# 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 $^{15}$√2 (=2$^{(1/15)}$), or 80 cents (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. 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 In addition, enharmonic equivalences from 12-EDO are no longer valid.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:
Size of intervals in 15 equal temperament :
| interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
| octave | 15 | 1200 | | 2:1 | 1200 | | 0 |
| perfect fifth | 9 | 720 | Play | 3:2 | 701.96 | Play | +18.04 |
| septimal tritone | 7 | 560 | Play | 7:5 | 582.51 | Play | −22.51 |
| 11:8 wide fourth | 7 | 560 | Play | 11:80 | 551.32 | Play | +08.68 |
| 15:11 wide fourth | 7 | 560 | Play | 15:11 | 536.95 | Play | +23.05 |
| perfect fourth | 6 | 480 | Play | 4:3 | 498.04 | Play | −18.04 |
| septimal major third | 5 | 400 | Play | 9:7 | 435.08 | Play | −35.08 |
| undecimal major third | 5 | 400 | Play | 14:11 | 417.51 | Play | −17.51 |
| major third | 5 | 400 | Play | 5:4 | 386.31 | Play | +13.69 |
| minor third | 4 | 320 | Play | 6:5 | 315.64 | Play | +04.36 |
| septimal minor third | 3 | 240 | Play | 7:6 | 266.87 | Play | −26.87 |
| septimal whole tone | 3 | 240 | Play | 8:7 | 231.17 | Play | +08.83 |
| major tone | 3 | 240 | Play | 9:8 | 203.91 | Play | +36.09 |
| minor tone | 2 | 160 | Play | 10:90 | 182.40 | Play | −22.40 |
| greater undecimal neutral second | 2 | 160 | Play | 11:10 | 165.00 | Play | −05.00 |
| lesser undecimal neutral second | 2 | 160 | Play | 12:11 | 150.63 | Play | +09.36 |
| just diatonic semitone | 1 | 080 | Play | 16:15 | 111.73 | Play | −31.73 |
| septimal chromatic semitone | 1 | 080 | Play | 21:20 | 084.46 | Play | −04.47 |
| just chromatic semitone | 1 | 080 | Play | 25:24 | 070.67 | 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 (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.

## Further subdivisions
### 45 equal temperament

45 equal temperament, which has a step size of 26.67 cents, is a threefold subdivision of 15 equal temperament. It has a perfect fifth of 693.33 cents, which is quite flat, but is still more accurate than the 720-cent fifth of 15-ET. Its best major third is 373.33 cents, which is slightly more accurate than the 400-cent one.
45-ET is an important example of a flattone temperament. Thus, it tempers out the syntonic comma like meantone temperaments do, but has a different mapping for intervals involving the seventh harmonic. Ordinarily, in meantone temperaments the 7:4 ratio is equated with the augmented sixth, whereas in 45-ET, this interval is instead equated with the diminished seventh, due to the smaller size of the chroma. This makes pieces involving chromatic alterations sound quite different.
45-ET is practically equivalent to an extended 2/5-comma meantone. Although it is harmonically less accurate than meantone temperaments like 31-ET or 19-ET, having a flatter fifth than both, it is still more accurate than 12-ET.

### 105 equal temperament

105 equal temperament (a sevenfold division of 15-ET) gives a meantone temperament with a fifth of 697.14 cents. It has a chromatic semitone equal to 80 cents, one step of 15-ET.

### 270 equal temperament

270 equal temperament does not see as much practical use as it is an extremely fine equal division with a step size of 4.44 cents, eighteen of which make up one step of 15-ET. Nonetheless, it is of academic interest due to being exceptionally accurate for an equal temperament of its size, particularly in the 15 odd limit.