of the scale.
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, a symmetric scale is a music scale which equally divides the octave. The concept and term appears to have been introduced by Joseph Schillinger
Joseph Schillinger
Joseph Schillinger was a composer, music theorist, and composition teacher. He was born in Kharkiv, Ukraine and died in New York City.-Life and career:...
and further developed by Nicolas Slonimsky
Nicolas Slonimsky
Nicolas Slonimsky was a Russian born American composer, conductor, musician, music critic, lexicographer and author. He described himself as a "diaskeuast" ; "a reviser or interpolator."- Life :...
as part of his famous "Thesaurus of Scales and Melodic Patterns". In twelve-tone equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, the octave can only be equally divided into two, three, four, six, or twelve parts, which consequently may be filled in by adding the same exact interval or sequence of intervals to each resulting note (called "interpolation of notes").
Examples include the octatonic scale
Octatonic scale
An octatonic scale is any eight-note musical scale. Among the most famous of these is a scale in which the notes ascend in alternating intervals of a whole step and a half step, creating a symmetric scale...
(also known as the symmetric diminished scale; its mirror image is known as the inverse symmetric diminished scale) and the two-semitone tritone scale:
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
to other notes, yet retain exactly the same notes as the original scale (Translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
).
This may be seen quite readily with the whole tone scale on C:
- {C, D, E, F, G, A, C}
If transposed up a whole tone to D, contains exactly the same notes in a different permutation:
- {D, E, F, G, A, C, D}
In the case of inversionally symmetrical scales, the inversion of the scale is identical. Thus the intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
between scale degrees are symmetrical
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
if read from the "top" (end) or "bottom" (beginning) of the scale (mirror symmetry
Reflection symmetry
Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...
). Examples include the Javanese slendro
Slendro
Slendro is a pentatonic scale, one of the two most common scales used in Indonesian gamelan music, the other being pélog.-Tuning:...
, the chromatic scale
Chromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
, whole-tone scale, Dorian
Dorian mode
Due to historical confusion, Dorian mode or Doric mode can refer to three very different musical modes or diatonic scales, the Greek, the medieval, and the modern.- Greek Dorian mode :...
scale, the Mixolydian 13 scale (fifth mode of the melodic minor), and the double harmonic scale.
Asymmetric scales are "far more common" than symmetric scales and this may be accounted for by the inability of symmetric scales to possess the property of uniqueness (containing each interval class a unique number of times) which assists with determining the location of notes in relation to the first note of the scale.
Further reading
- Yamaguchi, Masaya. 2006. The Complete Thesaurus of Musical Scales, revised edition. New York: Masaya Music Services. ISBN 0967635306.