114edo
114edo is the equal division of the octave into 114 parts, each of 10.52632 cents. In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and 117440512/117406179, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for shrutar temperament; it is in fact the optimal patent val for shrutar in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.
Period of 19-limit Shrutar
| Degree | Cents | Difference from 68edo |
|---|---|---|
| 2 | 21.05263 | 3.40557¢ |
| 3 | 31.57895 | -3.71517¢ |
| 5 | 52.63158 | -0.3096¢ |
| 7 | 73.68421 | 3.096¢ |
| 8 | 84.21053 | -4.02477¢ |
| 10 | 105.26316 | -0.619195¢ |
| 12 | 126.31579 | 2.78638¢ |
| 13 | 136.842105 | -4.334365¢ |
| 15 | 157.89474 | -0.9288¢ |
| 17 | 178.94737 | 2.47678¢ |
| 18 | 189.47369 | -4.644¢ |
| 20 | 210.52632 | -1.23839¢ |
| 22 | 231.57895 | 2.16718¢ |
| 23 | 242.10526 | -4.953560372 |
| 25 | 263.157895 | -1.548¢ |
| 27 | 284.21053 | 1.857585¢ |
| 29 | 305.26316 | 5.26316¢ |
| 30 | 315.78947 | -1.857585¢ |
| 32 | 336.842105 | 1.548¢ |
| 34 | 357.89474 | 4.95356¢ |
| 35 | 368.42105 | -2.16718¢ |
| 37 | 389.47368 | 1.23839¢ |
| 39 | 410.52632 | 4.64396¢ |
| 40 | 421.05263 | -2.47678¢ |
| 42 | 442.10526 | 0.92879¢ |
| 44 | 463.157895 | 4.334365¢ |
| 45 | 473.68421 | -2.78638¢ |
| 47 | 494.73684 | 0.619195¢ |
| 49 | 515.78947 | 4.02477¢ |
| 50 | 526.31579 | -3.095975¢ |
| 52 | 547.36842 | 0.3096¢ |
| 54 | 568.42105 | 3.71517¢ |
| 55 | 578.94737 | -3.40557¢ |
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
| Tones | Pattern | L:s |
|---|---|---|
| 5 | 4L 1s | 23:22 |
| 6 | 6edo | equal |
| 7 | 2L 5s | 17:16 |
| 8 | 2L 6s | 15:14 |
| 9 | 6L 3s | 13:12 |
| 10 | 4L 6s | 12:11 |
| 11 | 4L 7s | 11:10 |
| 12 | 6L 6s | 10:9 |
| 13 | 10L 3s | 9:8 |
| 14 | 2L 12s | |
| 15 | 9L 6s | 8:7 |
| 16 | 2L 14s | |
| 17 | 12L 5s | 7:6 |
| 18 | 6L 12s | |
| 19 | 19edo | equal |
| 20 | 14L 6s | 6:5 |
| 21 | 9L 12s | |
| 22 | 4L 18s | |
| 23 | 22L 1s | 5:4 |
| 24 | 18L 6s | |
| 25 | 14L 11s | |
| 26 | 10L 16s | |
| 27 | 6L 21s | |
| 28 | 2L 26s | |
| 29 | 27L 2s | 4:3 |
| 30 | 24L 6s | |
| 31 | 21L 10s | |
| 32 | 18L 14s | |
| 33 | 15L 18s | |
| 34 | 12L 22s | |
| 35 | 9L 26s | |
| 36 | 6L 30s | |
| 37 | 3L 34s | |
| 38 | 38edo | equal |
| 39 | 36L 3s | 3:2 |
| 40 | 34L 6s | |
| 41 | 32L 9s | |
| 42 | 30L 12s | |
| 43 | 28L 15s | |
| 44 | 26L 18s | |
| 45 | 24L 21s | |
| 46 | 22L 24s | |
| 47 | 20L 27s | |
| 48 | 18L 30L | |
| 49 | 16L 33s | |
| 50 | 14L 36s | |
| 51 | 12L 39s | |
| 52 | 10L 42s | |
| 53 | 8L 45s | |
| 54 | 6L 48s | |
| 55 | 4L 52s | |
| 56 | 2L 54s | |
| 57 | 57edo | equal |
| 58 | 56L 2s | 2:1 |
| 59 | 55L 4s | |
| 60 | 54L 6s | |
| 61 | 53L 8s | |
| 62 | 52L 10s | |
| 63 | 51L 12s | |
| 64 | 50L 14s | |
| 65 | 49L 16s | |
| 66 | 48L 18s | |
| 67 | 47L 20s | |
| 68 | 46L 22s | |
| 69 | 45L 24s | |
| 70 | 44L 26s | |
| 71 | 43L 28s | |
| 72 | 42L 30s | |
| 73 | 41L 32s | |
| 74 | 40L 34s | |
| 75 | 39L 36s | |
| 76 | 38L 38s | |
| 77 | 37L 40s | |
| 78 | 36L 42s | |
| 79 | 35L 44s | |
| 80 | 34L 46s | |
| 81 | 33L 48s | |
| 82 | 32L 50s | |
| 83 | 31L 52s | |
| 84 | 30L 54s | |
| 85 | 29L 56s | |
| 86 | 28L 58s | |
| 87 | 27L 60s | |
| 88 | 26L 62s | |
| 89 | 25L 64s | |
| 90 | 24L 66s | |
| 91 | 23L 68s |