961edo
| ← 960edo | 961edo | 962edo → |
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
| 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
| 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