111ed12
| ← 110ed12 | 111ed12 | 112ed12 → |
111 equal divisions of the 12th harmonic (abbreviated 111ed12) is a nonoctave tuning system that divides the interval of 12/1 into 111 equal parts of about 38.8 ¢ each. Each step represents a frequency ratio of 121/111, or the 111th root of 12.
Theory
111ed12 is nearly identical to 31edo, but with the 12th harmonic rather than the octave being just. The octave is about 1.45 cents stretched compared to 31edo. Like 31edo, 111ed12 is consistent through the 12-integer-limit, and like 80ed6, it optimizes for the 11-limit by trading the accuracy of the 5th and 7th harmonics for improved 3rd and 11th harmonics. The stretch is quite mild, but still considerable: the 11th harmonic is only 4.4 cents flat of just (in comparison, 31edo's 11th harmonic is 9.4 cents flat). Also improved is the 23rd harmonic, which is now only 2.4 cents flat of just.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.4 | -2.9 | +2.9 | +4.1 | -1.4 | +3.0 | +4.3 | -5.8 | +5.6 | -4.4 | +0.0 |
| Relative (%) | +3.7 | -7.5 | +7.5 | +10.7 | -3.7 | +7.7 | +11.2 | -14.9 | +14.4 | -11.3 | +0.0 | |
| Steps (reduced) |
31 (31) |
49 (49) |
62 (62) |
72 (72) |
80 (80) |
87 (87) |
93 (93) |
98 (98) |
103 (103) |
107 (107) |
111 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +16.5 | +4.4 | +1.2 | +5.8 | +17.1 | -4.3 | +18.3 | +7.0 | +0.1 | -2.9 | -2.4 | +1.4 |
| Relative (%) | +42.5 | +11.4 | +3.2 | +14.9 | +44.1 | -11.2 | +47.3 | +18.2 | +0.2 | -7.6 | -6.2 | +3.7 | |
| Steps (reduced) |
115 (4) |
118 (7) |
121 (10) |
124 (13) |
127 (16) |
129 (18) |
132 (21) |
134 (23) |
136 (25) |
138 (27) |
140 (29) |
142 (31) | |
Subsets and supersets
Since 111 factors into primes as 3 × 37, 111ed12 contains 3ed12 and 37ed12 as subset ed12's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 38.8 | 43/42, 44/43, 45/44, 46/45, 47/46 |
| 2 | 77.5 | 23/22, 45/43 |
| 3 | 116.3 | 31/29, 46/43, 47/44 |
| 4 | 155 | 35/32, 47/43 |
| 5 | 193.8 | 19/17, 47/42 |
| 6 | 232.5 | 8/7 |
| 7 | 271.3 | |
| 8 | 310.1 | |
| 9 | 348.8 | 11/9 |
| 10 | 387.6 | 5/4 |
| 11 | 426.3 | 23/18, 32/25 |
| 12 | 465.1 | 17/13 |
| 13 | 503.8 | |
| 14 | 542.6 | 26/19, 41/30 |
| 15 | 581.3 | 7/5 |
| 16 | 620.1 | |
| 17 | 658.9 | 19/13, 41/28 |
| 18 | 697.6 | |
| 19 | 736.4 | 26/17 |
| 20 | 775.1 | 36/23, 47/30 |
| 21 | 813.9 | 8/5 |
| 22 | 852.6 | 18/11 |
| 23 | 891.4 | |
| 24 | 930.2 | |
| 25 | 968.9 | 7/4 |
| 26 | 1007.7 | 34/19, 43/24 |
| 27 | 1046.4 | |
| 28 | 1085.2 | 43/23 |
| 29 | 1123.9 | 44/23 |
| 30 | 1162.7 | 45/23, 47/24 |
| 31 | 1201.4 | 2/1 |
| 32 | 1240.2 | 43/21, 45/22 |
| 33 | 1279 | 23/11, 44/21 |
| 34 | 1317.7 | 15/7 |
| 35 | 1356.5 | 35/16, 46/21 |
| 36 | 1395.2 | 47/21 |
| 37 | 1434 | |
| 38 | 1472.7 | |
| 39 | 1511.5 | |
| 40 | 1550.3 | |
| 41 | 1589 | |
| 42 | 1627.8 | 41/16 |
| 43 | 1666.5 | 34/13 |
| 44 | 1705.3 | |
| 45 | 1744 | |
| 46 | 1782.8 | 14/5 |
| 47 | 1821.5 | 43/15 |
| 48 | 1860.3 | 41/14 |
| 49 | 1899.1 | |
| 50 | 1937.8 | 46/15 |
| 51 | 1976.6 | 47/15 |
| 52 | 2015.3 | 16/5 |
| 53 | 2054.1 | 36/11 |
| 54 | 2092.8 | |
| 55 | 2131.6 | 24/7 |
| 56 | 2170.4 | 7/2 |
| 57 | 2209.1 | 43/12 |
| 58 | 2247.9 | 11/3 |
| 59 | 2286.6 | 15/4 |
| 60 | 2325.4 | 23/6 |
| 61 | 2364.1 | 47/12 |
| 62 | 2402.9 | |
| 63 | 2441.7 | 41/10 |
| 64 | 2480.4 | |
| 65 | 2519.2 | 30/7 |
| 66 | 2557.9 | |
| 67 | 2596.7 | |
| 68 | 2635.4 | |
| 69 | 2674.2 | |
| 70 | 2712.9 | |
| 71 | 2751.7 | |
| 72 | 2790.5 | |
| 73 | 2829.2 | 41/8 |
| 74 | 2868 | |
| 75 | 2906.7 | |
| 76 | 2945.5 | |
| 77 | 2984.2 | 28/5 |
| 78 | 3023 | |
| 79 | 3061.8 | 41/7 |
| 80 | 3100.5 | 6/1 |
| 81 | 3139.3 | |
| 82 | 3178 | |
| 83 | 3216.8 | |
| 84 | 3255.5 | |
| 85 | 3294.3 | |
| 86 | 3333 | |
| 87 | 3371.8 | |
| 88 | 3410.6 | 43/6 |
| 89 | 3449.3 | 22/3 |
| 90 | 3488.1 | 15/2 |
| 91 | 3526.8 | 23/3 |
| 92 | 3565.6 | 47/6 |
| 93 | 3604.3 | |
| 94 | 3643.1 | 41/5 |
| 95 | 3681.9 | |
| 96 | 3720.6 | |
| 97 | 3759.4 | |
| 98 | 3798.1 | |
| 99 | 3836.9 | |
| 100 | 3875.6 | |
| 101 | 3914.4 | |
| 102 | 3953.1 | |
| 103 | 3991.9 | |
| 104 | 4030.7 | 41/4 |
| 105 | 4069.4 | 21/2 |
| 106 | 4108.2 | |
| 107 | 4146.9 | |
| 108 | 4185.7 | |
| 109 | 4224.4 | |
| 110 | 4263.2 | |
| 111 | 4302 | 12/1 |