music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...

, a music scale can have certain symmetries, namely translational symmetry and inversional or mirror symmetry. The most prominent examples are scales which equally divides the

octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...

. The concept and term appears to have been introduced by

Joseph Schillinger Joseph Moiseyevich Schillinger (; (other sources: ) – 23 March 1943) was a composer, music theorist, and music composition, composition teacher who originated the Schillinger System of Musical Composition. He was born in Kharkiv, Kharkov, in the ...

and further developed by Nicolas Slonimsky as part of his famous ''Thesaurus of Scales and Melodic Patterns''. In twelve-tone

equal temperament An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...

, 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"). This leads to scales with translational symmetry which include the

octatonic scale An octatonic scale is any eight-note musical scale. However, the term most often refers to the ancohemitonic symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), ...

(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 In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation. Analogously, an operato ...

). This may be seen quite readily with the whole tone scale on C: * :

: If transposed up a

whole tone In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more deta ...

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. pp. 36-58. Thus the intervals between

scale degree In music theory, the scale degree is the position of a particular note on a scale relative to the tonic—the first and main note of the scale from which each octave is assumed to begin. Degrees are useful for indicating the size of intervals ...

s are

symmetrical Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

if read from the "top" (end) or "bottom" (beginning) of the scale ( mirror symmetry). Examples include the Neapolitan Major scale (fourth mode of the Major Locrian scale), the Javanese

slendro Slendro () is one of the essential tuning systems used in gamelan instruments that have pentatonic scale (music), scale. Based on Javanese people, Javanese mythology, the Slendro Gamelan tuning system is older than the ''pélog'' tuning system. ...

, the

chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...

, whole-tone scale, Dorian scale, the Aeolian Dominant scale (fifth mode of the

melodic minor In Classical_music, Western classical music theory, the minor scale refers to three Scale (music), scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending). ...

), and the double harmonic scale.

Asymmetric scales are "far more common" than symmetric scales and this may be accounted for by the inability of translational 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.

See also

References

Further reading