In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the octave into 15 equally large steps. Each step represents a frequency ratio of 21/15, 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 Indonesiangamelan.
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.1Easley 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:
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
^ David J. Benson, Music: A Mathematical Offering, Cambridge University Press, (2006), p. 385.
