316edo
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, 2197/2187, 4096/4095, 4225/4224, 6656/6655, and 10648/10647 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.58 | +1.03 | -0.47 | -0.69 | -1.29 | +1.37 | -1.31 | -1.69 | -0.46 | +1.80 |
| Relative (%) | +0.0 | +15.2 | +27.1 | -12.4 | -18.0 | -33.9 | +36.2 | -34.5 | -44.6 | -12.2 | +47.4 | |
| Steps (reduced) |
316 (0) |
501 (185) |
734 (102) |
887 (255) |
1093 (145) |
1169 (221) |
1292 (28) |
1342 (78) |
1429 (165) |
1535 (271) |
1566 (302) | |
Subsets and supersets
316 factors into 22 × 79, with subset edos 2, 4, 79, and 158.
Regular temperament properties
Template:Comma basis begin |- | 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, 2197/2187, 3025/3024, 3136/3125 | [⟨316 501 734 887 1093 1169]] | −0.016 | 0.293 | 7.72 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 51\316
| 193.67
| 28/25
| Didacus
|-
| 1
| 83\316
| 315.19
| 6/5
| Parakleismic (7-limit)
|-
| 1
| 84\316
| 322.78
| 3087/2560
| Seniority
|-
| 1
| 141\316
| 535.44
| 512/375
| Maquila (5-limit)
|-
| 1
| 149\316
| 565.82
| 18/13
| Threedic
|-
| 1
| 155\316
| 588.61
| 45927/32768
| Countritonic (7-limit)
|-
| 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)
| Unlit
Template:Rank-2 end
Template:Orf