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).  Play 
Diatonic scale on C in 15 equal temperament.  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).  Play 15-et ,  Play just , or  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 21/15, or 80 cents ( 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) just ratio just (cents) error audio
perfect fifth 9 720 3:2 701.96 +18.04  Play
septimal tritone 7 560 7:5 582.51 22.51  Play
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
perfect fourth 6 480 4:3 498.04 18.04  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
major third 5 400 5:4 386.31 +13.69  Play
minor third 4 320 6:5 315.64 +4.36  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
major tone 3 240 9:8 203.91 +36.09  Play
minor tone 2 160 10:9 182.40 22.40  Play
greater undecimal neutral second 2 160 11:10 165.00 5.00  [[:Media:|Play]]
lesser undecimal neutral second 2 160 12:11 150.63 +9.36  [[:Media:|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.


  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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