111edo
| ← 110edo | 111edo | 112edo → |
111 equal divisions of the octave (abbreviated 111edo or 111ed2), also called 111-tone equal temperament (111tet) or 111 equal temperament (111et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 111 equal parts of about 10.8 ¢ each. Each step represents a frequency ratio of 21/111, or the 111th root of 2.
Theory
111edo is consistent through to the 21-odd-limit, and is the smallest edo distinctly consistent through the 15-odd-limit, marking it as an important higher limit tuning. It has a sharp tendency, with primes 3 through 19 all tuned sharp. Since 111 = 3 × 37, 111edo shares the mappings for 5, 7, 11, and 13 with 37edo.
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.
It further tempers out among others 351/350, 352/351, 640/637, 676/675, 847/845, 1001/1000, 1188/1183, 1573/1568 in the 13-limit; 256/255, 325/324, 442/441 in the 17-limit; 286/285, 400/399, 476/475 in the 19-limit. It excels as a full 23-limit temperament, tempering out 253/252 and 276/275. The 23 is tuned a little flat, unlike the lower primes. 23/19, 23/21 and their octave complements are the only inconsistently mapped intervals in the 23-odd-limit.
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. Gene Ward Smith's trio in #Music section is in guanyin temperament, the rank-3 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 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.75 | +2.88 | +4.15 | +0.03 | +2.72 | +3.15 | +5.19 | -1.25 |
| Relative (%) | +0.0 | +6.9 | +26.6 | +38.4 | +0.3 | +25.1 | +29.2 | +48.0 | -11.5 | |
| Steps (reduced) |
111 (0) |
176 (65) |
258 (36) |
312 (90) |
384 (51) |
411 (78) |
454 (10) |
472 (28) |
502 (58) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.55 | +0.91 | -2.70 | +3.37 | -3.41 | +4.76 | +2.17 | +0.29 | -3.37 |
| Relative (%) | -23.6 | +8.4 | -24.9 | +31.2 | -31.5 | +44.1 | +20.1 | +2.7 | -31.2 | |
| Steps (reduced) |
539 (95) |
550 (106) |
578 (23) |
595 (40) |
602 (47) |
617 (62) |
636 (81) |
653 (98) |
658 (103) | |
Octave stretch
111edo can benefit from slightly compressing the octave if that is acceptable, using tunings such as 176edt or 287ed6. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit.
Subsets and supersets
Since 111 factors into primes as 3 × 37, 111edo contains 3edo and 37edo as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's 13-odd-limit. 333edo, which slices the step of 111edo in three, is a significant tuning.
Intervals
| # | Cents | Approximated ratios* | Ups and downs notation |
|---|---|---|---|
| 0 | 0.0 | 1/1 | D |
| 1 | 10.8 | 121/120, 126/125, 144/143, 169/168, 196/195, 225/224 | ^D, ^4E♭♭ |
| 2 | 21.6 | 64/63, 81/80, 91/90, 100/99, 105/104 | ^^D, ^5E♭♭ |
| 3 | 32.4 | 46/45, 50/49, 55/54, 56/55, 57/56, 65/64 | ^3D, v5E♭ |
| 4 | 43.2 | 36/35, 39/38, 40/39, 45/44, 49/48 | ^4D, v4E♭ |
| 5 | 54.1 | 33/32, 34/33, 35/34 | ^5D, v3E♭ |
| 6 | 64.9 | 26/25, 27/26, 28/27 | v5D♯, vvE♭ |
| 7 | 75.7 | 22/21, 23/22, 24/23, 25/24 | v4D♯, vE♭ |
| 8 | 86.5 | 20/19, 21/20 | v3D♯, E♭ |
| 9 | 97.3 | 18/17, 19/18 | vvD♯, ^E♭ |
| 10 | 108.1 | 16/15, 17/16 | vD♯, ^^E♭ |
| 11 | 118.9 | 15/14 | D♯, ^3E♭ |
| 12 | 129.7 | 14/13 | ^D♯, ^4E♭ |
| 13 | 140.5 | 13/12 | ^^D♯, ^5E♭ |
| 14 | 151.4 | 12/11 | ^3D♯, v5E |
| 15 | 162.2 | 11/10 | ^4D♯, v4E |
| 16 | 173.0 | 21/19 | ^5D♯, v3E |
| 17 | 183.8 | 10/9 | v5D𝄪, vvE |
| 18 | 194.6 | 19/17, 28/25 | v4D𝄪, vE |
| 19 | 205.4 | 9/8 | E |
| 20 | 216.2 | 17/15, 26/23 | ^E, ^4F♭ |
| 21 | 227.0 | 8/7 | ^^E, ^5F♭ |
| 22 | 237.8 | 23/20 | ^3E, v5F |
| 23 | 248.6 | 15/13 | ^4E, v4F |
| 24 | 259.5 | 22/19 | ^5E, v3F |
| 25 | 270.3 | 7/6 | v5E♯, vvF |
| 26 | 281.1 | 20/17 | v4E♯, vF |
| 27 | 291.9 | 13/11 | F |
| 28 | 302.7 | 19/16, 25/21 | ^F, ^4G♭♭ |
| 29 | 313.5 | 6/5 | ^^F, ^5G♭♭ |
| 30 | 324.3 | 23/19, 77/64 | ^3F, v5G♭ |
| 31 | 335.1 | 17/14, 40/33 | ^4F, v4G♭ |
| 32 | 345.9 | 11/9, 28/23, 39/32 | ^5F, v3G♭ |
| 33 | 356.8 | 16/13, 27/22 | v5F♯, vvG♭ |
| 34 | 367.6 | 21/17, 26/21 | v4F♯, vG♭ |
| 35 | 378.4 | 56/45 | v3F♯, G♭ |
| 36 | 389.2 | 5/4 | vvF♯, ^G♭ |
| 37 | 400.0 | 24/19, 34/27 | vF♯, ^^G♭ |
| 38 | 410.8 | 19/15 | F♯, ^3G♭ |
| 39 | 421.6 | 14/11, 23/18 | ^F♯, ^4G♭ |
| 40 | 432.4 | 9/7 | ^^F♯, ^5G♭ |
| 41 | 443.2 | 22/17 | ^3F♯, v5G |
| 42 | 454.1 | 13/10 | ^4F♯, v4G |
| 43 | 464.9 | 17/13 | ^5F♯, v3G |
| 44 | 475.7 | 21/16, 25/19 | v5F𝄪, vvG |
| 45 | 486.5 | 45/34, 65/49 | v4F𝄪, vG |
| 46 | 497.3 | 4/3 | G |
| 47 | 508.1 | 51/38 | ^G, ^4A♭♭ |
| 48 | 518.9 | 23/17, 27/20 | ^^G, ^5A♭♭ |
| 49 | 529.7 | 19/14 | ^3G, v5A♭ |
| 50 | 540.5 | 15/11, 26/19 | ^4G, v4A♭ |
| 51 | 551.4 | 11/8 | ^5G, v3A♭ |
| 52 | 562.2 | 18/13 | v5G♯, vvA♭ |
| 53 | 573.0 | 32/23 | v4G♯, vA♭ |
| 54 | 583.8 | 7/5 | v3G♯, A♭ |
| 55 | 594.6 | 24/17 | vvG♯, ^A♭ |
| … | … | … | … |
* As a 23-limit temperament, inconsistently mapped intervals in italic
Approximation to JI
Interval mappings
The following table shows how 15-odd-limit intervals are represented in 111edo. Prime harmonics are in bold.
As 111edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 11/8, 16/11 | 0.033 | 0.3 |
| 13/10, 20/13 | 0.160 | 1.5 |
| 15/14, 28/15 | 0.524 | 4.8 |
| 11/6, 12/11 | 0.714 | 6.6 |
| 3/2, 4/3 | 0.748 | 6.9 |
| 15/13, 26/15 | 0.908 | 8.4 |
| 13/9, 18/13 | 1.220 | 11.3 |
| 7/5, 10/7 | 1.272 | 11.8 |
| 9/5, 10/9 | 1.380 | 12.8 |
| 13/7, 14/13 | 1.431 | 13.2 |
| 11/9, 18/11 | 1.462 | 13.5 |
| 9/8, 16/9 | 1.495 | 13.8 |
| 13/12, 24/13 | 1.968 | 18.2 |
| 5/3, 6/5 | 2.128 | 19.7 |
| 9/7, 14/9 | 2.652 | 24.5 |
| 13/11, 22/13 | 2.682 | 24.8 |
| 13/8, 16/13 | 2.716 | 25.1 |
| 11/10, 20/11 | 2.842 | 26.3 |
| 5/4, 8/5 | 2.875 | 26.6 |
| 7/6, 12/7 | 3.399 | 31.4 |
| 15/11, 22/15 | 3.590 | 33.2 |
| 15/8, 16/15 | 3.623 | 33.5 |
| 11/7, 14/11 | 4.114 | 38.1 |
| 7/4, 8/7 | 4.147 | 38.4 |
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 655zpi | 111.059578 | 10.805011 | 9.038544 | 6.131468 | 1.394739 | 18.041165 | 1199.356259 | −0.643741 | 22 | 16 |
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 |
| 2.3.5.7.11.13.17.19.23 | 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399 | [⟨111 176 258 312 384 411 454 472 502]] | −0.628 | 0.586 | 5.43 |
- 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating 94 and 103h before being superseded by 121i.
Rank-2 temperaments
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
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 | 22/17 | Warrior |
| 1 | 43\111 | 464.86 | 17/13 | Semisept |
| 1 | 44\111 | 475.68 | 21/16 | 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) |
15/13 (12/11) |
Hemimist |
| 3 | 46\111 (9\111) |
497.30 (97.30) |
4/3 (18/17~19/18) |
Misty |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- Direct sunlight (subset of Sensi[19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
- Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)
Music
- Trio for SoftSaturn, NebulaSing and TromBonehead (archived 2010) – SoundCloud | details | play – in Guanyin[22], 111edo tuning