Equal division of the octave
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In music, an equal division of the octave, or EDO, is a system of musical tuning which divides the octave into n equal parts, each a frequency ratio of 21/n. In terms of cents it divides the octave, which is 1200 cents, into n parts of size 1200/n cents.
The division of the octave into n parts is often termed n-edo. The idea is very closely allied with the concept of n equal temperament, which is abbreviated n-tet or n-et, but saying "edo" carries no implication that the intervals are being used to approximate rational intervals other than the octave. From this point of view, 12-et is equal temperament, whereas 12-edo would be a more correct term for the use of the tuning system in dodecaphonic music.
Aside from the division into twelve parts, other notable divisions of the octave are into 19, 22, 31, 53 and 72 parts. These divisions are singled out because they have better than average ability to represent rational numbers with small numerators and denominators, and hence are usually most correctly thought of as equal temperaments, not merely equal divisions.
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External links
Also [A117536], [A117537], [A117539], [A117554], [A117555], [A117556], [A117557], [A117558], [A117559], [A117577], [A117578], and [A054540].
References
- James Murray Barbour, "Music and Ternary Continued Fractions", American Mathematical Monthly, 55 (1948), 545-555
- James Murray Barbour, Tuning and Temperament: A Historical Survey, Michigan State College Press, East Lansing, 1951; reprint Da Capo Press, New York, 1973, 228 pages; reprint Dover, New York, 2004
- Easley Blackwood, The Structure of Recognizable Diatonic Tunings Princeton University Press, Princeton NJ, 1985
- John H. Chalmers Jr. "Construction and Harmonization of Microtonal Scales in non-twelve-tone Equal Temperaments", Proceedings of the 8th International Computer Music Conference, ICMA, 534-555
- Ramon Fuller, "A Study of Microtonal Equal Temperaments", Journal of Music Theory 35 no. 1-2, (1991), 211-237
- Donald E. Hall, "A Systematic Evaluation of Equal Temperaments Through N=612", Interface 14 no. 1-2, (1985), 61-73
- Kees van Prooijen, "A Theory of Equal-Tempered Scales", Interface 7 no. 1, (1978), 45-56[link]
- William S. Stoney, "Theoretical Possibilities for Equally Tempered Musical Systems", The Computer and Music, Harry B. Lincoln (ed.), Cornell University Press, Ithaca, 1970, 163-171
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