111edo
The 111 equal divisions of the octave (111edo), or the 111(-tone) equal temperament (111tet, 111et) when viewed from a regular temperament perspective, is the equal division of the octave into 111 parts, each of size 10.81 cents.
Theory
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 tuning.
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 for the corresponding 11-limit rank-4 temperament.
In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the quartisma.
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 in #Music section 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 111 = 3 × 37.
Prime harmonics
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Regular temperament properties
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio (reduced) |
Temperament |
|---|---|---|---|---|
| 1 | 13\111 | 140.54 | 13/12 | Quanic |
| 1 | 35\111 | 378.38 | 56/45 | Subpental |
| 1 | 41\111 | 443.24 | 162/125 | Sensipent |
| 1 | 43\111 | 464.86 | 17/13 | Semisept |
| 1 | 44\111 | 475.68 | 21/16 | Vulture / buzzard |
| 1 | 46\111 | 497.30 | 4/3 | Kwai |
| 3 | 13\111 | 140.54 | 243/224 | Septichrome |
| 3 | 17\111 | 183.55 | 10/9 | Mirkat |
| 3 | 23\111 (14\111) |
248.65 (151.35) |
231/200 (12/11) |
Hemimist |
| 3 | 46\111 (9\111) |
497.30 (97.30) |
4/3 (18/17~19/18) |
Misty |
Scales
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments[clarification needed].
| 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 | 13L 1s | 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 |