1012edo

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← 1011edo1012edo1013edo →
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