# 58 equal temperament

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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,[1] 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. [2]

## List of intervals

 interval name size (steps) size (cents) just ratio just (cents) error perfect fifth 34 703.45 3:2 701.96 +1.49 greater septendecimal tritone 29 600 17:12 603.00 −3.00 lesser septendecimal tritone 24:17 597.00 +3.00 perfect fourth 24 496.55 4:3 498.04 −1.49 major third 19 393.10 5:4 386.31 +6.79 tridecimal semifourth 12 248.28 15:13 247.74 +0.54 septimal whole tone 11 227.59 8:7 231.17 −3.58 whole tone, major tone 10 206.90 9:8 203.91 +2.99 whole tone, minor tone 9 186.21 10:9 182.40 +3.81 greater undecimal neutral second 8 165.52 11:10 165.00 +0.52 lesser undecimal neutral second 7 144.83 12:11 150.64 −5.81 septimal diatonic semitone 6 124.14 15:14 119.44 +4.70 septendecimal semitone; 17th harmonic 5 103.45 17:16 104.96 −1.51 diatonic semitone 5 103.45 16:15 111.73 −8.28 septimal chromatic semitone 4 82.76 21:20 84.47 −1.71 chromatic semitone 3 62.07 25:24 70.67 −8.60 septimal third tone 3 62.07 28:27 62.96 −0.89 septimal quarter tone 2 41.38 36:35 48.77 −7.39 septimal diesis 2 41.38 49:48 35.70 +5.68 septimal comma 1 20.69 64:63 27.26 +6.57 syntonic comma 1 20.69 81:80 21.51 +0.82