# 961edo

 ← 960edo 961edo 962edo →
Prime factorization 312
Step size 1.2487¢
Fifth 562\961 (701.769¢)
Semitones (A1:m2) 90:73 (112.4¢ : 91.16¢)
Consistency limit 5
Distinct consistency limit 5

961 equal divisions of the octave (abbreviated 961edo or 961ed2), also called 961-tone equal temperament (961tet) or 961 equal temperament (961et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 961 equal parts of about 1.25 ¢ each. Each step represents a frequency ratio of 21/961, or the 961st root of 2.

## Theory

961edo has a reasonable 7-limit interpretation. The equal temperament tempers out the schisma in the 5-limit; 4375/4374, 65625/65536, and 14348907/14336000 in the 7-limit, supporting pontiac, the 395 & 566 temperament.

In the 11-limit, the 961e val 961 1523 2231 2698 3324] scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val 961 1523 2231 2697 3324], which tempers out 3025/3024 and 184877/184320. The patent val 961 1523 2231 2698 3325] tempers out 4000/3993 and 46656/46585.

It works much better as a 2.3.5.7.13.17 subgroup temperament, in which case it tempers out 10985/10976, 1275/1274, 2025/2023 and 4914/4913.

### Odd harmonics

Approximation of odd harmonics in 961edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.186 -0.466 +0.165 -0.372 +0.607 -0.153 +0.597 -0.065 -0.323 -0.021 -0.179
Relative (%) -14.9 -37.3 +13.2 -29.8 +48.6 -12.3 +47.8 -5.2 -25.8 -1.7 -14.3
Steps
(reduced)
1523
(562)
2231
(309)
2698
(776)
3046
(163)
3325
(442)
3556
(673)
3755
(872)
3928
(84)
4082
(238)
4221
(377)
4347
(503)

### Subsets and supersets

Since 961 factors into 312, 961edo has 31edo as its subset edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-1523 961 [961 1523]] 0.0587 0.0587 4.70
2.3.5 32805/32768, [-22 -137 103 [961 1523 2231]] 0.1060 0.0823 6.59
2.3.5.7 4375/4374, 32805/32768, [15 9 14 -22 [961 1523 2231 2698]] 0.0648 0.1008 8.01

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 399\961 498.231 4/3 Pontiac

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