15 equal temperament

File:15-tet scale on C.png

Easley Blackwood's 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).

File:15-tet diatonic scale on C.svg

Diatonic scale on C in 15 equal temperament.

File:Major chord on C in 15 equal temperament.png

Major chord (parsimonious trichord) on C in 15 equal temperament: all notes within 18 cents of just intonation (rather than 14 for 12 equal temperament). , <phonos file="Major chord on C in just intonation.mid">Play just</phonos>, or <phonos file="Major chord on C.mid">Play 12-et</phonos>

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 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".

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)	just ratio	just (cents)	error	audio
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	<phonos file="Major third on C.mid">Play</phonos>
undecimal major third	5	400	14:11	417.51	−17.51	<phonos file="Major third on C.mid">Play</phonos>
major third	5	400	5:4	386.31	+13.69	<phonos file="Major third on C.mid">Play</phonos>
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, one must stack two intervals of different sizes, whereas one can divide both the minor third and perfect fourth into two equal intervals.

External links

• Ivor Darreg, "15-TONE SCALE SYSTEM" (1991), Sonic-Arts.org.
• Brewt: "Fifteen note equal temperament tutorial".