117edo
| ← 116edo | 117edo | 118edo → |
117 equal divisions of the octave (abbreviated 117edo or 117ed2), also called 117-tone equal temperament (117tet) or 117 equal temperament (117et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 117 equal parts of about 10.3 ¢ each. Each step represents a frequency ratio of 21/117, or the 117th root of 2.
117edo is inconsistent to the 5-odd-limit and higher odd limits, with four mappings possible for the 11-limit: ⟨117 185 272 328 405] (patent val), ⟨117 186 272 329 405] (117bd), ⟨117 185 271 328 404] (117ce), and ⟨117 185 272 329 405] (117d).
Using the patent val, it tempers out 81/80 (syntonic comma) and [69 -1 -29⟩ in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting the 7-limit mohajira temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the rank-3 terpsichore temperament; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.
Using the 117d val, it tempers out 126/125, 225/224, and [29 3 0 -12⟩ in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit grosstone temperament.
Using the 117ce val, it tempers out 3125/3072 (magic comma) and [-31 24 -3⟩ in the 5-limit; 2401/2400, 3645/3584, and 4375/4374 in the 7-limit; 243/242, 441/440, and 1815/1792 in the 11-limit; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-limit.
Using the 117bd val, it tempers out 15625/15552 (kleisma) and [34 -17 -3⟩ in the 5-limit; 245/243, 3136/3125, and 51200/50421 in the 7-limit; 176/175, 1232/1215, 1375/1372, and 2560/2541 in the 11-limit; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.52 | +3.43 | -4.72 | +1.22 | +2.53 | +0.50 | -1.09 | -2.39 | -0.08 | +1.01 | -2.63 |
| Relative (%) | -44.1 | +33.4 | -46.1 | +11.9 | +24.7 | +4.9 | -10.6 | -23.3 | -0.8 | +9.9 | -25.7 | |
| Steps (reduced) |
185 (68) |
272 (38) |
328 (94) |
371 (20) |
405 (54) |
433 (82) |
457 (106) |
478 (10) |
497 (29) |
514 (46) |
529 (61) | |
Subsets and supersets
Since 117 factors into 32 × 13, 117edo has subset edos 3, 9, 13, and 39. 234edo, which doubles it, provides a correction for the approximation to harmonic 3.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation (Dual Flat Fifth 68\117) |
Ups and Downs Notation (Dual Sharp Fifth 69\117) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 10.256 | ^D, vvE♭♭ | ^D, v5E♭ | |
| 2 | 20.513 | ^^D, vE♭♭ | ^^D, v4E♭ | |
| 3 | 30.769 | ^3D, E♭♭ | ^3D, v3E♭ | |
| 4 | 41.026 | 41/40, 44/43 | ^4D, v7E♭ | ^4D, vvE♭ |
| 5 | 51.282 | 33/32, 34/33, 35/34 | ^5D, v6E♭ | ^5D, vE♭ |
| 6 | 61.538 | 29/28 | ^6D, v5E♭ | ^6D, E♭ |
| 7 | 71.795 | 24/23 | ^7D, v4E♭ | ^7D, v14E |
| 8 | 82.051 | 43/41 | D♯, v3E♭ | ^8D, v13E |
| 9 | 92.308 | ^D♯, vvE♭ | ^9D, v12E | |
| 10 | 102.564 | 35/33 | ^^D♯, vE♭ | ^10D, v11E |
| 11 | 112.821 | 16/15, 47/44 | ^3D♯, E♭ | ^11D, v10E |
| 12 | 123.077 | 44/41 | ^4D♯, v7E | ^12D, v9E |
| 13 | 133.333 | 40/37, 41/38 | ^5D♯, v6E | ^13D, v8E |
| 14 | 143.59 | 38/35 | ^6D♯, v5E | ^14D, v7E |
| 15 | 153.846 | 35/32, 47/43 | ^7D♯, v4E | D♯, v6E |
| 16 | 164.103 | 11/10 | D𝄪, v3E | ^D♯, v5E |
| 17 | 174.359 | ^D𝄪, vvE | ^^D♯, v4E | |
| 18 | 184.615 | ^^D𝄪, vE | ^3D♯, v3E | |
| 19 | 194.872 | E | ^4D♯, vvE | |
| 20 | 205.128 | ^E, vvF♭ | ^5D♯, vE | |
| 21 | 215.385 | 17/15, 43/38 | ^^E, vF♭ | E |
| 22 | 225.641 | 33/29 | ^3E, F♭ | ^E, v5F |
| 23 | 235.897 | 39/34, 47/41 | ^4E, v7F | ^^E, v4F |
| 24 | 246.154 | 15/13, 38/33 | ^5E, v6F | ^3E, v3F |
| 25 | 256.41 | ^6E, v5F | ^4E, vvF | |
| 26 | 266.667 | 7/6 | ^7E, v4F | ^5E, vF |
| 27 | 276.923 | 34/29 | E♯, v3F | F |
| 28 | 287.179 | 13/11, 46/39 | ^E♯, vvF | ^F, v5G♭ |
| 29 | 297.436 | 19/16 | ^^E♯, vF | ^^F, v4G♭ |
| 30 | 307.692 | 37/31 | F | ^3F, v3G♭ |
| 31 | 317.949 | ^F, vvG♭♭ | ^4F, vvG♭ | |
| 32 | 328.205 | 29/24 | ^^F, vG♭♭ | ^5F, vG♭ |
| 33 | 338.462 | ^3F, G♭♭ | ^6F, G♭ | |
| 34 | 348.718 | ^4F, v7G♭ | ^7F, v14G | |
| 35 | 358.974 | 16/13 | ^5F, v6G♭ | ^8F, v13G |
| 36 | 369.231 | 47/38 | ^6F, v5G♭ | ^9F, v12G |
| 37 | 379.487 | ^7F, v4G♭ | ^10F, v11G | |
| 38 | 389.744 | F♯, v3G♭ | ^11F, v10G | |
| 39 | 400 | 29/23 | ^F♯, vvG♭ | ^12F, v9G |
| 40 | 410.256 | 19/15 | ^^F♯, vG♭ | ^13F, v8G |
| 41 | 420.513 | ^3F♯, G♭ | ^14F, v7G | |
| 42 | 430.769 | 41/32 | ^4F♯, v7G | F♯, v6G |
| 43 | 441.026 | 40/31 | ^5F♯, v6G | ^F♯, v5G |
| 44 | 451.282 | ^6F♯, v5G | ^^F♯, v4G | |
| 45 | 461.538 | 30/23 | ^7F♯, v4G | ^3F♯, v3G |
| 46 | 471.795 | 46/35 | F𝄪, v3G | ^4F♯, vvG |
| 47 | 482.051 | 41/31 | ^F𝄪, vvG | ^5F♯, vG |
| 48 | 492.308 | ^^F𝄪, vG | G | |
| 49 | 502.564 | G | ^G, v5A♭ | |
| 50 | 512.821 | 35/26, 39/29, 43/32 | ^G, vvA♭♭ | ^^G, v4A♭ |
| 51 | 523.077 | 23/17 | ^^G, vA♭♭ | ^3G, v3A♭ |
| 52 | 533.333 | 49/36 | ^3G, A♭♭ | ^4G, vvA♭ |
| 53 | 543.59 | 26/19 | ^4G, v7A♭ | ^5G, vA♭ |
| 54 | 553.846 | ^5G, v6A♭ | ^6G, A♭ | |
| 55 | 564.103 | ^6G, v5A♭ | ^7G, v14A | |
| 56 | 574.359 | 39/28, 46/33 | ^7G, v4A♭ | ^8G, v13A |
| 57 | 584.615 | G♯, v3A♭ | ^9G, v12A | |
| 58 | 594.872 | 31/22 | ^G♯, vvA♭ | ^10G, v11A |
| 59 | 605.128 | 44/31 | ^^G♯, vA♭ | ^11G, v10A |
| 60 | 615.385 | ^3G♯, A♭ | ^12G, v9A | |
| 61 | 625.641 | 33/23 | ^4G♯, v7A | ^13G, v8A |
| 62 | 635.897 | ^5G♯, v6A | ^14G, v7A | |
| 63 | 646.154 | ^6G♯, v5A | G♯, v6A | |
| 64 | 656.41 | 19/13 | ^7G♯, v4A | ^G♯, v5A |
| 65 | 666.667 | 47/32 | G𝄪, v3A | ^^G♯, v4A |
| 66 | 676.923 | 34/23, 37/25 | ^G𝄪, vvA | ^3G♯, v3A |
| 67 | 687.179 | ^^G𝄪, vA | ^4G♯, vvA | |
| 68 | 697.436 | A | ^5G♯, vA | |
| 69 | 707.692 | ^A, vvB♭♭ | A | |
| 70 | 717.949 | ^^A, vB♭♭ | ^A, v5B♭ | |
| 71 | 728.205 | 35/23 | ^3A, B♭♭ | ^^A, v4B♭ |
| 72 | 738.462 | 23/15 | ^4A, v7B♭ | ^3A, v3B♭ |
| 73 | 748.718 | ^5A, v6B♭ | ^4A, vvB♭ | |
| 74 | 758.974 | 31/20, 45/29 | ^6A, v5B♭ | ^5A, vB♭ |
| 75 | 769.231 | ^7A, v4B♭ | ^6A, B♭ | |
| 76 | 779.487 | A♯, v3B♭ | ^7A, v14B | |
| 77 | 789.744 | 30/19, 41/26 | ^A♯, vvB♭ | ^8A, v13B |
| 78 | 800 | 46/29 | ^^A♯, vB♭ | ^9A, v12B |
| 79 | 810.256 | ^3A♯, B♭ | ^10A, v11B | |
| 80 | 820.513 | 45/28 | ^4A♯, v7B | ^11A, v10B |
| 81 | 830.769 | ^5A♯, v6B | ^12A, v9B | |
| 82 | 841.026 | 13/8 | ^6A♯, v5B | ^13A, v8B |
| 83 | 851.282 | ^7A♯, v4B | ^14A, v7B | |
| 84 | 861.538 | A𝄪, v3B | A♯, v6B | |
| 85 | 871.795 | 43/26, 48/29 | ^A𝄪, vvB | ^A♯, v5B |
| 86 | 882.051 | ^^A𝄪, vB | ^^A♯, v4B | |
| 87 | 892.308 | B | ^3A♯, v3B | |
| 88 | 902.564 | 32/19 | ^B, vvC♭ | ^4A♯, vvB |
| 89 | 912.821 | 22/13, 39/23 | ^^B, vC♭ | ^5A♯, vB |
| 90 | 923.077 | 29/17 | ^3B, C♭ | B |
| 91 | 933.333 | 12/7 | ^4B, v7C | ^B, v5C |
| 92 | 943.59 | ^5B, v6C | ^^B, v4C | |
| 93 | 953.846 | 26/15, 33/19 | ^6B, v5C | ^3B, v3C |
| 94 | 964.103 | ^7B, v4C | ^4B, vvC | |
| 95 | 974.359 | B♯, v3C | ^5B, vC | |
| 96 | 984.615 | 30/17 | ^B♯, vvC | C |
| 97 | 994.872 | ^^B♯, vC | ^C, v5D♭ | |
| 98 | 1005.128 | C | ^^C, v4D♭ | |
| 99 | 1015.385 | ^C, vvD♭♭ | ^3C, v3D♭ | |
| 100 | 1025.641 | 47/26 | ^^C, vD♭♭ | ^4C, vvD♭ |
| 101 | 1035.897 | 20/11 | ^3C, D♭♭ | ^5C, vD♭ |
| 102 | 1046.154 | ^4C, v7D♭ | ^6C, D♭ | |
| 103 | 1056.41 | 35/19 | ^5C, v6D♭ | ^7C, v14D |
| 104 | 1066.667 | 37/20 | ^6C, v5D♭ | ^8C, v13D |
| 105 | 1076.923 | 41/22 | ^7C, v4D♭ | ^9C, v12D |
| 106 | 1087.179 | 15/8 | C♯, v3D♭ | ^10C, v11D |
| 107 | 1097.436 | ^C♯, vvD♭ | ^11C, v10D | |
| 108 | 1107.692 | ^^C♯, vD♭ | ^12C, v9D | |
| 109 | 1117.949 | ^3C♯, D♭ | ^13C, v8D | |
| 110 | 1128.205 | 23/12 | ^4C♯, v7D | ^14C, v7D |
| 111 | 1138.462 | ^5C♯, v6D | C♯, v6D | |
| 112 | 1148.718 | 33/17 | ^6C♯, v5D | ^C♯, v5D |
| 113 | 1158.974 | 43/22 | ^7C♯, v4D | ^^C♯, v4D |
| 114 | 1169.231 | C𝄪, v3D | ^3C♯, v3D | |
| 115 | 1179.487 | ^C𝄪, vvD | ^4C♯, vvD | |
| 116 | 1189.744 | ^^C𝄪, vD | ^5C♯, vD | |
| 117 | 1200 | 2/1 | D | D |