836edo
| ← 835edo | 836edo | 837edo → |
836 equal divisions of the octave (abbreviated 836edo or 836ed2), also called 836-tone equal temperament (836tet) or 836 equal temperament (836et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 836 equal parts of about 1.44 ¢ each. Each step represents a frequency ratio of 21/836, or the 836th root of 2.
Theory
836edo is a strong 11-limit system, having the lowest absolute error and 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.
Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the patent val, it tempers out 2200/2197, 4096/4095, 31250/31213 in the 13-limit; and 1275/1274, 2500/2499, 2601/2600 in the 17-limit. It provides the optimal patent val for 13-limit quasithird. Using the 836f val, it tempers out 1716/1715, 2080/2079, 15379/15360 in the 13-limit; and 2431/2430, 2500/2499, 4914/4913, 5832/5831, 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.
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 (%) | +0.0 | -2.9 | -13.2 | +5.1 | -8.5 | +43.2 | -11.9 | -26.7 | +30.2 | -27.2 | +29.2 | |
| 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. 1672edo, which doubles it, provides a good correction for harmonic 13.
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 |
| 2.3.5.7.11.17 | 2500/2499, 3025/3024, 4375/4374, 57375/57344, 108086/108045 | [⟨836 1325 1941 2347 2892 3417]] | +0.0264 | 0.0337 | 2.35 |
| 2.3.5.7.11.13 | 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213 | [⟨836 1325 1941 2347 2892 3094]] (836) | −0.0085 | 0.0785 | 5.47 |
| 2.3.5.7.11.13.17 | 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374 | [⟨836 1325 1941 2347 2892 3094 3417]] (836) | −0.0014 | 0.0747 | 5.21 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256 | [⟨836 1325 1941 2347 2892 3093]] (836f) | +0.0561 | 0.0805 | 5.60 |
| 2.3.5.7.11.13.17 | 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4914/4913, 11271/11264 | [⟨836 1325 1941 2347 2892 3093 3417]] (836f) | +0.0541 | 0.0747 | 5.20 |
- 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* | Cents* | 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 |
| 22 | 347\836 (5\836) |
498.09 (7.18) |
4/3 ([16 -13 2⟩) |
Major arcana |
| 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct