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 $\sqrt{2|15}$ (=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.

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
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 In addition, enharmonic equivalences from 12-EDO are no longer valid.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.

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.