319edo

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← 318edo319edo320edo →
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.

If we instead adopt the 2.9.… subgroup interpretation, then 2.9.15.21 is a good subgroup to start with.

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