58 equal temperament


In music, 58 equal temperament divides the octave into 58 equal parts of approximately 20.69 cents each. It is notable as the simplest equal division of the octave to faithfully represent the 17-limit, and the first that distinguishes between all the elements of the 11-limit tonality diamond. The next-smallest equal temperament to do both these things is 72 equal temperament.
Compared to 72-EDO, which is also consistent in the 17-limit, 58-EDO's approximations of most intervals are not quite as good. One obvious exception is the perfect fifth, and another is the tridecimal minor third, which is significantly better in 58-EDO than in 72-EDO. The two systems temper out different commas; 72-EDO tempers out the comma 169:168, thus equating the 14:13 and 13:12 intervals. On the other hand, 58-EDO tempers out 144:143 instead of 169:168, so 14:13 and 13:12 are left distinct, but 13:12 and 12:11 are equated.
58-EDO, unlike 72-EDO, is not a multiple of 12, so the only interval that it shares with 12-EDO is the 600-cent tritone. On the other hand, 58-EDO has fewer pitches than 72-EDO and is therefore simpler.

History and use

The medieval Italian music theorist Marchetto da Padova proposed a system that is approximately 29-EDO, which is a subset of 58-EDO, in 1318.

Interval size