836edo
| ← 835edo | 836edo | 837edo → |
836 equal divisions of the octave (836edo), or 836-tone equal temperament (836tet), 836 equal temperament (836et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 836 equal parts of about 1.44 ¢ each.
Theory
836edo is a strong 11-limit system, having the lowest absolute error, beating 612edo.
The equal temperament tempers out the counterschisma and the enneadeca in the 5-limit; 4375/4374, 703125/702464 in the 7-limit; 3025/3024 and 9801/9800 in the 11-limit. It supports enneadecal in the 7-limit as well as hemienneadecal in the 11-limit. It also tunes orga and quasithird. In addition, it is divisible by 44 and in light of that it tunes ruthenium in the 7-limit and also 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | -0.041 | -0.189 | +0.074 | -0.122 | +0.621 | -0.171 | -0.384 | +0.434 | -0.391 | +0.419 |
| relative (%) | +0 | -3 | -13 | +5 | -8 | +43 | -12 | -27 | +30 | -27 | +29 | |
| Steps (reduced) |
836 (0) |
1325 (489) |
1941 (269) |
2347 (675) |
2892 (384) |
3094 (586) |
3417 (73) |
3551 (207) |
3782 (438) |
4061 (717) |
4142 (798) | |
Subsets and supersets
Since 836 factors into 22 × 11 × 19, 836edo has subset edos 2, 4, 11, 19, 22, 38, 44, 76, 209, 418.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1325 836⟩ | [⟨836 1325]] | +0.0130 | 0.0130 | 0.90 |
| 2.3.5 | [-14 -19 19⟩, [-69 45 -1⟩ | [⟨836 1325 1941]] | +0.0358 | 0.0340 | 2.37 |
| 2.3.5.7 | 4375/4374, 703125/702464, [41 -4 2 -14⟩ | [⟨836 1325 1941 2347]] | +0.0203 | 0.0399 | 2.78 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 234375/234256, [22 -4 2 -6 -1⟩ | [⟨836 1325 1941 2347 2892]] | +0.0233 | 0.0362 | 2.52 |
- 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past 612 and before 1084.
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 347\836 | 498.09 | 4/3 | Counterschismic |
| 2 | 161\836 | 231.10 | 8/7 | Orga (836f) |
| 2 | 265\836 (56\836) |
380.38 (80.38) |
81/65 (22/21) |
Quasithird (836) |
| 19 | 347\836 (5\836) |
498.09 (7.18) |
4/3 (225/224) |
Enneadecal |
| 38 | 347\836 (5\836) |
498.09 (7.18) |
4/3 (225/224) |
Hemienneadecal |
| 44 | 347\836 (5\836) |
498.09 (7.18) |
4/3 (18375/18304) |
Ruthenium |