15 equal temperament
In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is a tempered scale derived by dividing the octave into 15 equal steps (equal frequency ratios). Each step represents a frequency ratio of 21/15, or 80 cents ( Play (help·info)). Because 15 factors into 3 times 5, it can be seen as being made up of three scales of 5 equal divisions of the octave, each of which resembles the Slendro scale in Indonesian gamelan. 15 equal temperament is not a meantone system.
History and use
Guitars have been constructed for 15-ET tuning. The American musician Wendy Carlos used 15-ET as one of two scales in the track Afterlife from the album Tales of Heaven and Hell. Easley Blackwood, Jr. has written and recorded a suite for 15-ET guitar. Blackwood believes that 15 equal temperament, "is likely to bring about a considerable enrichment of both classical and popular repertoire in a variety of styles".
Here are the sizes of some common intervals in 15-ET:
|interval name||size (steps)||size (cents)||just ratio||just (cents)||error||audio|
|11:8 wide fourth||7||560||11:8||551.32||+8.68||Play|
|15:11 wide fourth||7||560||15:11||536.95||+23.05||Play|
|septimal major third||5||400||9:7||435.08||−35.08||Play|
|undecimal major third||5||400||14:11||417.51||−17.51||Play|
|septimal minor third||3||240||7:6||266.87||−26.87||Play|
|septimal whole tone||3||240||8:7||231.17||+8.83||Play|
|greater undecimal neutral second||2||160||11:10||165.00||−5.00||Play|
|lesser undecimal neutral second||2||160||12:11||150.63||+9.36||Play|
|just diatonic semitone||1||80||16:15||111.73||−31.73||Play|
|septimal chromatic semitone||1||80||21:20||84.46||−4.47||Play|
|just chromatic semitone||1||80||25:24||70.67||+9.33||Play|
15-ET matches the 7th and 11th harmonics well, but only matches the 3rd and 5th harmonics roughly. The perfect fifth is more out of tune than in 12-ET, 19-ET, or 22-ET, and the major third in 15-ET is the same as the major third in 12-ET, but the other intervals matched are more in tune. 15-ET is the smallest tuning that matches the 11th harmonic at all and still has a usable perfect fifth, but its match to intervals utilizing the 11th harmonic is poorer than 22-ET, which also has more in-tune fifths and major thirds.
Although it contains a perfect fifth as well as major and minor thirds, the remainder of the harmonic and melodic language of 15-ET is quite different from 12-ET, and thus 15-ET could be described as xenharmonic. Unlike 12-ET and 19-ET, 15-ET matches the 11:8 and 16:11 ratios. 15-ET also has a neutral second and septimal whole tone. To construct a major third, one must stack two intervals of different sizes, whereas one can divide both the minor third and perfect fourth into two equal intervals.
- Myles Leigh Skinner (2007). Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky, p.52. ISBN 9780542998478.
- Skinner (2007), p.58n11. Cites Cohn, Richard (1997). "Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations", Journal of Music Theory 41/1.
- David J. Benson, Music: A Mathematical Offering, Cambridge University Press, (2006), p. 385. ISBN 9780521853873.
- Easley Blackwood, Jeffrey Kust, Easley Blackwood: Microtonal, Cedille (1996) ASIN: B0000018Z8.
- Skinner (2007), p.75.
- Ivor Darreg, "15-TONE SCALE SYSTEM" (1991), Sonic-Arts.org.
- Brewt: "Fifteen note equal temperament tutorial".