# 1012edo

 ← 1011edo 1012edo 1013edo →
Prime factorization 22 × 11 × 23
Step size 1.18577¢
Fifth 592\1012 (701.976¢) (→148\253)
Semitones (A1:m2) 96:76 (113.8¢ : 90.12¢)
Consistency limit 15
Distinct consistency limit 15
Special properties

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 1.19 ¢ each. Each step represents a frequency ratio of 21/1012, 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

Approximation of prime harmonics in 1012edo
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 (%) +0.0 +1.8 +20.9 -4.3 +5.5 +15.5 +48.8 +9.7 +15.5 -27.7 +35.3
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 22 × 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

Table of rank-2 temperaments by generator
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