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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.75 | +2.88 | +4.15 | +0.03 | +2.72 | +3.15 | +5.19 | -1.25 | -2.55 | +0.91 |
| Relative (%) | +0.0 | +6.9 | +26.6 | +38.4 | +0.3 | +25.1 | +29.2 | +48.0 | -11.5 | -23.6 | +8.4 | |
| Steps (reduced) |
111 (0) |
176 (65) |
258 (36) |
312 (90) |
384 (51) |
411 (78) |
454 (10) |
472 (28) |
502 (58) |
539 (95) |
550 (106) | |
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 8ve |
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 | 14\111 | 151.35 | 12/11 | Browser |
| 1 | 16\111 | 172.97 | 400/363 | Undetrita |
| 1 | 20\111 | 216.22 | 17/15 | Tremka |
| 1 | 23\111 | 248.65 | 15/13 | Hemikwai |
| 1 | 31\111 | 335.14 | 17/14 | Cohemimabila |
| 1 | 35\111 | 378.38 | 56/45 | Subpental |
| 1 | 41\111 | 443.24 | 162/125 | Sensipent / warrior |
| 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 |