114edo
Jump to navigation
Jump to search
Prime factorization
2 × 3 × 19
Step size
10.5263¢
Fifth
67\114 (705.263¢)
Semitones (A1:m2)
13:7 (136.8¢ : 73.68¢)
Consistency limit
7
Distinct consistency limit
7
| ← 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.
In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and 117440512/117406179, 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 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 three bisector temperament.
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 | +30 | -4 | -37 | -38 | +15 | -39 | +3 | -26 | +28 | +31 | |
| Steps (reduced) |
181 (67) |
265 (37) |
320 (92) |
361 (19) |
394 (52) |
422 (80) |
445 (103) |
466 (10) |
484 (28) |
501 (45) |
516 (60) | |
Intervals
"| Steps | Cents | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 10.5263 | ^D, v6E♭ | |
| 2 | 21.0526 | ^^D, v5E♭ | |
| 3 | 31.5789 | ^3D, v4E♭ | 49/48, 56/55 |
| 4 | 42.1053 | ^4D, v3E♭ | 40/39 |
| 5 | 52.6316 | ^5D, vvE♭ | 33/32, 36/35, 65/63 |
| 6 | 63.1579 | ^6D, vE♭ | |
| 7 | 73.6842 | ^7D, E♭ | 25/24 |
| 8 | 84.2105 | ^8D, v12E | 21/20 |
| 9 | 94.7368 | ^9D, v11E | 55/52 |
| 10 | 105.263 | ^10D, v10E | 35/33, 52/49 |
| 11 | 115.789 | ^11D, v9E | |
| 12 | 126.316 | ^12D, v8E | 14/13 |
| 13 | 136.842 | D♯, v7E | 13/12, 27/25 |
| 14 | 147.368 | ^D♯, v6E | |
| 15 | 157.895 | ^^D♯, v5E | 35/32 |
| 16 | 168.421 | ^3D♯, v4E | |
| 17 | 178.947 | ^4D♯, v3E | 10/9, 72/65 |
| 18 | 189.474 | ^5D♯, vvE | 39/35, 49/44 |
| 19 | 200 | ^6D♯, vE | 55/49 |
| 20 | 210.526 | E | |
| 21 | 221.053 | ^E, v6F | |
| 22 | 231.579 | ^^E, v5F | 8/7, 55/48 |
| 23 | 242.105 | ^3E, v4F | |
| 24 | 252.632 | ^4E, v3F | |
| 25 | 263.158 | ^5E, vvF | 7/6, 64/55 |
| 26 | 273.684 | ^6E, vF | |
| 27 | 284.211 | F | 33/28 |
| 28 | 294.737 | ^F, v6G♭ | |
| 29 | 305.263 | ^^F, v5G♭ | 25/21 |
| 30 | 315.789 | ^3F, v4G♭ | 6/5 |
| 31 | 326.316 | ^4F, v3G♭ | |
| 32 | 336.842 | ^5F, vvG♭ | 40/33 |
| 33 | 347.368 | ^6F, vG♭ | 49/40 |
| 34 | 357.895 | ^7F, G♭ | 16/13 |
| 35 | 368.421 | ^8F, v12G | 26/21 |
| 36 | 378.947 | ^9F, v11G | |
| 37 | 389.474 | ^10F, v10G | 5/4 |
| 38 | 400 | ^11F, v9G | 63/50 |
| 39 | 410.526 | ^12F, v8G | 33/26, 80/63 |
| 40 | 421.053 | F♯, v7G | 14/11 |
| 41 | 431.579 | ^F♯, v6G | 50/39 |
| 42 | 442.105 | ^^F♯, v5G | |
| 43 | 452.632 | ^3F♯, v4G | 13/10 |
| 44 | 463.158 | ^4F♯, v3G | 55/42, 64/49 |
| 45 | 473.684 | ^5F♯, vvG | 21/16 |
| 46 | 484.211 | ^6F♯, vG | |
| 47 | 494.737 | G | 4/3 |
| 48 | 505.263 | ^G, v6A♭ | |
| 49 | 515.789 | ^^G, v5A♭ | 35/26, 66/49 |
| 50 | 526.316 | ^3G, v4A♭ | 65/48 |
| 51 | 536.842 | ^4G, v3A♭ | |
| 52 | 547.368 | ^5G, vvA♭ | 11/8, 48/35 |
| 53 | 557.895 | ^6G, vA♭ | |
| 54 | 568.421 | ^7G, A♭ | 25/18 |
| 55 | 578.947 | ^8G, v12A | 7/5 |
| 56 | 589.474 | ^9G, v11A | |
| 57 | 600 | ^10G, v10A | |
| 58 | 610.526 | ^11G, v9A | |
| 59 | 621.053 | ^12G, v8A | 10/7 |
| 60 | 631.579 | G♯, v7A | 36/25 |
| 61 | 642.105 | ^G♯, v6A | |
| 62 | 652.632 | ^^G♯, v5A | 16/11, 35/24 |
| 63 | 663.158 | ^3G♯, v4A | |
| 64 | 673.684 | ^4G♯, v3A | |
| 65 | 684.211 | ^5G♯, vvA | 49/33, 52/35 |
| 66 | 694.737 | ^6G♯, vA | |
| 67 | 705.263 | A | 3/2 |
| 68 | 715.789 | ^A, v6B♭ | |
| 69 | 726.316 | ^^A, v5B♭ | 32/21 |
| 70 | 736.842 | ^3A, v4B♭ | 49/32 |
| 71 | 747.368 | ^4A, v3B♭ | 20/13 |
| 72 | 757.895 | ^5A, vvB♭ | 65/42 |
| 73 | 768.421 | ^6A, vB♭ | 39/25 |
| 74 | 778.947 | ^7A, B♭ | 11/7 |
| 75 | 789.474 | ^8A, v12B | 52/33, 63/40 |
| 76 | 800 | ^9A, v11B | |
| 77 | 810.526 | ^10A, v10B | 8/5 |
| 78 | 821.053 | ^11A, v9B | |
| 79 | 831.579 | ^12A, v8B | 21/13 |
| 80 | 842.105 | A♯, v7B | 13/8 |
| 81 | 852.632 | ^A♯, v6B | 80/49 |
| 82 | 863.158 | ^^A♯, v5B | 33/20 |
| 83 | 873.684 | ^3A♯, v4B | |
| 84 | 884.211 | ^4A♯, v3B | 5/3 |
| 85 | 894.737 | ^5A♯, vvB | 42/25 |
| 86 | 905.263 | ^6A♯, vB | |
| 87 | 915.789 | B | 56/33 |
| 88 | 926.316 | ^B, v6C | |
| 89 | 936.842 | ^^B, v5C | 12/7, 55/32 |
| 90 | 947.368 | ^3B, v4C | |
| 91 | 957.895 | ^4B, v3C | |
| 92 | 968.421 | ^5B, vvC | 7/4 |
| 93 | 978.947 | ^6B, vC | |
| 94 | 989.474 | C | |
| 95 | 1000 | ^C, v6D♭ | |
| 96 | 1010.53 | ^^C, v5D♭ | 70/39 |
| 97 | 1021.05 | ^3C, v4D♭ | 9/5, 65/36 |
| 98 | 1031.58 | ^4C, v3D♭ | |
| 99 | 1042.11 | ^5C, vvD♭ | 64/35 |
| 100 | 1052.63 | ^6C, vD♭ | |
| 101 | 1063.16 | ^7C, D♭ | 24/13, 50/27 |
| 102 | 1073.68 | ^8C, v12D | 13/7 |
| 103 | 1084.21 | ^9C, v11D | |
| 104 | 1094.74 | ^10C, v10D | 49/26, 66/35 |
| 105 | 1105.26 | ^11C, v9D | |
| 106 | 1115.79 | ^12C, v8D | 40/21 |
| 107 | 1126.32 | C♯, v7D | 48/25 |
| 108 | 1136.84 | ^C♯, v6D | |
| 109 | 1147.37 | ^^C♯, v5D | 35/18, 64/33 |
| 110 | 1157.89 | ^3C♯, v4D | 39/20 |
| 111 | 1168.42 | ^4C♯, v3D | 55/28 |
| 112 | 1178.95 | ^5C♯, vvD | |
| 113 | 1189.47 | ^6C♯, vD | |
| 114 | 1200 | D | 2/1 |
Period of 19-limit Shrutar
| Degree | Cents | Difference from 68edo |
|---|---|---|
| 2 | 21.05263 | 3.40557¢ |
| 3 | 31.57895 | -3.71517¢ |
| 5 | 52.63158 | -0.3096¢ |
| 7 | 73.68421 | 3.096¢ |
| 8 | 84.21053 | -4.02477¢ |
| 10 | 105.26316 | -0.619195¢ |
| 12 | 126.31579 | 2.78638¢ |
| 13 | 136.842105 | -4.334365¢ |
| 15 | 157.89474 | -0.9288¢ |
| 17 | 178.94737 | 2.47678¢ |
| 18 | 189.47369 | -4.644¢ |
| 20 | 210.52632 | -1.23839¢ |
| 22 | 231.57895 | 2.16718¢ |
| 23 | 242.10526 | -4.953560372 |
| 25 | 263.157895 | -1.548¢ |
| 27 | 284.21053 | 1.857585¢ |
| 29 | 305.26316 | 5.26316¢ |
| 30 | 315.78947 | -1.857585¢ |
| 32 | 336.842105 | 1.548¢ |
| 34 | 357.89474 | 4.95356¢ |
| 35 | 368.42105 | -2.16718¢ |
| 37 | 389.47368 | 1.23839¢ |
| 39 | 410.52632 | 4.64396¢ |
| 40 | 421.05263 | -2.47678¢ |
| 42 | 442.10526 | 0.92879¢ |
| 44 | 463.157895 | 4.334365¢ |
| 45 | 473.68421 | -2.78638¢ |
| 47 | 494.73684 | 0.619195¢ |
| 49 | 515.78947 | 4.02477¢ |
| 50 | 526.31579 | -3.095975¢ |
| 52 | 547.36842 | 0.3096¢ |
| 54 | 568.42105 | 3.71517¢ |
| 55 | 578.94737 | -3.40557¢ |