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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 (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.

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., up-G = up-A-flat). Playⓘ
Downmajor 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ⓘ

History and use

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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

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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

Ups and Downs Notation,[6] uses up and down arrows, written as a caret and a lower-case "v", usually in a sans-serif font. One arrow equals one edostep. In note names, the arrows come first, to facilitate chord naming. This yields this chromatic scale:

B/C, ^C/^D♭, vC♯/vD,

D, ^D/^E♭, vD♯/vE,

E/F, ^F/^G♭, vF♯/vG,

G, ^G/^A♭, vG♯/vA,

A, ^A/^B♭, vA♯/vB, B/C

Chords are spelled differently. C–E♭–G is technically a C minor chord, but in fact it sounds like a sus2 chord C–D–G. The usual minor chord with 6/5 is the upminor chord. It's spelled as C–^E♭–G and named as C^m. Compare with ^Cm (^C–^E♭–^G).

Likewise the usual major chord with 5/4 is actually a downmajor chord. It's spelled as C–vE–G and named as Cv.

Porcupine Notation significantly changes chord spellings (e.g. the major triad is now C–E♯–G♯). In addition, enharmonic equivalences from 12-EDO are no longer valid. 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

One possible decatonic notation uses the digits 0-9. Each of the 3 circles of 5 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

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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ⓘ 11:8 551.32 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 (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.

Further subdivisions

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45 equal temperament

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45 equal temperament, which has a step size of 26.67 cents, is a threefold subdivision of 15 equal temperament. It has a perfect fifth of 693.33 cents, which is quite flat, but is still more accurate than the 720-cent fifth of 15-ET. Its best major third is 373.33 cents, which is slightly more accurate than the 400-cent one.

45-ET is an important example of a flattone temperament. Thus, it tempers out the syntonic comma like meantone temperaments do, but has a different mapping for intervals involving the seventh harmonic. Ordinarily, in meantone temperaments the 7:4 ratio is equated with the augmented sixth, whereas in 45-ET, this interval is instead equated with the diminished seventh, due to the smaller size of the chroma. This makes pieces involving chromatic alterations sound quite different.

45-ET is practically equivalent to an extended 2/5-comma meantone. Although it is harmonically less accurate than meantone temperaments like 31-ET or 19-ET, having a flatter fifth than both, it is still more accurate than 12-ET. [7]

105 equal temperament

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105 equal temperament (a sevenfold division of 15-ET) gives a meantone temperament with a fifth of 697.14 cents. It has a chromatic semitone equal to 80 cents, one step of 15-ET. [8]

270 equal temperament

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270 equal temperament does not see as much practical use as it is an extremely fine equal division with a step size of 4.44 cents, eighteen of which make up one step of 15-ET. Nonetheless, it is of academic interest due to being exceptionally accurate for an equal temperament of its size, particularly in the 15 odd limit.[9]

References

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  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.
  6. ^ "Ups_and_downs_notation", on Xenharmonic Wiki. Accessed 2023-8-12.
  7. ^ "45edo".
  8. ^ "105edo".
  9. ^ "270edo".
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