# 319edo

 ← 318edo 319edo 320edo →
Prime factorization 11 × 29
Step size 3.76176¢
Fifth 187\319 (703.448¢) (→17\29)
Semitones (A1:m2) 33:22 (124.1¢ : 82.76¢)
Dual sharp fifth 187\319 (703.448¢) (→17\29)
Dual flat fifth 186\319 (699.687¢)
Dual major 2nd 54\319 (203.135¢)
Consistency limit 7
Distinct consistency limit 7

319 equal divisions of the octave (abbreviated 319edo or 319ed2), also called 319-tone equal temperament (319tet) or 319 equal temperament (319et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 319 equal parts of about 3.76 ¢ each. Each step represents a frequency ratio of 21/319, or the 319th root of 2.

## Theory

319edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway its steps. The full 11-limit patent val is nonetheless a reasonable interpretation since the lower harmonics all tend sharp. Using this val, it tempers out 6144/6125, 10976/10935, and [9 -10 9 -5 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 15488/15435, 59290/59049, and 65536/65219 in the 11-limit. It supports mystery in the 5-limit and protolangwidge.

### Odd harmonics

Approximation of odd harmonics in 319edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.49 +1.15 +1.71 -0.78 +1.66 -1.66 -1.12 +0.37 -0.33 -0.56 -0.06
Relative (%) +39.7 +30.5 +45.4 -20.6 +44.1 -44.0 -29.8 +9.9 -8.9 -14.9 -1.6
Steps
(reduced)
506
(187)
741
(103)
896
(258)
1011
(54)
1104
(147)
1180
(223)
1246
(289)
1304
(28)
1355
(79)
1401
(125)
1443
(167)

### Subsets and supersets

Since 319 factors into 11 × 29, 319edo has 11edo and 29edo as its subsets. 638edo, which doubles it, gives a good correction to the harmonics 3, 5, and 7.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [-1011 319 [319 1011]] +0.1223 0.1223 3.25
2.9.15 [-51 5 9, [-16 26 -17 [319 1011 1246]] +0.1869 0.1353 3.60