15 equal temperament

In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the 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 ( Play (help·info)). 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 which use 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.^[1] Easley Blackwood Jr. has written and recorded a suite for 15-ET guitar.^[2]

Interval size

Here are the sizes of some common intervals in 15-ET:

interval name	size (steps)	size (cents)	just ratio	just (cents)	error	audio
perfect fifth	9	720	3:2	701.96	+18.04	Play
septimal tritone	7	560	7:5	582.51	−22.51	Play
11:8 wide fourth	7	560	11:8	551.32	+8.68	Play
15:11 wide fourth	7	560	15:11	536.95	+23.05	Play
perfect fourth	6	480	4:3	498.04	−18.04	Play
septimal major third	5	400	9:7	435.08	−35.08	Play
undecimal major third	5	400	14:11	417.51	−17.51	Play
major third	5	400	5:4	386.31	+13.69	Play
minor third	4	320	6:5	315.64	+4.36	Play
septimal minor third	3	240	7:6	266.87	−26.87	Play
septimal whole tone	3	240	8:7	231.17	+8.83	Play
major tone	3	240	9:8	203.91	+36.09	Play
minor tone	2	160	10:9	182.40	−22.40	Play
greater undecimal neutral second	2	160	11:10	165.00	−5.00	Play
lesser undecimal neutral second	2	160	12:11	150.63	+9.36	Play
just diatonic semitone	1	80	16:15	111.73	−31.73	Play
septimal chromatic semitone	1	80	21:20	84.46	−4.47	Play
just chromatic semitone	1	80	25:24	70.67	+9.33	Play

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, and does not match the 7:5 or 10:7 tritone. 15-ET also has a neutral second and septimal whole tone. In order 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.

References

External links

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