316edo

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The 316 equal divisions of the octave (316edo), or the 316(-tone) equal temperament (316tet, 316et), divides the octave into 316 equal parts of 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.

Prime harmonics

Approximation of prime intervals in 316 EDO
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

Table of rank-2 temperaments by generator
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
(4428675/4194304)
Undim