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 about 10.811 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. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to 37edo, with which it shares the mappings for 5, 7, 11, and 13.
It is also significant for lower limits, especially in terms of what it tempers out in its patent val; 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-limit versions of semisept, the 31&80 temperament, and buzzard, the 53&58 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.
Prime harmonics
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Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [176 -111⟩ | [⟨111 176]] | -0.236 | 0.236 | 2.18 |
| 2.3.5 | 78732/78125, 67108864/66430125 | [⟨111 176 258]] | -0.570 | 0.510 | 4.72 |
| 2.3.5.7 | 1728/1715, 3136/3125, 5120/5103 | [⟨111 176 258 312]] | -0.797 | 0.591 | 5.47 |
| 2.3.5.7.11 | 176/175, 540/539, 1331/1323, 5120/5103 | [⟨111 176 258 312 384]] | -0.639 | 0.615 | 5.69 |
| 2.3.5.7.11.13 | 176/175, 351/350, 540/539, 676/675, 1331/1323 | [⟨111 176 258 312 384 411]] | -0.655 | 0.562 | 5.21 |
| 2.3.5.7.11.13.17 | 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 | [⟨111 176 258 312 384 411 454]] | -0.672 | 0.523 | 4.84 |
| 2.3.5.7.11.13.17.19 | 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 | [⟨111 176 258 312 384 411 454 472]] | -0.740 | 0.521 | 4.83 |
Rank-2 temperaments
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed.
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio (reduced) |
Temperament |
|---|---|---|---|---|
| 1 | 11\111 | 118.92 | 15/14 | Subsedia |
| 1 | 13\111 | 140.54 | 13/12 | Quanic |
| 1 | 16\111 | 172.97 | 400/363 | Undetrita |
| 1 | 31\111 | 335.14 | 17/14 | Cohemimabila |
| 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 |
| 1 | 49\111 | 529.73 | 19/14 | Tuskaloosa |
| 1 | 55\111 | 594.59 | 55/39 | Gaster |
| 3 | 7\111 | 75.68 | 24/23 | Terture |
| 3 | 12\111 | 129.73 | 14/13 | Trimabila |
| 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 10.811 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 |