# 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,^{[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 |

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 |

## See also

- Harry Partch's 43-tone scale; 58-ET is the smallest equal temperament that can reasonably approximate this scale