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(1/15)), 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. ⓘ
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".
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, 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 In addition, enharmonic equivalences from 12-EDO are no longer valid.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.
Here are the sizes of some common intervals in 15-ET:
|interval name||size (steps)||size (cents)||midi||just ratio||just (cents)||midi||error|
|11:8 wide fourth||7||560||ⓘ||11:8||551.32||ⓘ||+8.68|
|15:11 wide fourth||7||560||ⓘ||15:11||536.95||ⓘ||+23.05|
|septimal major third||5||400||ⓘ||9:7||435.08||ⓘ||−35.08|
|undecimal major third||5||400||ⓘ||14:11||417.51||ⓘ||−17.51|
|septimal minor third||3||240||ⓘ||7:6||266.87||ⓘ||−26.87|
|septimal whole tone||3||240||ⓘ||8:7||231.17||ⓘ||+8.83|
|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 (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.