# 59edo

← 58edo | 59edo | 60edo → |

**59 equal divisions of the octave** (abbreviated **59edo**), or **59-tone equal temperament** (**59tet**), **59 equal temperament** (**59et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each. Each step of 59edo represents a frequency ratio of 2^{1/59}, or the 59th root of 2.

## Theory

59edo's best fifth is stretched about 9.91 cents from the just interval, and yet its major third is nearly pure (stretched only 0.127 cents), as the denominator of a convergent to log_{2}5. It is a good porcupine tuning, giving in fact the optimal patent val for 11-limit porcupine. This patent val tempers out 250/243 in the 5-limit, 64/63 and 16875/16807 in the 7-limit, and 55/54, 100/99 and 176/175 in the 11-limit. As every other step of 118edo, 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 2*59 subgroup, on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50&59 temperament with a subminor third generator provides an interesting temperament.

Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to garibaldi temperament with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for flattone temperament in the 59bc val, a variant of meantone with flat fifths.

59edo is the 17th prime edo.

Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | absolute (¢) | +9.91 | +0.13 | +7.45 | -0.52 | -2.17 | -6.63 | +10.04 | -3.26 | +7.57 | -2.98 | +2.23 | +0.25 | +9.39 |

relative (%) | +49 | +1 | +37 | -3 | -11 | -33 | +49 | -16 | +37 | -15 | +11 | +1 | +46 | |

Steps (reduced) |
94 (35) |
137 (19) |
166 (48) |
187 (10) |
204 (27) |
218 (41) |
231 (54) |
241 (5) |
251 (15) |
259 (23) |
267 (31) |
274 (38) |
281 (45) |

## Intervals

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## Instruments

**Lumatone**

See Lumatone mapping for 59edo