music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

, a symmetric scale is a music scale which equally divides the

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

. The concept and term appears to have been introduced by

Joseph Schillinger Joseph Moiseyevich Schillinger (Russian: Иосиф Моисеевич Шиллингер, (other sources: ) – 23 March 1943) was a composer, music theorist, and composition teacher who originated the Schillinger System of Musical Composition ...

and further developed by

Nicolas Slonimsky Nicolas Slonimsky ( – December 25, 1995), born Nikolai Leonidovich Slonimskiy (russian: Никола́й Леони́дович Сло́нимский), was a Russian-born American conductor, author, pianist, composer and lexicographer. B ...

as part of his famous ''Thesaurus of Scales and Melodic Patterns''. In twelve-tone

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

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

(also known as the ''symmetric diminished'' scale; its mirror image is known as the ''inverse symmetric diminished'' scale) and the two-semitone tritone scale: Two-semitone tritone scale on C (1)

As explained above, both are composed of repeating sub-units within an octave. This property allows these scales to be

transposed In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...

to other notes, yet retain exactly the same notes as the original scale (

Translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...

). 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 Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...

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 and ...

s are

symmetrical Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

if read from the "top" (end) or "bottom" (beginning) of the scale (

mirror symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...

). Examples include the Ukrainian Dorian b9 scale (sixth mode of the Hungarian Major scale), the Jazz Minor b5 scale (third mode of the Hungarian Major Inverse), the Neapolitan Major scale (fourth mode of the Major Locrian scale), the Javanese

slendro Slendro ( jv, ꦱ꧀ꦭꦺꦤ꧀ꦢꦿꦺꦴ, ban, slendro, translit=Sléndro) ( su, salendro, translit=Saléndro) is one of the essential tuning systems used in gamelan instruments that have pentatonic scale. Based on Javanese mythology, ...

, 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 In music, a whole-tone scale is a scale in which each note is separated from its neighbors by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole-tone scales, both six-note or ''hexatonic'' s ...

, Dorian scale, the Aeolian Dominant scale (fifth mode of the

melodic minor In music theory, the minor scale is three scale patterns – the natural minor scale (or Aeolian mode), the harmonic minor scale, and the melodic minor scale (ascending or descending) – rather than just two as with the major scale, which al ...

), the Harmonic Minor scale, the Major Locrian Major 7th/Harmonic Major b5 scale, the Chromatic Lydian scale (fourth mode of Blues Leading-Tone scale), the Phrygian Major Lydian scale (fourth mode of Neapolitan Major b5 scale), and the

double harmonic scale In music, the double harmonic major scaleStetina, Troy (1999). ''The Ultimate Scale Book'', p. 59. . is a scale whose gaps may sound unfamiliar to Western listeners. This is also known as Mayamalavagowla, Bhairav Raga, Byzantine scale, Arabic (Hi ...

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.

See also

References

Further reading