# 1012edo

← 1011edo | 1012edo | 1013edo → |

^{2}× 11 × 23**1012 equal divisions of the octave** (abbreviated **1012edo** or **1012ed2**), also called **1012-tone equal temperament** (**1012tet**) or **1012 equal temperament** (**1012et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1012 equal parts of about 1.186 ¢ each. Each step represents a frequency ratio of 2^{1/1012}, or the 1012th root of 2.

## Theory

1012edo is a strong 13-limit system, distinctly consistent through the 15-odd-limit. It is a zeta peak edo, though not zeta integral nor zeta gap. A basis for the 13-limit commas consists of 2401/2400, 4096/4095, 6656/6655, 9801/9800 and [2 6 -1 2 0 4⟩.

In the 5-limit, 1012edo is enfactored, with the same tuning as 506edo, supporting vishnu, monzismic, and lafa. In the 7-limit, it tempers out the breedsma, 2401/2400, and tunes the osiris temperament. Furthermore, noting its exceptional strength in the 2.3.7 subgroup, it is a septiruthenian system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the optimal patent val for quarvish temperament in the 7-limit and also in the 11-limit.

### Other techniques

In addition to containing 22edo and 23edo, it contains a 22L 1s scale produced by generator of 45\1012 associated with 33/32, and is associated with the 45 & 1012 temperament, making it concoctic. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, [18 15 -12 -1 0 -3⟩.

In the 2.3.7.11.101, it tempers out 7777/7776 and is a tuning for the neutron star temperament.

### Prime harmonics

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

Error | absolute (¢) | +0.000 | +0.021 | +0.248 | -0.051 | +0.065 | +0.184 | +0.578 | +0.115 | +0.184 | -0.328 | +0.419 |

relative (%) | +0 | +2 | +21 | -4 | +6 | +16 | +49 | +10 | +16 | -28 | +35 | |

Steps (reduced) |
1012 (0) |
1604 (592) |
2350 (326) |
2841 (817) |
3501 (465) |
3745 (709) |
4137 (89) |
4299 (251) |
4578 (530) |
4916 (868) |
5014 (966) |

### Subsets and supersets

Since 1012 factors into 2^{2} × 11 × 23, 1012edo has subset edos 2, 4, 11, 22, 23, 44, 46, 92, 253, 506. 2024edo, which divides the edostep in two, provides a good correction for the 17th harmonic.

## Regular temperament properties

### Rank-2 temperaments

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

1 | 361\1012 | 428.066 | 2800/2187 | Osiris |

2 | 491\1012 | 498.023 | 7/5 | Quarvish |

44 | 420\1012 (6\1012) |
498.023 (7.115) |
4/3 (18375/18304) |
Ruthenium |

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