319edo
| ← 318edo | 319edo | 320edo → |
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
| 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 (%) | +40 | +30 | +45 | -21 | +44 | -44 | -30 | +10 | -9 | -15 | -2 | |
| 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 |