961edo

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← 960edo961edo962edo →
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 1.249 ¢ 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 (%) -15 -37 +13 -30 +49 -12 +48 -5 -26 -2 -14
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

Scales