15 Equal Temperament
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15 Equal Temperament
Easley Blackwood's[1] notation system for 15 equal temperament: intervals are notated similarly to those they approximate and there are different enharmonic equivalents (e.g., G-up = A-flat-up). About this sound Play 
Diatonic scale on C in 15 equal temperament. About this sound Play 
Major chord (parsimonious trichord[2]) on C in 15 equal temperament: all notes within 18 cents of just intonation (rather than 14 for 12 equal temperament). About this sound Play 15-et , About this sound Play just , or About this sound Play 12-et 

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 , or 80 cents (About this sound Play ). 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.[3]Easley Blackwood, Jr. has written and recorded a suite for 15-ET guitar.[4] 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".[5]

Interval size

Here are the sizes of some common intervals in 15-ET:

Size of intervals in 15 equal temperament
interval name size (steps) size (cents) midi just ratio just (cents) midi error
perfect fifth 9 720 About this sound Play 3:2 701.96 About this sound Play +18.04
septimal tritone 7 560 About this sound Play 7:5 582.51 About this sound Play -22.51
11:8 wide fourth 7 560 About this sound Play 11:80 551.32 About this sound Play +08.68
15:11 wide fourth 7 560 About this sound Play 15:11 536.95 About this sound Play +23.05
perfect fourth 6 480 About this sound Play 4:3 498.04 About this sound Play -18.04
septimal major third 5 400 About this sound Play 9:7 435.08 About this sound Play -35.08
undecimal major third 5 400 About this sound Play 14:11 417.51 About this sound Play -17.51
major third 5 400 About this sound Play 5:4 386.31 About this sound Play +13.69
minor third 4 320 About this sound Play 6:5 315.64 About this sound Play +04.36
septimal minor third 3 240 About this sound Play 7:6 266.87 About this sound Play -26.87
septimal whole tone 3 240 About this sound Play 8:7 231.17 About this sound Play +08.83
major tone 3 240 About this sound Play 9:8 203.91 About this sound Play +36.09
minor tone 2 160 About this sound Play 10:90 182.40 About this sound Play -22.40
greater undecimal neutral second 2 160 About this sound Play 11:10 165.00 About this sound Play -05.00
lesser undecimal neutral second 2 160 About this sound Play 12:11 150.63 About this sound Play +09.36
just diatonic semitone 1 080 About this sound Play 16:15 111.73 About this sound Play -31.73
septimal chromatic semitone 1 080 About this sound Play 21:20 084.46 About this sound Play -04.47
just chromatic semitone 1 080 About this sound Play 25:24 070.67 About this sound Play +09.33

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. 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.


  1. ^ Myles Leigh Skinner (2007). Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky, p.52. ISBN 9780542998478.
  2. ^ Skinner (2007), p.58n11. Cites Cohn, Richard (1997). "Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations", Journal of Music Theory 41/1.
  3. ^ David J. Benson, Music: A Mathematical Offering, Cambridge University Press, (2006), p. 385. ISBN 9780521853873.
  4. ^ Easley Blackwood, Jeffrey Kust, Easley Blackwood: Microtonal, Cedille (1996) ASIN: B0000018Z8.
  5. ^ Skinner (2007), p.75.

External links

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