114edo
| ← 113edo | 114edo | 115edo → |
114 equal divisions of the octave (abbreviated 114edo or 114ed2), also called 114-tone equal temperament (114tet) or 114 equal temperament (114et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 114 equal parts of about 10.5 ¢ each. Each step represents a frequency ratio of 21/114, or the 114th root of 2.
Theory
In the 5-limit the equal temperament tempers out 2048/2025 (diaschisma), in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and notably the quartisma, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for the shrutar temperament; it is in fact the optimal patent val for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 bisector temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.31 | +3.16 | -0.40 | -3.91 | -3.95 | +1.58 | -4.06 | +0.31 | -2.78 | +2.90 | +3.30 |
| Relative (%) | +31.4 | +30.0 | -3.8 | -37.1 | -37.5 | +15.0 | -38.6 | +2.9 | -26.4 | +27.6 | +31.4 | |
| Steps (reduced) |
181 (67) |
265 (37) |
320 (92) |
361 (19) |
394 (52) |
422 (80) |
445 (103) |
466 (10) |
484 (28) |
501 (45) |
516 (60) | |
Subsets and supersets
Since 114 factors into 2 × 3 × 19, 114 edo has subset edos 2, 3, 6, 19, 38, and 57.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 10.5 | ^D, v6E♭ | |
| 2 | 21.1 | ^^D, v5E♭ | |
| 3 | 31.6 | ^3D, v4E♭ | |
| 4 | 42.1 | 40/39, 41/40, 42/41, 43/42 | ^4D, v3E♭ |
| 5 | 52.6 | 33/32, 34/33 | ^5D, vvE♭ |
| 6 | 63.2 | ^6D, vE♭ | |
| 7 | 73.7 | 24/23 | v6D♯, E♭ |
| 8 | 84.2 | 21/20, 43/41 | v5D♯, ^E♭ |
| 9 | 94.7 | 37/35 | v4D♯, ^^E♭ |
| 10 | 105.3 | 17/16 | v3D♯, ^3E♭ |
| 11 | 115.8 | 31/29, 46/43, 47/44 | vvD♯, ^4E♭ |
| 12 | 126.3 | 14/13, 43/40 | vD♯, ^5E♭ |
| 13 | 136.8 | 13/12, 40/37 | D♯, ^6E♭ |
| 14 | 147.4 | 37/34 | ^D♯, v6E |
| 15 | 157.9 | 23/21, 34/31 | ^^D♯, v5E |
| 16 | 168.4 | 32/29, 43/39 | ^3D♯, v4E |
| 17 | 178.9 | 41/37 | ^4D♯, v3E |
| 18 | 189.5 | 29/26, 48/43 | ^5D♯, vvE |
| 19 | 200 | 37/33, 46/41 | ^6D♯, vE |
| 20 | 210.5 | 26/23, 35/31 | E |
| 21 | 221.1 | 42/37 | ^E, v6F |
| 22 | 231.6 | 8/7 | ^^E, v5F |
| 23 | 242.1 | 23/20 | ^3E, v4F |
| 24 | 252.6 | 22/19, 37/32 | ^4E, v3F |
| 25 | 263.2 | ^5E, vvF | |
| 26 | 273.7 | 34/29, 41/35, 48/41 | ^6E, vF |
| 27 | 284.2 | 33/28, 46/39 | F |
| 28 | 294.7 | ^F, v6G♭ | |
| 29 | 305.3 | 31/26, 37/31 | ^^F, v5G♭ |
| 30 | 315.8 | 6/5 | ^3F, v4G♭ |
| 31 | 326.3 | 29/24, 35/29 | ^4F, v3G♭ |
| 32 | 336.8 | 17/14 | ^5F, vvG♭ |
| 33 | 347.4 | ^6F, vG♭ | |
| 34 | 357.9 | 16/13, 43/35 | v6F♯, G♭ |
| 35 | 368.4 | 26/21, 47/38 | v5F♯, ^G♭ |
| 36 | 378.9 | 46/37 | v4F♯, ^^G♭ |
| 37 | 389.5 | v3F♯, ^3G♭ | |
| 38 | 400 | 29/23 | vvF♯, ^4G♭ |
| 39 | 410.5 | vF♯, ^5G♭ | |
| 40 | 421.1 | 37/29 | F♯, ^6G♭ |
| 41 | 431.6 | ^F♯, v6G | |
| 42 | 442.1 | 31/24, 40/31 | ^^F♯, v5G |
| 43 | 452.6 | 13/10, 48/37 | ^3F♯, v4G |
| 44 | 463.2 | 17/13 | ^4F♯, v3G |
| 45 | 473.7 | 46/35 | ^5F♯, vvG |
| 46 | 484.2 | 37/28, 41/31 | ^6F♯, vG |
| 47 | 494.7 | G | |
| 48 | 505.3 | ^G, v6A♭ | |
| 49 | 515.8 | 31/23, 35/26 | ^^G, v5A♭ |
| 50 | 526.3 | 42/31 | ^3G, v4A♭ |
| 51 | 536.8 | ^4G, v3A♭ | |
| 52 | 547.4 | 48/35 | ^5G, vvA♭ |
| 53 | 557.9 | 29/21, 40/29 | ^6G, vA♭ |
| 54 | 568.4 | 25/18, 43/31 | v6G♯, A♭ |
| 55 | 578.9 | v5G♯, ^A♭ | |
| 56 | 589.5 | v4G♯, ^^A♭ | |
| 57 | 600 | 41/29 | v3G♯, ^3A♭ |
| 58 | 610.5 | 37/26, 47/33 | vvG♯, ^4A♭ |
| 59 | 621.1 | vG♯, ^5A♭ | |
| 60 | 631.6 | 36/25 | G♯, ^6A♭ |
| 61 | 642.1 | 29/20, 42/29 | ^G♯, v6A |
| 62 | 652.6 | 35/24 | ^^G♯, v5A |
| 63 | 663.2 | ^3G♯, v4A | |
| 64 | 673.7 | 31/21 | ^4G♯, v3A |
| 65 | 684.2 | 46/31 | ^5G♯, vvA |
| 66 | 694.7 | ^6G♯, vA | |
| 67 | 705.3 | A | |
| 68 | 715.8 | ^A, v6B♭ | |
| 69 | 726.3 | 35/23 | ^^A, v5B♭ |
| 70 | 736.8 | 26/17 | ^3A, v4B♭ |
| 71 | 747.4 | 20/13, 37/24 | ^4A, v3B♭ |
| 72 | 757.9 | 31/20, 48/31 | ^5A, vvB♭ |
| 73 | 768.4 | 39/25 | ^6A, vB♭ |
| 74 | 778.9 | v6A♯, B♭ | |
| 75 | 789.5 | 41/26 | v5A♯, ^B♭ |
| 76 | 800 | 46/29 | v4A♯, ^^B♭ |
| 77 | 810.5 | v3A♯, ^3B♭ | |
| 78 | 821.1 | 37/23 | vvA♯, ^4B♭ |
| 79 | 831.6 | 21/13 | vA♯, ^5B♭ |
| 80 | 842.1 | 13/8 | A♯, ^6B♭ |
| 81 | 852.6 | ^A♯, v6B | |
| 82 | 863.2 | 28/17 | ^^A♯, v5B |
| 83 | 873.7 | 48/29 | ^3A♯, v4B |
| 84 | 884.2 | 5/3 | ^4A♯, v3B |
| 85 | 894.7 | 47/28 | ^5A♯, vvB |
| 86 | 905.3 | ^6A♯, vB | |
| 87 | 915.8 | 39/23 | B |
| 88 | 926.3 | 29/17, 41/24 | ^B, v6C |
| 89 | 936.8 | 43/25 | ^^B, v5C |
| 90 | 947.4 | 19/11 | ^3B, v4C |
| 91 | 957.9 | 40/23 | ^4B, v3C |
| 92 | 968.4 | 7/4 | ^5B, vvC |
| 93 | 978.9 | 37/21 | ^6B, vC |
| 94 | 989.5 | 23/13 | C |
| 95 | 1000 | 41/23 | ^C, v6D♭ |
| 96 | 1010.5 | 43/24 | ^^C, v5D♭ |
| 97 | 1021.1 | ^3C, v4D♭ | |
| 98 | 1031.6 | 29/16 | ^4C, v3D♭ |
| 99 | 1042.1 | 31/17, 42/23 | ^5C, vvD♭ |
| 100 | 1052.6 | ^6C, vD♭ | |
| 101 | 1063.2 | 24/13, 37/20 | v6C♯, D♭ |
| 102 | 1073.7 | 13/7 | v5C♯, ^D♭ |
| 103 | 1084.2 | 43/23 | v4C♯, ^^D♭ |
| 104 | 1094.7 | 32/17 | v3C♯, ^3D♭ |
| 105 | 1105.3 | vvC♯, ^4D♭ | |
| 106 | 1115.8 | 40/21 | vC♯, ^5D♭ |
| 107 | 1126.3 | 23/12 | C♯, ^6D♭ |
| 108 | 1136.8 | ^C♯, v6D | |
| 109 | 1147.4 | 33/17 | ^^C♯, v5D |
| 110 | 1157.9 | 39/20, 41/21 | ^3C♯, v4D |
| 111 | 1168.4 | ^4C♯, v3D | |
| 112 | 1178.9 | ^5C♯, vvD | |
| 113 | 1189.5 | ^6C♯, vD | |
| 114 | 1200 | 2/1 | D |