111edo
111edo is the equal division of the octave into 111 parts, each of size 10.81 cents.
111edo is consistent through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent optimal patent val tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and 117440512/117406179. It is a particularly good tuning for the 11- or 13-versions of semisept, the 31&111 temperament, and buzzard, the 58&111 temperament. The Trio piece below is in guanyin temperament, the planar temperament tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.
The prime factorization is
[math]\displaystyle{ 111 = 3 \cdot 37 }[/math]
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments. As 10*11edo, it is the first edo which allows one to use consecutive smaller edos as circulating temperaments.
| Tones | Pattern | L:s |
|---|---|---|
| 5 | 1L 4s | 23:22 |
| 6 | 3L 3s | 19:18 |
| 7 | 6L 1s | 16:15 |
| 8 | 7L 1s | 14:13 |
| 9 | 3L 6s | 13:12 |
| 10 | 1L 9s | 12:11 |
| 11 | 1L 10s | 11:10 |
| 12 | 3L 9s | 10:9 |
| 13 | 6L 7s | 9:8 |
| 14 | 11L 3s | 8:7 |
| 15 | 6L 9s | |
| 16 | 15L 1s | 7:6 |
| 17 | 9L 8s | |
| 18 | 3L 15s | |
| 19 | 16L 3s | 6:5 |
| 20 | 11L 9s | |
| 21 | 6L 15s | |
| 22 | 1L 21s | |
| 23 | 19L 4s | 5:4 |
| 24 | 15L 9s | |
| 25 | 11L 14s | |
| 26 | 7L 19s | |
| 27 | 3L 24s | |
| 28 | 27L 1s | 4:3 |
| 29 | 24L 5s | |
| 30 | 21L 9s | |
| 31 | 18L 13s | |
| 32 | 15L 17s | |
| 33 | 12L 21s | |
| 34 | 9L 25s | |
| 35 | 6L 29s | |
| 36 | 3L 33s | |
| 37 | 37edo | equal |
| 38 | 35L 3s | 3:2 |
| 39 | 33L 6s | |
| 40 | 31L 9s | |
| 41 | 29L 12s | |
| 42 | 27L 15s | |
| 43 | 25L 18s | |
| 44 | 23L 21s | |
| 45 | 21L 24s | |
| 46 | 19L 27s | |
| 47 | 17L 30s | |
| 48 | 15L 33s | |
| 49 | 13L 36s | |
| 50 | 11L 39s | |
| 51 | 9L 42s | |
| 52 | 7L 45s | |
| 53 | 5L 48s | |
| 54 | 3L 51s | |
| 55 | 1L 54s | |
| 56 | 55L 1s | 2:1 |
| 57 | 54L 3s | |
| 58 | 53L 5s | |
| 59 | 52L 7s | |
| 60 | 51L 9s | |
| 61 | 50L 11s | |
| 62 | 49L 13s | |
| 63 | 48L 15s | |
| 64 | 47L 17s | |
| 65 | 46L 19s | |
| 66 | 45L 21s | |
| 67 | 44L 23s | |
| 68 | 43L 25s | |
| 69 | 42L 27s | |
| 70 | 41L 29s | |
| 71 | 40L 31s | |
| 72 | 39L 33s | |
| 73 | 38L 35s | |
| 74 | 37L 37s | |
| 75 | 36L 39s | |
| 76 | 35L 41s | |
| 77 | 34L 43s | |
| 78 | 33L 45s | |
| 79 | 32L 47s | |
| 80 | 31L 49s | |
| 81 | 30L 51s | |
| 82 | 29L 53s | |
| 83 | 28L 55s | |
| 84 | 27L 57s | |
| 85 | 26L 59s | |
| 86 | 25L 61s | |
| 87 | 24L 63s | |
| 88 | 23L 65s |
Music
Trio for SoftSaturn, NebulaSing and TromBonehead play by Gene Ward Smith