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. Each step represents a frequency ratio of , or 80 cents. 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. Easley Blackwood, Jr. has written and recorded a suite for 15-ET guitar. 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".Notation
'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:| interval name | size | size | midi | just ratio | just | midi | error |
| octave | 15 | 1200 | 2:1 | 1200 | 0 | ||
| perfect fifth | 9 | 720 | 3:2 | 701.96 | +18.04 | ||
| septimal tritone | 7 | 560 | 7:5 | 582.51 | −22.51 | ||
| 11:8 wide fourth | 7 | 560 | 11:8 | 551.32 | +8.68 | ||
| 15:11 wide fourth | 7 | 560 | 15:11 | 536.95 | +23.05 | ||
| perfect fourth | 6 | 480 | 4:3 | 498.04 | −18.04 | ||
| septimal major third | 5 | 400 | 9:7 | 435.08 | −35.08 | ||
| undecimal major third | 5 | 400 | 14:11 | 417.51 | −17.51 | ||
| major third | 5 | 400 | 5:4 | 386.31 | +13.69 | ||
| minor third | 4 | 320 | 6:5 | 315.64 | +4.36 | ||
| septimal minor third | 3 | 240 | 7:6 | 266.87 | −26.87 | ||
| septimal whole tone | 3 | 240 | 8:7 | 231.17 | +8.83 | ||
| major tone | 3 | 240 | 9:8 | 203.91 | +36.09 | ||
| minor tone | 2 | 160 | 10:9 | 182.40 | −22.40 | ||
| greater undecimal neutral second | 2 | 160 | 11:10 | 165.00 | −5.00 | ||
| lesser undecimal neutral second | 2 | 160 | 12:11 | 150.63 | +9.36 | ||
| just diatonic semitone | 1 | 80 | 16:15 | 111.73 | −31.73 | ||
| septimal chromatic semitone | 1 | 80 | 21:20 | 84.46 | −4.47 | ||
| just chromatic semitone | 1 | 80 | 25:24 | 70.67 | +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. 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.