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 | 10.526 | ^D, v6E♭ | |
| 2 | 21.053 | ^^D, v5E♭ | |
| 3 | 31.579 | 49/48, 56/55 | ^3D, v4E♭ |
| 4 | 42.105 | 40/39 | ^4D, v3E♭ |
| 5 | 52.632 | 33/32, 36/35, 65/63 | ^5D, vvE♭ |
| 6 | 63.158 | ^6D, vE♭ | |
| 7 | 73.684 | 25/24 | ^7D, E♭ |
| 8 | 84.211 | 21/20 | ^8D, v12E |
| 9 | 94.737 | 55/52 | ^9D, v11E |
| 10 | 105.263 | 35/33, 52/49 | ^10D, v10E |
| 11 | 115.789 | ^11D, v9E | |
| 12 | 126.316 | 14/13 | ^12D, v8E |
| 13 | 136.842 | 13/12, 27/25 | D♯, v7E |
| 14 | 147.368 | ^D♯, v6E | |
| 15 | 157.895 | 35/32 | ^^D♯, v5E |
| 16 | 168.421 | ^3D♯, v4E | |
| 17 | 178.947 | 10/9, 72/65 | ^4D♯, v3E |
| 18 | 189.474 | 39/35, 49/44 | ^5D♯, vvE |
| 19 | 200 | 55/49 | ^6D♯, vE |
| 20 | 210.526 | E | |
| 21 | 221.053 | ^E, v6F | |
| 22 | 231.579 | 8/7, 55/48 | ^^E, v5F |
| 23 | 242.105 | ^3E, v4F | |
| 24 | 252.632 | ^4E, v3F | |
| 25 | 263.158 | 7/6, 64/55 | ^5E, vvF |
| 26 | 273.684 | ^6E, vF | |
| 27 | 284.211 | 33/28 | F |
| 28 | 294.737 | ^F, v6G♭ | |
| 29 | 305.263 | 25/21 | ^^F, v5G♭ |
| 30 | 315.789 | 6/5 | ^3F, v4G♭ |
| 31 | 326.316 | ^4F, v3G♭ | |
| 32 | 336.842 | 40/33 | ^5F, vvG♭ |
| 33 | 347.368 | 49/40 | ^6F, vG♭ |
| 34 | 357.895 | 16/13 | ^7F, G♭ |
| 35 | 368.421 | 26/21 | ^8F, v12G |
| 36 | 378.947 | ^9F, v11G | |
| 37 | 389.474 | 5/4 | ^10F, v10G |
| 38 | 400 | 63/50 | ^11F, v9G |
| 39 | 410.526 | 33/26, 80/63 | ^12F, v8G |
| 40 | 421.053 | 14/11 | F♯, v7G |
| 41 | 431.579 | 50/39 | ^F♯, v6G |
| 42 | 442.105 | ^^F♯, v5G | |
| 43 | 452.632 | 13/10 | ^3F♯, v4G |
| 44 | 463.158 | 55/42, 64/49 | ^4F♯, v3G |
| 45 | 473.684 | 21/16 | ^5F♯, vvG |
| 46 | 484.211 | ^6F♯, vG | |
| 47 | 494.737 | 4/3 | G |
| 48 | 505.263 | ^G, v6A♭ | |
| 49 | 515.789 | 35/26, 66/49 | ^^G, v5A♭ |
| 50 | 526.316 | 65/48 | ^3G, v4A♭ |
| 51 | 536.842 | ^4G, v3A♭ | |
| 52 | 547.368 | 11/8, 48/35 | ^5G, vvA♭ |
| 53 | 557.895 | ^6G, vA♭ | |
| 54 | 568.421 | 25/18 | ^7G, A♭ |
| 55 | 578.947 | 7/5 | ^8G, v12A |
| 56 | 589.474 | ^9G, v11A | |
| 57 | 600 | ^10G, v10A | |
| 58 | 610.526 | ^11G, v9A | |
| 59 | 621.053 | 10/7 | ^12G, v8A |
| 60 | 631.579 | 36/25 | G♯, v7A |
| 61 | 642.105 | ^G♯, v6A | |
| 62 | 652.632 | 16/11, 35/24 | ^^G♯, v5A |
| 63 | 663.158 | ^3G♯, v4A | |
| 64 | 673.684 | ^4G♯, v3A | |
| 65 | 684.211 | 49/33, 52/35 | ^5G♯, vvA |
| 66 | 694.737 | ^6G♯, vA | |
| 67 | 705.263 | 3/2 | A |
| 68 | 715.789 | ^A, v6B♭ | |
| 69 | 726.316 | 32/21 | ^^A, v5B♭ |
| 70 | 736.842 | 49/32 | ^3A, v4B♭ |
| 71 | 747.368 | 20/13 | ^4A, v3B♭ |
| 72 | 757.895 | 65/42 | ^5A, vvB♭ |
| 73 | 768.421 | 39/25 | ^6A, vB♭ |
| 74 | 778.947 | 11/7 | ^7A, B♭ |
| 75 | 789.474 | 52/33, 63/40 | ^8A, v12B |
| 76 | 800 | ^9A, v11B | |
| 77 | 810.526 | 8/5 | ^10A, v10B |
| 78 | 821.053 | ^11A, v9B | |
| 79 | 831.579 | 21/13 | ^12A, v8B |
| 80 | 842.105 | 13/8 | A♯, v7B |
| 81 | 852.632 | 80/49 | ^A♯, v6B |
| 82 | 863.158 | 33/20 | ^^A♯, v5B |
| 83 | 873.684 | ^3A♯, v4B | |
| 84 | 884.211 | 5/3 | ^4A♯, v3B |
| 85 | 894.737 | 42/25 | ^5A♯, vvB |
| 86 | 905.263 | ^6A♯, vB | |
| 87 | 915.789 | 56/33 | B |
| 88 | 926.316 | ^B, v6C | |
| 89 | 936.842 | 12/7, 55/32 | ^^B, v5C |
| 90 | 947.368 | ^3B, v4C | |
| 91 | 957.895 | ^4B, v3C | |
| 92 | 968.421 | 7/4 | ^5B, vvC |
| 93 | 978.947 | ^6B, vC | |
| 94 | 989.474 | C | |
| 95 | 1000 | ^C, v6D♭ | |
| 96 | 1010.526 | 70/39 | ^^C, v5D♭ |
| 97 | 1021.053 | 9/5, 65/36 | ^3C, v4D♭ |
| 98 | 1031.579 | ^4C, v3D♭ | |
| 99 | 1042.105 | 64/35 | ^5C, vvD♭ |
| 100 | 1052.632 | ^6C, vD♭ | |
| 101 | 1063.158 | 24/13, 50/27 | ^7C, D♭ |
| 102 | 1073.684 | 13/7 | ^8C, v12D |
| 103 | 1084.211 | ^9C, v11D | |
| 104 | 1094.737 | 49/26, 66/35 | ^10C, v10D |
| 105 | 1105.263 | ^11C, v9D | |
| 106 | 1115.789 | 40/21 | ^12C, v8D |
| 107 | 1126.316 | 48/25 | C♯, v7D |
| 108 | 1136.842 | ^C♯, v6D | |
| 109 | 1147.368 | 35/18, 64/33 | ^^C♯, v5D |
| 110 | 1157.895 | 39/20 | ^3C♯, v4D |
| 111 | 1168.421 | 55/28 | ^4C♯, v3D |
| 112 | 1178.947 | ^5C♯, vvD | |
| 113 | 1189.474 | ^6C♯, vD | |
| 114 | 1200 | 2/1 | D |