116edo
| ← 115edo | 116edo | 117edo → |
116 equal divisions of the octave (abbreviated 116edo or 116ed2), also called 116-tone equal temperament (116tet) or 116 equal temperament (116et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 116 equal parts of about 10.3 ¢ each. Each step represents a frequency ratio of 21/116, or the 116th root of 2.
116edo is only consistent to the 5-odd-limit, and is not quite accurate for its size. It can be viewed as splitting 58edo's step in two, and the enfactored 116cef val comes out on top accuracy in the 7-, 11-, and 13-limit. In the 5-limit, however, the patent val ⟨116 184 269] beats the enfactored 116c val ⟨116 184 270] by a thin margin, and it tempers out 20000/19683 (tetracot comma) and 2197265625/2147483648 (wizard comma).
In the 7-, 11- and 13-limit, the patent val ⟨116 184 269 326 401 429] comes in second best after the enfactored 116cef val ⟨116 184 270 326 402 430] , and it tempers out 225/224, 15625/15309, and 51200/50421 in the 7-limit; 385/384, 540/539, 4000/3993, and 6655/6561 in the 11-limit; 169/168, 275/273, 352/351, and 640/637 in the 13-limit. 116edo provides the optimal patent val for the submajor temperament in the 11- and 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.49 | -3.56 | +3.59 | -3.04 | -2.60 | -1.51 | +2.49 | +2.76 | +4.91 | +3.24 |
| Relative (%) | +0.0 | +14.4 | -34.4 | +34.7 | -29.4 | -25.1 | -14.6 | +24.0 | +26.7 | +47.4 | +31.3 | |
| Steps (reduced) |
116 (0) |
184 (68) |
269 (37) |
326 (94) |
401 (53) |
429 (81) |
474 (10) |
493 (29) |
525 (61) |
564 (100) |
575 (111) | |
Subsets and supersets
Since 116 factors into 22 × 29, 116edo has subset edos 2, 4, 29, and 58. 232edo, which doubles it, is a notable tuning.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 10.3 | ^D, ^5E♭♭ | |
| 2 | 20.7 | ^^D, v6E♭ | |
| 3 | 31 | ^3D, v5E♭ | |
| 4 | 41.4 | 41/40, 44/43 | ^4D, v4E♭ |
| 5 | 51.7 | 33/32, 34/33, 35/34 | ^5D, v3E♭ |
| 6 | 62.1 | 28/27, 29/28 | ^6D, vvE♭ |
| 7 | 72.4 | 24/23 | v5D♯, vE♭ |
| 8 | 82.8 | 43/41 | v4D♯, E♭ |
| 9 | 93.1 | 19/18, 39/37 | v3D♯, ^E♭ |
| 10 | 103.4 | 17/16, 35/33 | vvD♯, ^^E♭ |
| 11 | 113.8 | 16/15, 31/29, 47/44 | vD♯, ^3E♭ |
| 12 | 124.1 | 29/27, 43/40, 44/41 | D♯, ^4E♭ |
| 13 | 134.5 | 40/37 | ^D♯, ^5E♭ |
| 14 | 144.8 | 37/34 | ^^D♯, v6E |
| 15 | 155.2 | 35/32, 47/43 | ^3D♯, v5E |
| 16 | 165.5 | 11/10 | ^4D♯, v4E |
| 17 | 175.9 | 31/28, 41/37 | ^5D♯, v3E |
| 18 | 186.2 | 39/35 | ^6D♯, vvE |
| 19 | 196.6 | 37/33 | v5D𝄪, vE |
| 20 | 206.9 | 44/39 | E |
| 21 | 217.2 | 17/15 | ^E, ^5F♭ |
| 22 | 227.6 | ^^E, v6F | |
| 23 | 237.9 | 31/27, 39/34, 47/41 | ^3E, v5F |
| 24 | 248.3 | 15/13 | ^4E, v4F |
| 25 | 258.6 | 36/31, 43/37 | ^5E, v3F |
| 26 | 269 | ^6E, vvF | |
| 27 | 279.3 | 20/17, 27/23, 47/40 | v5E♯, vF |
| 28 | 289.7 | 13/11 | F |
| 29 | 300 | 44/37 | ^F, ^5G♭♭ |
| 30 | 310.3 | ^^F, v6G♭ | |
| 31 | 320.7 | ^3F, v5G♭ | |
| 32 | 331 | 23/19, 40/33 | ^4F, v4G♭ |
| 33 | 341.4 | 28/23, 39/32 | ^5F, v3G♭ |
| 34 | 351.7 | 38/31 | ^6F, vvG♭ |
| 35 | 362.1 | 37/30 | v5F♯, vG♭ |
| 36 | 372.4 | 36/29 | v4F♯, G♭ |
| 37 | 382.8 | v3F♯, ^G♭ | |
| 38 | 393.1 | vvF♯, ^^G♭ | |
| 39 | 403.4 | 24/19 | vF♯, ^3G♭ |
| 40 | 413.8 | 33/26, 47/37 | F♯, ^4G♭ |
| 41 | 424.1 | 23/18 | ^F♯, ^5G♭ |
| 42 | 434.5 | 9/7 | ^^F♯, v6G |
| 43 | 444.8 | 22/17, 31/24 | ^3F♯, v5G |
| 44 | 455.2 | 13/10 | ^4F♯, v4G |
| 45 | 465.5 | 17/13 | ^5F♯, v3G |
| 46 | 475.9 | ^6F♯, vvG | |
| 47 | 486.2 | 45/34 | v5F𝄪, vG |
| 48 | 496.6 | 4/3 | G |
| 49 | 506.9 | ^G, ^5A♭♭ | |
| 50 | 517.2 | 31/23 | ^^G, v6A♭ |
| 51 | 527.6 | 19/14, 42/31 | ^3G, v5A♭ |
| 52 | 537.9 | 15/11 | ^4G, v4A♭ |
| 53 | 548.3 | 48/35 | ^5G, v3A♭ |
| 54 | 558.6 | 29/21, 47/34 | ^6G, vvA♭ |
| 55 | 569 | v5G♯, vA♭ | |
| 56 | 579.3 | v4G♯, A♭ | |
| 57 | 589.7 | 38/27, 45/32 | v3G♯, ^A♭ |
| 58 | 600 | vvG♯, ^^A♭ | |
| 59 | 610.3 | 27/19, 37/26, 47/33 | vG♯, ^3A♭ |
| 60 | 620.7 | G♯, ^4A♭ | |
| 61 | 631 | ^G♯, ^5A♭ | |
| 62 | 641.4 | 42/29 | ^^G♯, v6A |
| 63 | 651.7 | 35/24 | ^3G♯, v5A |
| 64 | 662.1 | 22/15 | ^4G♯, v4A |
| 65 | 672.4 | 28/19, 31/21 | ^5G♯, v3A |
| 66 | 682.8 | 46/31 | ^6G♯, vvA |
| 67 | 693.1 | v5G𝄪, vA | |
| 68 | 703.4 | 3/2 | A |
| 69 | 713.8 | ^A, ^5B♭♭ | |
| 70 | 724.1 | ^^A, v6B♭ | |
| 71 | 734.5 | 26/17 | ^3A, v5B♭ |
| 72 | 744.8 | 20/13 | ^4A, v4B♭ |
| 73 | 755.2 | 17/11, 48/31 | ^5A, v3B♭ |
| 74 | 765.5 | 14/9 | ^6A, vvB♭ |
| 75 | 775.9 | 36/23, 47/30 | v5A♯, vB♭ |
| 76 | 786.2 | v4A♯, B♭ | |
| 77 | 796.6 | 19/12 | v3A♯, ^B♭ |
| 78 | 806.9 | vvA♯, ^^B♭ | |
| 79 | 817.2 | vA♯, ^3B♭ | |
| 80 | 827.6 | 29/18 | A♯, ^4B♭ |
| 81 | 837.9 | ^A♯, ^5B♭ | |
| 82 | 848.3 | 31/19 | ^^A♯, v6B |
| 83 | 858.6 | 23/14 | ^3A♯, v5B |
| 84 | 869 | 33/20, 38/23, 43/26 | ^4A♯, v4B |
| 85 | 879.3 | ^5A♯, v3B | |
| 86 | 889.7 | ^6A♯, vvB | |
| 87 | 900 | 37/22 | v5A𝄪, vB |
| 88 | 910.3 | 22/13 | B |
| 89 | 920.7 | 17/10, 46/27 | ^B, ^5C♭ |
| 90 | 931 | ^^B, v6C | |
| 91 | 941.4 | 31/18 | ^3B, v5C |
| 92 | 951.7 | 26/15, 45/26 | ^4B, v4C |
| 93 | 962.1 | ^5B, v3C | |
| 94 | 972.4 | ^6B, vvC | |
| 95 | 982.8 | 30/17 | v5B♯, vC |
| 96 | 993.1 | 39/22 | C |
| 97 | 1003.4 | ^C, ^5D♭♭ | |
| 98 | 1013.8 | ^^C, v6D♭ | |
| 99 | 1024.1 | 47/26 | ^3C, v5D♭ |
| 100 | 1034.5 | 20/11 | ^4C, v4D♭ |
| 101 | 1044.8 | ^5C, v3D♭ | |
| 102 | 1055.2 | ^6C, vvD♭ | |
| 103 | 1065.5 | 37/20 | v5C♯, vD♭ |
| 104 | 1075.9 | 41/22 | v4C♯, D♭ |
| 105 | 1086.2 | 15/8 | v3C♯, ^D♭ |
| 106 | 1096.6 | 32/17 | vvC♯, ^^D♭ |
| 107 | 1106.9 | 36/19 | vC♯, ^3D♭ |
| 108 | 1117.2 | C♯, ^4D♭ | |
| 109 | 1127.6 | 23/12 | ^C♯, ^5D♭ |
| 110 | 1137.9 | 27/14 | ^^C♯, v6D |
| 111 | 1148.3 | 33/17 | ^3C♯, v5D |
| 112 | 1158.6 | 43/22 | ^4C♯, v4D |
| 113 | 1169 | ^5C♯, v3D | |
| 114 | 1179.3 | ^6C♯, vvD | |
| 115 | 1189.7 | v5C𝄪, vD | |
| 116 | 1200 | 2/1 | D |