316edo
| ← 315edo | 316edo | 317edo → |
The 316 equal divisions of the octave (316edo), or the 316(-tone) equal temperament (316tet, 316et) when viewed from a regular temperament perspective, divides the octave into 316 equal parts of about 3.80 cents each.
Theory
While not highly accurate for its size, 316et is the point where a few important temperaments meet, and is distinctly consistent in the 11-odd-limit. It tempers out the parakleisma, [8 14 -13⟩, the undim comma, [41 -20 -4⟩, and the maquila comma, [49 -6 -17⟩ in the 5-limit; 3136/3125, 4375/4374, 10976/10935 in the 7-limit; 3025/3024, 3388/3375, 9801/9800 and 14641/14580 in the 11-limit; and using the patent val, 1716/1715, 2080/2079 and 4096/4095 in the 13-limit; notably supporting abigail and semiparakleismic.
It provides the optimal patent val for the rank-4 temperament tempering out 3388/3375, and triglav, which also tempers out 3025/3024.
316 factors into 22 × 79, with subset edos 2, 4, 79, and 158.
Prime harmonics
| Prime number | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | |
|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +0.58 | +1.03 | -0.47 | -0.69 | -1.29 | +1.37 | -1.31 |
| relative (%) | +0 | +15 | +27 | -12 | -18 | -34 | +36 | -35 | |
| Steps (reduced) | 316 (0) | 501 (185) | 734 (102) | 887 (255) | 1093 (145) | 1169 (221) | 1292 (28) | 1342 (78) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [501 -316⟩ | [⟨316 501]] | -0.182 | 0.182 | 4.79 |
| 2.3.5 | [8 14 -13⟩, [41 -20 -4⟩ | [⟨316 501 734]] | -0.269 | 0.193 | 5.08 |
| 2.3.5.7 | 3136/3125, 4375/4374, [-26 -1 1 9⟩ | [⟨316 501 734 887]] | -0.160 | 0.252 | 6.64 |
| 2.3.5.7.11 | 3025/3024, 3136/3125, 4375/4374, 131072/130977 | [⟨316 501 734 887 1093]] | -0.088 | 0.267 | 7.04 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 3136/3125, 4096/4095 | [⟨316 501 734 887 1093 1169]] | -0.016 | 0.293 | 7.72 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 51\316 | 193.67 | 28/25 | Didacus |
| 1 | 83\316 | 315.19 | 6/5 | Parakleismic |
| 1 | 84\316 | 322.78 | 3087/2560 | Seniority |
| 1 | 141\316 | 535.44 | 512/375 | Maquila |
| 2 | 55\316 | 208.86 | 44/39 | Abigail |
| 2 | 83\316 (75\316) |
315.19 (284.81) |
6/5 (33/28) |
Semiparakleismic |
| 4 | 131\316 (27\316) |
497.47 (102.53) |
4/3 (35/33) |
Undim / unlit |