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) | |
Octave stretch
117edo’s approximations of 3/1, 5/1, 7/1 and 17/1 are all noticeably improved by APS7hekt, a compressed-octave version of 117edo. The trade-off is an unnoticeably worse 2/1 and 11/1, but noticeably worse 13/1.
There are also several nearby Zeta peak index (ZPI) tunings which can be used for this same purpose: 696zpi, 697zpi, 698zpi, 699zpi, 700zpi, 701zpi and 702zpi.
The details of each of those ZPI tunings are visible in User:Contribution’s gallery of Zeta Peak Indexes (1 - 10 000). Warning: due to its length, that page may slow down your device while it is open. The effect will go away after you close the page.
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 | 10.3 | ^D, vvE♭♭ | ^D, v5E♭ | |
| 2 | 20.5 | ^^D, vE♭♭ | ^^D, v4E♭ | |
| 3 | 30.8 | ^3D, E♭♭ | ^3D, v3E♭ | |
| 4 | 41 | 41/40, 44/43 | ^4D, ^E♭♭ | ^4D, vvE♭ |
| 5 | 51.3 | 33/32, 34/33, 35/34 | v3D♯, ^^E♭♭ | ^5D, vE♭ |
| 6 | 61.5 | 29/28 | vvD♯, ^3E♭♭ | ^6D, E♭ |
| 7 | 71.8 | 24/23 | vD♯, v4E♭ | ^7D, ^E♭ |
| 8 | 82.1 | 43/41 | D♯, v3E♭ | v7D♯, ^^E♭ |
| 9 | 92.3 | ^D♯, vvE♭ | v6D♯, ^3E♭ | |
| 10 | 102.6 | 35/33 | ^^D♯, vE♭ | v5D♯, ^4E♭ |
| 11 | 112.8 | 16/15, 47/44 | ^3D♯, E♭ | v4D♯, ^5E♭ |
| 12 | 123.1 | 44/41 | ^4D♯, ^E♭ | v3D♯, ^6E♭ |
| 13 | 133.3 | 40/37, 41/38 | v3D𝄪, ^^E♭ | vvD♯, ^7E♭ |
| 14 | 143.6 | 38/35 | vvD𝄪, ^3E♭ | vD♯, v7E |
| 15 | 153.8 | 35/32, 47/43 | vD𝄪, v4E | D♯, v6E |
| 16 | 164.1 | 11/10 | D𝄪, v3E | ^D♯, v5E |
| 17 | 174.4 | ^D𝄪, vvE | ^^D♯, v4E | |
| 18 | 184.6 | ^^D𝄪, vE | ^3D♯, v3E | |
| 19 | 194.9 | E | ^4D♯, vvE | |
| 20 | 205.1 | ^E, vvF♭ | ^5D♯, vE | |
| 21 | 215.4 | 17/15, 43/38 | ^^E, vF♭ | E |
| 22 | 225.6 | 33/29 | ^3E, F♭ | ^E, v5F |
| 23 | 235.9 | 39/34, 47/41 | ^4E, ^F♭ | ^^E, v4F |
| 24 | 246.2 | 15/13, 38/33 | v3E♯, ^^F♭ | ^3E, v3F |
| 25 | 256.4 | vvE♯, ^3F♭ | ^4E, vvF | |
| 26 | 266.7 | 7/6 | vE♯, v4F | ^5E, vF |
| 27 | 276.9 | 34/29 | E♯, v3F | F |
| 28 | 287.2 | 13/11, 46/39 | ^E♯, vvF | ^F, v5G♭ |
| 29 | 297.4 | 19/16 | ^^E♯, vF | ^^F, v4G♭ |
| 30 | 307.7 | 37/31 | F | ^3F, v3G♭ |
| 31 | 317.9 | ^F, vvG♭♭ | ^4F, vvG♭ | |
| 32 | 328.2 | 29/24 | ^^F, vG♭♭ | ^5F, vG♭ |
| 33 | 338.5 | ^3F, G♭♭ | ^6F, G♭ | |
| 34 | 348.7 | ^4F, ^G♭♭ | ^7F, ^G♭ | |
| 35 | 359 | 16/13 | v3F♯, ^^G♭♭ | v7F♯, ^^G♭ |
| 36 | 369.2 | 47/38 | vvF♯, ^3G♭♭ | v6F♯, ^3G♭ |
| 37 | 379.5 | vF♯, v4G♭ | v5F♯, ^4G♭ | |
| 38 | 389.7 | F♯, v3G♭ | v4F♯, ^5G♭ | |
| 39 | 400 | 29/23 | ^F♯, vvG♭ | v3F♯, ^6G♭ |
| 40 | 410.3 | 19/15 | ^^F♯, vG♭ | vvF♯, ^7G♭ |
| 41 | 420.5 | ^3F♯, G♭ | vF♯, v7G | |
| 42 | 430.8 | 41/32 | ^4F♯, ^G♭ | F♯, v6G |
| 43 | 441 | 40/31 | v3F𝄪, ^^G♭ | ^F♯, v5G |
| 44 | 451.3 | vvF𝄪, ^3G♭ | ^^F♯, v4G | |
| 45 | 461.5 | 30/23 | vF𝄪, v4G | ^3F♯, v3G |
| 46 | 471.8 | 46/35 | F𝄪, v3G | ^4F♯, vvG |
| 47 | 482.1 | 41/31 | ^F𝄪, vvG | ^5F♯, vG |
| 48 | 492.3 | ^^F𝄪, vG | G | |
| 49 | 502.6 | G | ^G, v5A♭ | |
| 50 | 512.8 | 35/26, 39/29, 43/32 | ^G, vvA♭♭ | ^^G, v4A♭ |
| 51 | 523.1 | 23/17 | ^^G, vA♭♭ | ^3G, v3A♭ |
| 52 | 533.3 | 49/36 | ^3G, A♭♭ | ^4G, vvA♭ |
| 53 | 543.6 | 26/19 | ^4G, ^A♭♭ | ^5G, vA♭ |
| 54 | 553.8 | v3G♯, ^^A♭♭ | ^6G, A♭ | |
| 55 | 564.1 | vvG♯, ^3A♭♭ | ^7G, ^A♭ | |
| 56 | 574.4 | 39/28, 46/33 | vG♯, v4A♭ | v7G♯, ^^A♭ |
| 57 | 584.6 | G♯, v3A♭ | v6G♯, ^3A♭ | |
| 58 | 594.9 | 31/22 | ^G♯, vvA♭ | v5G♯, ^4A♭ |
| 59 | 605.1 | 44/31 | ^^G♯, vA♭ | v4G♯, ^5A♭ |
| 60 | 615.4 | ^3G♯, A♭ | v3G♯, ^6A♭ | |
| 61 | 625.6 | 33/23 | ^4G♯, ^A♭ | vvG♯, ^7A♭ |
| 62 | 635.9 | v3G𝄪, ^^A♭ | vG♯, v7A | |
| 63 | 646.2 | vvG𝄪, ^3A♭ | G♯, v6A | |
| 64 | 656.4 | 19/13 | vG𝄪, v4A | ^G♯, v5A |
| 65 | 666.7 | 47/32 | G𝄪, v3A | ^^G♯, v4A |
| 66 | 676.9 | 34/23, 37/25 | ^G𝄪, vvA | ^3G♯, v3A |
| 67 | 687.2 | ^^G𝄪, vA | ^4G♯, vvA | |
| 68 | 697.4 | A | ^5G♯, vA | |
| 69 | 707.7 | ^A, vvB♭♭ | A | |
| 70 | 717.9 | ^^A, vB♭♭ | ^A, v5B♭ | |
| 71 | 728.2 | 35/23 | ^3A, B♭♭ | ^^A, v4B♭ |
| 72 | 738.5 | 23/15 | ^4A, ^B♭♭ | ^3A, v3B♭ |
| 73 | 748.7 | v3A♯, ^^B♭♭ | ^4A, vvB♭ | |
| 74 | 759 | 31/20, 45/29 | vvA♯, ^3B♭♭ | ^5A, vB♭ |
| 75 | 769.2 | vA♯, v4B♭ | ^6A, B♭ | |
| 76 | 779.5 | A♯, v3B♭ | ^7A, ^B♭ | |
| 77 | 789.7 | 30/19, 41/26 | ^A♯, vvB♭ | v7A♯, ^^B♭ |
| 78 | 800 | 46/29 | ^^A♯, vB♭ | v6A♯, ^3B♭ |
| 79 | 810.3 | ^3A♯, B♭ | v5A♯, ^4B♭ | |
| 80 | 820.5 | 45/28 | ^4A♯, ^B♭ | v4A♯, ^5B♭ |
| 81 | 830.8 | v3A𝄪, ^^B♭ | v3A♯, ^6B♭ | |
| 82 | 841 | 13/8 | vvA𝄪, ^3B♭ | vvA♯, ^7B♭ |
| 83 | 851.3 | vA𝄪, v4B | vA♯, v7B | |
| 84 | 861.5 | A𝄪, v3B | A♯, v6B | |
| 85 | 871.8 | 43/26, 48/29 | ^A𝄪, vvB | ^A♯, v5B |
| 86 | 882.1 | ^^A𝄪, vB | ^^A♯, v4B | |
| 87 | 892.3 | B | ^3A♯, v3B | |
| 88 | 902.6 | 32/19 | ^B, vvC♭ | ^4A♯, vvB |
| 89 | 912.8 | 22/13, 39/23 | ^^B, vC♭ | ^5A♯, vB |
| 90 | 923.1 | 29/17 | ^3B, C♭ | B |
| 91 | 933.3 | 12/7 | ^4B, ^C♭ | ^B, v5C |
| 92 | 943.6 | v3B♯, ^^C♭ | ^^B, v4C | |
| 93 | 953.8 | 26/15, 33/19 | vvB♯, ^3C♭ | ^3B, v3C |
| 94 | 964.1 | vB♯, v4C | ^4B, vvC | |
| 95 | 974.4 | B♯, v3C | ^5B, vC | |
| 96 | 984.6 | 30/17 | ^B♯, vvC | C |
| 97 | 994.9 | ^^B♯, vC | ^C, v5D♭ | |
| 98 | 1005.1 | C | ^^C, v4D♭ | |
| 99 | 1015.4 | ^C, vvD♭♭ | ^3C, v3D♭ | |
| 100 | 1025.6 | 47/26 | ^^C, vD♭♭ | ^4C, vvD♭ |
| 101 | 1035.9 | 20/11 | ^3C, D♭♭ | ^5C, vD♭ |
| 102 | 1046.2 | ^4C, ^D♭♭ | ^6C, D♭ | |
| 103 | 1056.4 | 35/19 | v3C♯, ^^D♭♭ | ^7C, ^D♭ |
| 104 | 1066.7 | 37/20 | vvC♯, ^3D♭♭ | v7C♯, ^^D♭ |
| 105 | 1076.9 | 41/22 | vC♯, v4D♭ | v6C♯, ^3D♭ |
| 106 | 1087.2 | 15/8 | C♯, v3D♭ | v5C♯, ^4D♭ |
| 107 | 1097.4 | ^C♯, vvD♭ | v4C♯, ^5D♭ | |
| 108 | 1107.7 | ^^C♯, vD♭ | v3C♯, ^6D♭ | |
| 109 | 1117.9 | ^3C♯, D♭ | vvC♯, ^7D♭ | |
| 110 | 1128.2 | 23/12 | ^4C♯, ^D♭ | vC♯, v7D |
| 111 | 1138.5 | v3C𝄪, ^^D♭ | C♯, v6D | |
| 112 | 1148.7 | 33/17 | vvC𝄪, ^3D♭ | ^C♯, v5D |
| 113 | 1159 | 43/22 | vC𝄪, v4D | ^^C♯, v4D |
| 114 | 1169.2 | C𝄪, v3D | ^3C♯, v3D | |
| 115 | 1179.5 | ^C𝄪, vvD | ^4C♯, vvD | |
| 116 | 1189.7 | ^^C𝄪, vD | ^5C♯, vD | |
| 117 | 1200 | 2/1 | D | D |