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 (help·info)

Diatonic scale on C in 15 equal temperament.

Play (help·info)

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 (help·info),

Play just (help·info), or

Play 12-et (help·info)

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 (Play (help·info)). 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[]

^ 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.

External links[]

Ivor Darreg, "15-TONE SCALE SYSTEM" (1991), Sonic-Arts.org.
Brewt: "Fifteen note equal temperament tutorial".
Noah Jordan: "The Devil" (piano work).

[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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