# 577edo

← 576edo | 577edo | 578edo → |

**577 equal divisions of the octave** (abbreviated **577edo** or **577ed2**), also called **577-tone equal temperament** (**577tet**) or **577 equal temperament** (**577et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 577 equal parts of about 2.08 ¢ each. Each step represents a frequency ratio of 2^{1/577}, or the 577th root of 2.

## Theory

577edo is consistent to the 7-odd-limit, but its harmonic 3 is about halfway its steps. In the 2.9.5.7.11 subgroup interpretation, it tempers out 26873856/26796875, 184528125/184473632 and 1640558367/1638400000 in the 7-limit; 5632/5625, 102487/102400, 151263/151250, and 472392/471625 in the 11-limit.

### Odd harmonics

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

Error | Absolute (¢) | +0.991 | +0.515 | +0.325 | -0.097 | -0.191 | -0.320 | -0.574 | -0.969 | -0.113 | -0.764 | -0.198 |

Relative (%) | +47.7 | +24.7 | +15.6 | -4.7 | -9.2 | -15.4 | -27.6 | -46.6 | -5.4 | -36.7 | -9.5 | |

Steps (reduced) |
915 (338) |
1340 (186) |
1620 (466) |
1829 (98) |
1996 (265) |
2135 (404) |
2254 (523) |
2358 (50) |
2451 (143) |
2534 (226) |
2610 (302) |

### Subsets and supersets

577edo is the 106th prime edo. 1154edo, which doubles it, gives a good correction to the harmonic 3, but it does poorly in the harmonics 5 and 7. 2308edo, which quadruples it, also gives a good correction to the harmonic 3 and is consistent to the 11-odd-limit.

## Regular temperament properties

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

Absolute (¢) | Relative (%) | ||||

2.9 | [-1829 577⟩ | [⟨577 1829]] | +0.0153 | 0.0153 | 0.74 |

2.9.5 | [-7 11 -12⟩, [125 -5 -47⟩ | [⟨577 1829 1340]] | -0.0637 | 0.1124 | 5.40 |

2.9.5.7 | 26873856/26796875, 184528125/184473632, 1640558367/1638400000 | [⟨577 1829 1340 1620]] | -0.0767 | 0.0999 | 4.80 |

2.9.5.7.11 | 5632/5625, 102487/102400, 151263/151250, 472392/471625 | [⟨577 1829 1340 1620 1996]] | -0.0503 | 0.1038 | 4.99 |

2.9.5.7.11.13 | 1001/1000, 10648/10647, 10985/10976, 75712/75625, 472392/471625 | [⟨577 1829 1340 1620 1996 2135]] | -0.0275 | 0.1076 | 5.17 |