# 3125edo

← 3124edo | 3125edo | 3126edo → |

^{5}**3125 equal divisions of the octave** (abbreviated **3125edo** or **3125ed2**), also called **3125-tone equal temperament** (**3125tet**) or **3125 equal temperament** (**3125et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3125 equal parts of exactly 0.384 ¢ each. Each step represents a frequency ratio of 2^{1/3125}, or the 3125th root of 2.

## Theory

3125edo is distinctly consistent through the 15-odd-limit. A basis for its 7-limit commas is 78125000/78121827, 645700815/645657712 and 281484423828125/281474976710656. In the 11-limit, 151263/151250, 820125/819896, 21437500/21434787 and 117440512/117406179 are tempered out – it should be noted this edo is so far the only one known to have been confirmed as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five 33/32 quartertones and one 7/6 subminor third. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.

In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general, described as 1789 & 3125.

### Prime harmonics

Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +0.000 | -0.003 | -0.010 | +0.006 | +0.106 | +0.048 | -0.123 | +0.087 | -0.050 | -0.073 | +0.052 |

Relative (%) | +0.0 | -0.8 | -2.5 | +1.6 | +27.6 | +12.6 | -32.1 | +22.7 | -13.1 | -19.1 | +13.7 | |

Steps (reduced) |
3125 (0) |
4953 (1828) |
7256 (1006) |
8773 (2523) |
10811 (1436) |
11564 (2189) |
12773 (273) |
13275 (775) |
14136 (1636) |
15181 (2681) |
15482 (2982) |

### Subsets and supersets

3125 = 5^{5}, and as such it is the 5th edo of the form *n*^{n}. It has subset edos 5, 25, 125, and 625.

## Regular temperament properties

3125et is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower relative error.

### Rank-2 temperaments

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

1 | 139\3125 | 53.376 | 33/32 | Prequartismic |

1 | 411\3125 | 157.824 | 36756909/33554432 | Hemiegads |

1 | 577\3125 | 221.568 | 8388608/7381125 | Fortune |

1 | 822\3125 | 315.648 | 6/5 | Egads |

1 | 894\3125 | 343.296 | 8000/6561 | Raider |

1 | 1359\3125 | 521.856 | 80275/59392 | Estates general |

1 | 1412\3125 | 542.208 | 16807/12288 | Revopent |

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct