58 equal temperament

In music, 58 equal temperament (also called 58-ET or 58edo) 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 equal temperament, which is also consistent in the 17-limit, 58-ET's approximations of most intervals are not quite as good (although still workable). One obvious exception is the perfect fifth (slightly better in 58-ET), and another is the tridecimal minor third (11:13), which is significantly better in 58-ET than in 72-ET. The two systems temper out different commas; 72-ET tempers out the comma 169:168, thus equating the 14:13 and 13:12 intervals. On the other hand, 58-ET 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-ET, unlike 72-ET, is not a multiple of 12, so the only interval (up to octave equivalency) that it shares with 12-ET is the 600-cent tritone (which functions as both 17:12 and 24:17). On the other hand, 58-ET has fewer pitches than 72-ET and is therefore simpler.

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