# 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, 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.

## Interval size

| interval name | size (steps) | size (cents) | just ratio | just (cents) | error (cents) |
| 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 | 29 | 600 | 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 | 17 | 351.72 | 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 |