# 58 equal temperament

In music, 58 equal temperament (also called 58-ET or 58-EDO) 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-EDO, which is also consistent in the 17-limit, 58-EDO's approximations of most intervals are not quite as good (although still workable). One obvious exception is the perfect fifth (slightly better in 58-EDO), and another is the tridecimal minor third (11:13), 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 (up to octave equivalency) that it shares with 12-EDO is the 600-cent tritone (which functions as both 17:12 and 24:17). 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.[2]

## Interval size

 interval name size (steps) size (cents) just ratio just (cents) error octave 58 1200 2:1 1200 0 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 septimal tritone 28 579.31 7:5 582.51 −3.20 eleventh harmonic 27 558.62 11:8 551.32 +7.30 15:11 wide fourth 26 537.93 15:11 536.95 +0.98 perfect fourth 24 496.55 4:3 498.04 −1.49 septimal narrow fourth 23 475.86 21:16 470.78 +5.08 tridecimal major third 22 455.17 13:10 454.21 +0.96 septimal major third 21 434.48 9:7 435.08 −0.60 undecimal major third 20 413.79 14:11 417.51 −3.72 major third 19 393.10 5:4 386.31 +6.79 tridecimal neutral third 17 351.72 16:13 359.47 −7.75 undecimal neutral third 11:9 347.41 +4.31 minor third 15 310.34 6:5 315.64 −5.30 tridecimal minor third 14 289.66 13:11 289.21 +0.45 septimal minor third 13 268.97 7:6 266.87 +2.10 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