# 677edo

← 676edo | 677edo | 678edo → |

**677 equal divisions of the octave** (abbreviated **677edo** or **677ed2**), also called **677-tone equal temperament** (**677tet**) or **677 equal temperament** (**677et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 677 equal parts of about 1.77 ¢ each. Each step represents a frequency ratio of 2^{1/677}, or the 677th root of 2.

## Theory

While it does well as a 2.3.5.11 tuning system, it's notable for its high accuracy among EDOs of about its size and lower with the first several metallic ratios. Among those, it tunes acoustic phi (the golden ratio) and the acoustic copper ratio each with less than 1% relative error. The first nine metallic ratios are all tuned within 20% of an edostep.

### Prime harmonics

Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
---|---|---|---|---|---|---|---|---|---|---|

Error | absolute (¢) | +0.00 | -0.03 | +0.10 | +0.75 | -0.06 | -0.35 | -0.38 | +0.27 | -0.80 |

relative (%) | +0 | -2 | +5 | +42 | -4 | -20 | -21 | +15 | -45 | |

Steps (reduced) |
677 (0) |
1073 (396) |
1572 (218) |
1901 (547) |
2342 (311) |
2505 (474) |
2767 (59) |
2876 (168) |
3062 (354) |

### Subsets and supersets

677edo is the 123rd prime EDO.

## Regular temperament properties

Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|

Absolute (¢) | Relative (%) | ||||

2.3 | [-1073 677⟩ | [⟨677 1073]] | +0.0110 | 0.0110 | 0.62 |

2.3.5 | [38 -2 -15⟩, [-31 43 -16⟩ | [⟨677 1073 1572]] | -0.0066 | 0.0264 | 1.49 |

2.3.5.7 | 703125/702464, 589824/588245, 14348907/14336000 | [⟨677 1073 1572 1901]] | -0.0714 | 0.1145 | 6.46 |

2.3.5.7.11 | 3025/3024, 24057/24010, 131072/130977, 759375/758912 | [⟨677 1073 1572 1901 2342]] | -0.0535 | 0.1084 | 6.12 |

### Rank-2 temperaments

Periods per 8ve |
Generator (reduced)* |
Cents (reduced)* |
Associated Ratio* |
Temperaments |
---|---|---|---|---|

1 | 109\677 | 193.21 | 262144/234375 | Luna |

1 | 125\677 | 221.57 | 8388608/7381125 | Fortune |

1 | 281\677 | 498.08 | 4/3 | Counterschismic |