Английская Википедия:15 equal temperament

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Шаблон:Short description

Файл:15-tet scale on C.png
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). Шаблон:Audio
Файл:15-tet diatonic scale on C.svg
Downmajor scale on C in 15 equal temperament. Шаблон:Audio
Файл:Major chord on C in 15 equal temperament.png
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). Шаблон:Audio, Шаблон:Audio, or Шаблон:Audio

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 Шаблон:Radic (=2Шаблон:Sup), or 80 cents (Шаблон:Audio). 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:

BШаблон:Music/C, CШаблон:Music/DШаблон:Music, D, DШаблон:Music, EШаблон:Music, E, EШаблон:Music/F, FШаблон:Music/GШаблон:Music, G, GШаблон:Music, AШаблон:Music, A, AШаблон:Music, BШаблон:Music, B, BШаблон:Music/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/^DШаблон:Music, vCШаблон:Music/vD,

D, ^D/^EШаблон:Music, vDШаблон:Music/vE,

E/F, ^F/^GШаблон:Music, vFШаблон:Music/vG,

G, ^G/^AШаблон:Music, vGШаблон:Music/vA,

A, ^A/^BШаблон:Music, vAШаблон:Music/vB, B/C

Chords are spelled differently. C–EШаблон:Music–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–^EШаблон:Music–G and named as C^m. Compare with ^Cm (^C–^EШаблон:Music–^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, CШаблон:Music/DШаблон:Music, D, DШаблон:Music/EШаблон:Music, E, EШаблон:Music/FШаблон:Music, F, FШаблон:Music/GШаблон:Music, G, GШаблон:Music, AШаблон:Music, A, AШаблон:Music/BШаблон:Music, B, BШаблон:Music, 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, 1Шаблон:Music/2Шаблон:Music, 2, 3, 3Шаблон:Music/4Шаблон:Music, 4, 5, 5Шаблон:Music/6Шаблон:Music, 6, 7, 7Шаблон:Music/8Шаблон:Music, 8, 9, 9Шаблон:Music/0Шаблон:Music, 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 Шаблон:Audio 3:2 701.96 Шаблон:Audio +18.04
septimal tritone 7 560 Шаблон:Audio 7:5 582.51 Шаблон:Audio −22.51
11:8 wide fourth 7 560 Шаблон:Audio 11:8Шаблон:0 551.32 Шаблон:Audio +Шаблон:08.68
15:11 wide fourth 7 560 Шаблон:Audio 15:11 536.95 Шаблон:Audio +23.05
perfect fourth 6 480 Шаблон:Audio 4:3 498.04 Шаблон:Audio −18.04
septimal major third 5 400 Шаблон:Audio 9:7 435.08 Шаблон:Audio −35.08
undecimal major third 5 400 Шаблон:Audio 14:11 417.51 Шаблон:Audio −17.51
major third 5 400 Шаблон:Audio 5:4 386.31 Шаблон:Audio +13.69
minor third 4 320 Шаблон:Audio 6:5 315.64 Шаблон:Audio +Шаблон:04.36
septimal minor third 3 240 Шаблон:Audio 7:6 266.87 Шаблон:Audio −26.87
septimal whole tone 3 240 Шаблон:Audio 8:7 231.17 Шаблон:Audio +Шаблон:08.83
major tone 3 240 Шаблон:Audio 9:8 203.91 Шаблон:Audio +36.09
minor tone 2 160 Шаблон:Audio 10:9Шаблон:0 182.40 Шаблон:Audio −22.40
greater undecimal neutral second 2 160 Шаблон:Audio 11:10 165.00 Шаблон:Audio Шаблон:05.00
lesser undecimal neutral second 2 160 Шаблон:Audio 12:11 150.63 Шаблон:Audio +Шаблон:09.36
just diatonic semitone 1 Шаблон:080 Шаблон:Audio 16:15 111.73 Шаблон:Audio −31.73
septimal chromatic semitone 1 Шаблон:080 Шаблон:Audio 21:20 Шаблон:084.46 Шаблон:Audio Шаблон:04.47
just chromatic semitone 1 Шаблон:080 Шаблон:Audio 25:24 Шаблон:070.67 Шаблон:Audio +Шаблон: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

Шаблон:Reflist

External links

Шаблон:Microtonal music Шаблон:Musical tuning

  1. Myles Leigh Skinner (2007). Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky, p.52. Шаблон:ISBN.
  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. Шаблон:ISBN.
  4. Easley Blackwood, Jeffrey Kust, Easley Blackwood: Microtonal, Cedille (1996) ASIN: B0000018Z8.
  5. Skinner (2007), p.75.
  6. Шаблон:Xenharmonic wiki Accessed 2023-8-12.