15 Equal Temperament
Get 15 Equal Temperament essential facts below. View Videos or join the 15 Equal Temperament discussion. Add 15 Equal Temperament to your PopFlock.com topic list for future reference or share this resource on social media.
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). About this sound Play 
Diatonic scale on C in 15 equal temperament. About this sound 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). About this sound Play 15-et , About this sound Play just , or About this sound Play 12-et 

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 , or 80 cents (About this sound 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.

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]

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
perfect fifth 9 720 About this sound Play 3:2 701.96 About this sound Play +18.04
septimal tritone 7 560 About this sound Play 7:5 582.51 About this sound Play -22.51
11:8 wide fourth 7 560 About this sound Play 11:80 551.32 About this sound Play +08.68
15:11 wide fourth 7 560 About this sound Play 15:11 536.95 About this sound Play +23.05
perfect fourth 6 480 About this sound Play 4:3 498.04 About this sound Play -18.04
septimal major third 5 400 About this sound Play 9:7 435.08 About this sound Play -35.08
undecimal major third 5 400 About this sound Play 14:11 417.51 About this sound Play -17.51
major third 5 400 About this sound Play 5:4 386.31 About this sound Play +13.69
minor third 4 320 About this sound Play 6:5 315.64 About this sound Play +04.36
septimal minor third 3 240 About this sound Play 7:6 266.87 About this sound Play -26.87
septimal whole tone 3 240 About this sound Play 8:7 231.17 About this sound Play +08.83
major tone 3 240 About this sound Play 9:8 203.91 About this sound Play +36.09
minor tone 2 160 About this sound Play 10:90 182.40 About this sound Play -22.40
greater undecimal neutral second 2 160 About this sound Play 11:10 165.00 About this sound Play -05.00
lesser undecimal neutral second 2 160 About this sound Play 12:11 150.63 About this sound Play +09.36
just diatonic semitone 1 080 About this sound Play 16:15 111.73 About this sound Play -31.73
septimal chromatic semitone 1 080 About this sound Play 21:20 084.46 About this sound Play -04.47
just chromatic semitone 1 080 About this sound Play 25:24 070.67 About this sound 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. 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

  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.

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

15_equal_temperament
 



 


 
Music Scenes