# 836edo

 ← 835edo 836edo 837edo →
Prime factorization 22 × 11 × 19
Step size 1.43541¢
Fifth 489\836 (701.914¢)
Semitones (A1:m2) 79:63 (113.4¢ : 90.43¢)
Consistency limit 11
Distinct consistency limit 11

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

Approximation of prime harmonics in 836edo
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

Table of rank-2 temperaments by generator
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)
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)