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 15√ (=2(1/15)), 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".
Easley Blackwood, Jr.'s notation of 15-EDO creates this chromatic scale:
B♯/C, C♯/D♭, D, D♯, E♭, E, E♯/F, F♯/G♭, G, G♯, A♭, A, A♯, B♭, B, B♯/C
An alternate form of notation, which is sometimes called "Porcupine Notation," can be used. It yields the following chromatic scale:
C, C♯/D♭, D, D♯/E♭, E, E♯/F♭, F, F♯/G♭, G, G♯, A♭, A, A♯/B♭, B, B♯, C
A notation that uses the numerals is also possible, in which each chain of fifths is notated either by the odd numbers, the even numbers, or with accidentals.
1, 1♯/2♭, 2, 3, 3♯/4♭, 4, 5, 5♯/6♭, 6, 7, 7♯/8♭, 8, 9, 9♯/0♭, 0, 1
In this article, unless specified otherwise, Blackwood's notation will be used.
Here are the sizes of some common intervals in 15-ET:
|interval name||size (steps)||size (cents)||midi||just ratio||just (cents)||midi||error|
|11:8 wide fourth||7||560||Play||11:8||551.32||Play||+8.68|
|15:11 wide fourth||7||560||Play||15:11||536.95||Play||+23.05|
|septimal major third||5||400||Play||9:7||435.08||Play||−35.08|
|undecimal major third||5||400||Play||14:11||417.51||Play||−17.51|
|septimal minor third||3||240||Play||7:6||266.87||Play||−26.87|
|septimal whole tone||3||240||Play||8:7||231.17||Play||+8.83|
|greater undecimal neutral second||2||160||Play||11:10||165.00||Play||−5.00|
|lesser undecimal neutral second||2||160||Play||12:11||150.63||Play||+9.36|
|just diatonic semitone||1||80||Play||16:15||111.73||Play||−31.73|
|septimal chromatic semitone||1||80||Play||21:20||84.46||Play||−4.47|
|just chromatic semitone||1||80||Play||25:24||70.67||Play||+9.33|
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 (except for the septimal tritones). 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 in 15-ET, 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.