# 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 (=2^{(1/15)}), or 80 cents (

## 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:

interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
---|---|---|---|---|---|---|---|

octave | 15 | 1200 | 2:1 | 1200 | 0 | ||

perfect fifth | 9 | 720 | 3:2 | 701.96 | +18.04 | ||

septimal tritone | 7 | 560 | 7:5 | 582.51 | −22.51 | ||

11:8 wide fourth | 7 | 560 | 11:8 | 551.32 | + | 8.68||

15:11 wide fourth | 7 | 560 | 15:11 | 536.95 | +23.05 | ||

perfect fourth | 6 | 480 | 4:3 | 498.04 | −18.04 | ||

septimal major third | 5 | 400 | 9:7 | 435.08 | −35.08 | ||

undecimal major third | 5 | 400 | 14:11 | 417.51 | −17.51 | ||

major third | 5 | 400 | 5:4 | 386.31 | +13.69 | ||

minor third | 4 | 320 | 6:5 | 315.64 | + | 4.36||

septimal minor third | 3 | 240 | 7:6 | 266.87 | −26.87 | ||

septimal whole tone | 3 | 240 | 8:7 | 231.17 | + | 8.83||

major tone | 3 | 240 | 9:8 | 203.91 | +36.09 | ||

minor tone | 2 | 160 | 10:9 | 182.40 | −22.40 | ||

greater undecimal neutral second | 2 | 160 | 11:10 | 165.00 | − | 5.00||

lesser undecimal neutral second | 2 | 160 | 12:11 | 150.63 | + | 9.36||

just diatonic semitone | 1 | 80 | 16:15 | 111.73 | −31.73 | ||

septimal chromatic semitone | 1 | 80 | 21:20 | 84.46 | − | 4.47||

just chromatic semitone | 1 | 80 | 25:24 | 70.67 | + | 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.

## References

- 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.