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 152 (=2(1/15)), 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]

Notation

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.

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
octave 15 1200 2:1 1200 0
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 11:80 551.32 Play +08.68
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 +04.36
septimal minor third 3 240 Play 7:6 266.87 Play −26.87
septimal whole tone 3 240 Play 8:7 231.17 Play +08.83
major tone 3 240 Play 9:8 203.91 Play +36.09
minor tone 2 160 Play 10:90 182.40 Play −22.40
greater undecimal neutral second 2 160 Play 11:10 165.00 Play 05.00
lesser undecimal neutral second 2 160 Play 12:11 150.63 Play +09.36
just diatonic semitone 1 080 Play 16:15 111.73 Play −31.73
septimal chromatic semitone 1 080 Play 21:20 084.46 Play 04.47
just chromatic semitone 1 080 Play 25:24 070.67 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 (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.

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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