113edo
113edo is the equal division of the octave into 113 parts of 10.6195 cents each.
Theory
113edo tempers out 1600000/1594323 and 34171875/33554432 in the 5-limit; 225/224, 1029/1024 and 1071875/1062882 in the 7-limit; 243/242, 385/384, and 441/440 in the 11-limit; 325/324, 364/363, 729/728, and 1625/1617 in the 13-limit. It supports the 5-limit amity temperament, 7-limit amicable temperament, 7- and 11-limit miracle temperament, and 13-limit manna temperament.
113edo is the 30th prime EDO.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-179 113⟩ | [⟨113 179]] | +0.338 | 0.338 | 3.18 |
| 2.3.5 | 1600000/1594323, 34171875/33554432 | [⟨113 179 262]] | +0.801 | 0.712 | 6.70 |
| 2.3.5.7 | 225/224, 1029/1024, 1071875/1062882 | [⟨113 179 262 317]] | +0.820 | 0.617 | 5.81 |
| 2.3.5.7.11 | 225/224, 243/242, 385/384, 980000/970299 | [⟨113 179 262 317 391]] | +0.604 | 0.700 | 6.59 |
| 2.3.5.7.11.13 | 225/224, 243/242, 325/324, 385/384, 1875/1859 | [⟨113 179 262 317 391 418]] | +0.575 | 0.643 | 6.05 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 6\113 | 63.717 | 28/27 | Sycamore / betic |
| 1 | 8\113 | 84.956 | 21/20 | Amicable |
| 1 | 11\113 | 116.814 | 15/14~16/15 | Miracle / manna |
| 1 | 13\113 | 138.053 | 27/25 | Quartemka |
| 1 | 22\113 | 233.628 | 8/7 | Slendric |
| 1 | 27\113 | 286.726 | 13/11 | Gamity |
| 1 | 29\113 | 307.965 | 3200/2673 | Familia |
| 1 | 32\113 | 339.823 | 243/200 | Amity / houborizic |
| 1 | 34\113 | 360.062 | 16/13 | Phicordial |
| 1 | 37\113 | 392.920 | 2744/2187 | Emmthird |
| 1 | 47\113 | 499.115 | 4/3 | Gracecordial |
Scales
Since 113edo has a step of 10.6195 cents, it also allows one to use its MOS scales as circulating temperaments[clarification needed]. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament.
| Tones | Pattern | L:s |
|---|---|---|
| 5 | 3L 2s | 23:22 |
| 6 | 5L 1s | 19:18 |
| 7 | 1L 6s | 17:16 |
| 8 | 1L 7s | 15:14 |
| 9 | 5L 4s | 13:12 |
| 10 | 3L 7s | 12:11 |
| 11 | 3L 8s | 11:10 |
| 12 | 5L 7s | 10:9 |
| 13 | 9L 4s | 9:8 |
| 14 | 1L 13s | |
| 15 | 7L 8s | 8:7 |
| 16 | 1L 15s | |
| 17 | 11L 6s | 7:6 |
| 18 | 5L 13s | |
| 19 | 18L 1s | 6:5 |
| 20 | 13L 7s | |
| 21 | 8L 13s | |
| 22 | 3L 19s | |
| 23 | 21L 2s | 5:4 |
| 24 | 17L 7s | |
| 25 | 13L 12s | |
| 26 | 9L 17s | |
| 27 | 5L 22s | |
| 28 | 1L 27s | |
| 29 | 26L 3s | 4:3 |
| 30 | 23L 7s | |
| 31 | 20L 11s | |
| 32 | 17L 15s | |
| 33 | 14L 19s | |
| 34 | 11L 23s | |
| 35 | 8L 27s | |
| 36 | 5L 31s | |
| 37 | 2L 35s | |
| 38 | 37L 1s | 3:2 |
| 39 | 35L 4s | |
| 40 | 33L 7s | |
| 41 | 31L 10s | |
| 42 | 29L 13s | |
| 43 | 27L 16s | |
| 44 | 25L 19s | |
| 45 | 23L 22s | |
| 46 | 21L 25s | |
| 47 | 19L 28s | |
| 48 | 17L 31s | |
| 49 | 15L 34s | |
| 50 | 13L 37s | |
| 51 | 11L 40s | |
| 52 | 9L 43s | |
| 53 | 7L 46s | |
| 54 | 5L 49s | |
| 55 | 3L 52s | |
| 56 | 1L 55s | |
| 57 | 56L 1s | 2:1 |
| 58 | 55L 3s | |
| 59 | 54L 5s | |
| 60 | 53L 7s | |
| 61 | 52L 9s | |
| 62 | 51L 11s | |
| 63 | 50L 13s | |
| 64 | 49L 15s | |
| 65 | 48L 17s | |
| 66 | 47L 19s | |
| 67 | 46L 21s | |
| 68 | 45L 23s | |
| 69 | 44L 25s | |
| 70 | 43L 27s | |
| 71 | 42L 29s | |
| 72 | 41L 31s | |
| 73 | 40L 33s | |
| 74 | 39L 35s | |
| 75 | 38L 37s | |
| 76 | 37L 39s | |
| 77 | 36L 41s | |
| 78 | 35L 43s | |
| 79 | 34L 45s | |
| 80 | 33L 47s | |
| 81 | 32L 49s | |
| 82 | 31L 51s | |
| 83 | 30L 53s | |
| 84 | 29L 55s | |
| 85 | 28L 57s | |
| 86 | 27L 59s | |
| 87 | 26L 61s | |
| 88 | 25L 63s | |
| 89 | 24L 65s | |
| 90 | 23L 67s |