88edo
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88 equal divisions of the octave (abbreviated 88edo or 88ed2), also called 88-tone equal temperament (88tet) or 88 equal temperament (88et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 88 equal parts of about 13.6 ¢ each. Each step represents a frequency ratio of 21/88, or the 88th root of 2.
Theory
Using two different approximations to the perfect fifth (one of 51 steps and one of 52 steps), it is compatible with both meantone temperament and the particular variety of superpyth temperament(s) supported by 22 equal temperament, respectively. The meantone fifth is 0.0384 cents flatter than that of Lucy Tuning and, thus, audibly indistinguishable from it. It also gives the optimal patent val for the 11-limit mothra and euterpe temperaments.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -6.50 | -4.50 | -0.64 | +0.64 | -5.86 | +4.93 | +2.64 | +4.14 | +2.49 | +6.49 | -1.00 |
| relative (%) | -48 | -33 | -5 | +5 | -43 | +36 | +19 | +30 | +18 | +48 | -7 | |
| Steps (reduced) |
139 (51) |
204 (28) |
247 (71) |
279 (15) |
304 (40) |
326 (62) |
344 (80) |
360 (8) |
374 (22) |
387 (35) |
398 (46) | |
Interval table
| Steps | Cents | Ups and downs notation (dual flat fifth 51\88) |
Ups and downs notation (dual sharp fifth 52\88) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 13.6364 | ^D, v3E♭♭ | ^D, v3E♭ | |
| 2 | 27.2727 | ^^D, vvE♭♭ | ^^D, vvE♭ | 65/64, 78/77 |
| 3 | 40.9091 | ^3D, vE♭♭ | ^3D, vE♭ | 40/39, 49/48 |
| 4 | 54.5455 | ^4D, E♭♭ | ^4D, E♭ | |
| 5 | 68.1818 | D♯, v4E♭ | ^5D, v11E | 25/24, 80/77 |
| 6 | 81.8182 | ^D♯, v3E♭ | ^6D, v10E | 21/20, 22/21 |
| 7 | 95.4545 | ^^D♯, vvE♭ | ^7D, v9E | |
| 8 | 109.091 | ^3D♯, vE♭ | ^8D, v8E | |
| 9 | 122.727 | ^4D♯, E♭ | ^9D, v7E | |
| 10 | 136.364 | D𝄪, v4E | ^10D, v6E | |
| 11 | 150 | ^D𝄪, v3E | ^11D, v5E | 12/11, 35/32 |
| 12 | 163.636 | ^^D𝄪, vvE | D♯, v4E | 11/10 |
| 13 | 177.273 | ^3D𝄪, vE | ^D♯, v3E | |
| 14 | 190.909 | E | ^^D♯, vvE | 39/35, 49/44 |
| 15 | 204.545 | ^E, v3F♭ | ^3D♯, vE | 44/39 |
| 16 | 218.182 | ^^E, vvF♭ | E | 25/22 |
| 17 | 231.818 | ^3E, vF♭ | ^E, v3F | 8/7, 55/48, 63/55 |
| 18 | 245.455 | ^4E, F♭ | ^^E, vvF | |
| 19 | 259.091 | E♯, v4F | ^3E, vF | 65/56 |
| 20 | 272.727 | ^E♯, v3F | F | |
| 21 | 286.364 | ^^E♯, vvF | ^F, v3G♭ | |
| 22 | 300 | ^3E♯, vF | ^^F, vvG♭ | 25/21 |
| 23 | 313.636 | F | ^3F, vG♭ | 6/5 |
| 24 | 327.273 | ^F, v3G♭♭ | ^4F, G♭ | |
| 25 | 340.909 | ^^F, vvG♭♭ | ^5F, v11G | 39/32 |
| 26 | 354.545 | ^3F, vG♭♭ | ^6F, v10G | 16/13, 49/40 |
| 27 | 368.182 | ^4F, G♭♭ | ^7F, v9G | |
| 28 | 381.818 | F♯, v4G♭ | ^8F, v8G | 5/4 |
| 29 | 395.455 | ^F♯, v3G♭ | ^9F, v7G | 44/35, 49/39, 63/50 |
| 30 | 409.091 | ^^F♯, vvG♭ | ^10F, v6G | |
| 31 | 422.727 | ^3F♯, vG♭ | ^11F, v5G | 14/11 |
| 32 | 436.364 | ^4F♯, G♭ | F♯, v4G | 77/60 |
| 33 | 450 | F𝄪, v4G | ^F♯, v3G | |
| 34 | 463.636 | ^F𝄪, v3G | ^^F♯, vvG | 55/42, 64/49, 72/55 |
| 35 | 477.273 | ^^F𝄪, vvG | ^3F♯, vG | 33/25 |
| 36 | 490.909 | ^3F𝄪, vG | G | 65/49 |
| 37 | 504.545 | G | ^G, v3A♭ | |
| 38 | 518.182 | ^G, v3A♭♭ | ^^G, vvA♭ | |
| 39 | 531.818 | ^^G, vvA♭♭ | ^3G, vA♭ | 15/11 |
| 40 | 545.455 | ^3G, vA♭♭ | ^4G, A♭ | 48/35 |
| 41 | 559.091 | ^4G, A♭♭ | ^5G, v11A | |
| 42 | 572.727 | G♯, v4A♭ | ^6G, v10A | 25/18, 39/28 |
| 43 | 586.364 | ^G♯, v3A♭ | ^7G, v9A | 7/5 |
| 44 | 600 | ^^G♯, vvA♭ | ^8G, v8A | |
| 45 | 613.636 | ^3G♯, vA♭ | ^9G, v7A | 10/7 |
| 46 | 627.273 | ^4G♯, A♭ | ^10G, v6A | 36/25, 56/39 |
| 47 | 640.909 | G𝄪, v4A | ^11G, v5A | |
| 48 | 654.545 | ^G𝄪, v3A | G♯, v4A | 35/24 |
| 49 | 668.182 | ^^G𝄪, vvA | ^G♯, v3A | 22/15 |
| 50 | 681.818 | ^3G𝄪, vA | ^^G♯, vvA | |
| 51 | 695.455 | A | ^3G♯, vA | |
| 52 | 709.091 | ^A, v3B♭♭ | A | |
| 53 | 722.727 | ^^A, vvB♭♭ | ^A, v3B♭ | 50/33 |
| 54 | 736.364 | ^3A, vB♭♭ | ^^A, vvB♭ | 49/32, 55/36 |
| 55 | 750 | ^4A, B♭♭ | ^3A, vB♭ | 77/50 |
| 56 | 763.636 | A♯, v4B♭ | ^4A, B♭ | |
| 57 | 777.273 | ^A♯, v3B♭ | ^5A, v11B | 11/7 |
| 58 | 790.909 | ^^A♯, vvB♭ | ^6A, v10B | |
| 59 | 804.545 | ^3A♯, vB♭ | ^7A, v9B | 35/22, 78/49 |
| 60 | 818.182 | ^4A♯, B♭ | ^8A, v8B | 8/5, 77/48 |
| 61 | 831.818 | A𝄪, v4B | ^9A, v7B | |
| 62 | 845.455 | ^A𝄪, v3B | ^10A, v6B | 13/8, 80/49 |
| 63 | 859.091 | ^^A𝄪, vvB | ^11A, v5B | 64/39 |
| 64 | 872.727 | ^3A𝄪, vB | A♯, v4B | |
| 65 | 886.364 | B | ^A♯, v3B | 5/3 |
| 66 | 900 | ^B, v3C♭ | ^^A♯, vvB | 42/25 |
| 67 | 913.636 | ^^B, vvC♭ | ^3A♯, vB | |
| 68 | 927.273 | ^3B, vC♭ | B | |
| 69 | 940.909 | ^4B, C♭ | ^B, v3C | |
| 70 | 954.545 | B♯, v4C | ^^B, vvC | |
| 71 | 968.182 | ^B♯, v3C | ^3B, vC | 7/4 |
| 72 | 981.818 | ^^B♯, vvC | C | 44/25 |
| 73 | 995.455 | ^3B♯, vC | ^C, v3D♭ | 39/22 |
| 74 | 1009.09 | C | ^^C, vvD♭ | 70/39 |
| 75 | 1022.73 | ^C, v3D♭♭ | ^3C, vD♭ | |
| 76 | 1036.36 | ^^C, vvD♭♭ | ^4C, D♭ | 20/11 |
| 77 | 1050 | ^3C, vD♭♭ | ^5C, v11D | 11/6, 64/35 |
| 78 | 1063.64 | ^4C, D♭♭ | ^6C, v10D | |
| 79 | 1077.27 | C♯, v4D♭ | ^7C, v9D | |
| 80 | 1090.91 | ^C♯, v3D♭ | ^8C, v8D | |
| 81 | 1104.55 | ^^C♯, vvD♭ | ^9C, v7D | |
| 82 | 1118.18 | ^3C♯, vD♭ | ^10C, v6D | 21/11, 40/21 |
| 83 | 1131.82 | ^4C♯, D♭ | ^11C, v5D | 48/25, 77/40 |
| 84 | 1145.45 | C𝄪, v4D | C♯, v4D | |
| 85 | 1159.09 | ^C𝄪, v3D | ^C♯, v3D | 39/20 |
| 86 | 1172.73 | ^^C𝄪, vvD | ^^C♯, vvD | 77/39 |
| 87 | 1186.36 | ^3C𝄪, vD | ^3C♯, vD | |
| 88 | 1200 | D | D | 2/1 |