Английская Википедия: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 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 In addition, enharmonic equivalences from 12-EDO are no longer valid.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:
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
External links
- Ivor Darreg, "15-TONE SCALE SYSTEM" (1991), Sonic-Arts.org.
- Brewt: "Fifteen note equal temperament tutorial".
- Noah Jordan: "The Devil" (piano work).
- Claudi Meneghin: "Tocada" (for Two Organs).
Шаблон:Microtonal music Шаблон:Musical tuning
- ↑ Myles Leigh Skinner (2007). Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky, p.52. Шаблон:ISBN.
- ↑ 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. Шаблон:ISBN.
- ↑ Easley Blackwood, Jeffrey Kust, Easley Blackwood: Microtonal, Cedille (1996) ASIN: B0000018Z8.
- ↑ Skinner (2007), p.75.
- ↑ Шаблон:Xenharmonic wiki Accessed 2023-8-12.
