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 15√2, 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) midi just ratio just (cents) midi error
perfect fifth 9 720  Play 3:2 701.96  Play +18.04
septimal tritone 7 560  Play 7:5 582.51  Play −22.51
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
perfect fourth 6 480  Play 4:3 498.04  Play −18.04
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
major third 5 400  Play 5:4 386.31  Play +13.69
minor third 4 320  Play 6:5 315.64  Play +4.36
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
major tone 3 240  Play 9:8 203.91  Play +36.09
minor tone 2 160  Play 10:9 182.40  Play −22.40
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. 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.

References

  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.
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