88edo: Difference between revisions
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== Theory == | == Theory == | ||
Using two different approximations to the perfect fifth (one of 51 steps and one of 52 steps), | Using two different approximations to the [[3/2|perfect fifth]] (one of 51 steps and one of 52 steps), 88edo is compatible with both [[meantone]] and the particular variety of [[superpyth]] supported by [[22edo|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 === | === Odd harmonics === | ||
Revision as of 06:24, 21 June 2024
| ← 87edo | 88edo | 89edo → |
Theory
Using two different approximations to the perfect fifth (one of 51 steps and one of 52 steps), 88edo is compatible with both meantone and the particular variety of superpyth 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 (%) | -47.7 | -33.0 | -4.7 | +4.7 | -43.0 | +36.1 | +19.4 | +30.3 | +18.2 | +47.6 | -7.3 | |
| 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 | Approximate ratios | Ups and downs notation (Dual flat fifth 51\88) |
Ups and downs notation (Dual sharp fifth 52\88) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 13.6 | ^D, ^^E♭♭♭ | ^D, v3E♭ | |
| 2 | 27.3 | ^^D, vvE♭♭ | ^^D, vvE♭ | |
| 3 | 40.9 | 41/40, 42/41 | vvD♯, vE♭♭ | ^3D, vE♭ |
| 4 | 54.5 | 32/31 | vD♯, E♭♭ | ^4D, E♭ |
| 5 | 68.2 | 25/24 | D♯, ^E♭♭ | ^5D, ^E♭ |
| 6 | 81.8 | 21/20, 22/21 | ^D♯, ^^E♭♭ | ^6D, ^^E♭ |
| 7 | 95.5 | 37/35 | ^^D♯, vvE♭ | v5D♯, ^3E♭ |
| 8 | 109.1 | vvD𝄪, vE♭ | v4D♯, ^4E♭ | |
| 9 | 122.7 | vD𝄪, E♭ | v3D♯, ^5E♭ | |
| 10 | 136.4 | 40/37 | D𝄪, ^E♭ | vvD♯, v6E |
| 11 | 150 | 12/11 | ^D𝄪, ^^E♭ | vD♯, v5E |
| 12 | 163.6 | 11/10 | ^^D𝄪, vvE | D♯, v4E |
| 13 | 177.3 | 31/28, 41/37 | vvD♯𝄪, vE | ^D♯, v3E |
| 14 | 190.9 | 19/17, 29/26 | E | ^^D♯, vvE |
| 15 | 204.5 | ^E, ^^F♭♭ | ^3D♯, vE | |
| 16 | 218.2 | 42/37 | ^^E, vvF♭ | E |
| 17 | 231.8 | 8/7 | vvE♯, vF♭ | ^E, v3F |
| 18 | 245.5 | vE♯, F♭ | ^^E, vvF | |
| 19 | 259.1 | E♯, ^F♭ | ^3E, vF | |
| 20 | 272.7 | 34/29, 41/35 | ^E♯, ^^F♭ | F |
| 21 | 286.4 | ^^E♯, vvF | ^F, v3G♭ | |
| 22 | 300 | 19/16, 25/21 | vvE𝄪, vF | ^^F, vvG♭ |
| 23 | 313.6 | 6/5 | F | ^3F, vG♭ |
| 24 | 327.3 | ^F, ^^G♭♭♭ | ^4F, G♭ | |
| 25 | 340.9 | 28/23, 39/32 | ^^F, vvG♭♭ | ^5F, ^G♭ |
| 26 | 354.5 | 38/31 | vvF♯, vG♭♭ | ^6F, ^^G♭ |
| 27 | 368.2 | vF♯, G♭♭ | v5F♯, ^3G♭ | |
| 28 | 381.8 | F♯, ^G♭♭ | v4F♯, ^4G♭ | |
| 29 | 395.5 | 39/31 | ^F♯, ^^G♭♭ | v3F♯, ^5G♭ |
| 30 | 409.1 | ^^F♯, vvG♭ | vvF♯, v6G | |
| 31 | 422.7 | vvF𝄪, vG♭ | vF♯, v5G | |
| 32 | 436.4 | vF𝄪, G♭ | F♯, v4G | |
| 33 | 450 | F𝄪, ^G♭ | ^F♯, v3G | |
| 34 | 463.6 | 17/13 | ^F𝄪, ^^G♭ | ^^F♯, vvG |
| 35 | 477.3 | ^^F𝄪, vvG | ^3F♯, vG | |
| 36 | 490.9 | vvF♯𝄪, vG | G | |
| 37 | 504.5 | G | ^G, v3A♭ | |
| 38 | 518.2 | 31/23 | ^G, ^^A♭♭♭ | ^^G, vvA♭ |
| 39 | 531.8 | ^^G, vvA♭♭ | ^3G, vA♭ | |
| 40 | 545.5 | 26/19 | vvG♯, vA♭♭ | ^4G, A♭ |
| 41 | 559.1 | vG♯, A♭♭ | ^5G, ^A♭ | |
| 42 | 572.7 | 32/23, 39/28 | G♯, ^A♭♭ | ^6G, ^^A♭ |
| 43 | 586.4 | ^G♯, ^^A♭♭ | v5G♯, ^3A♭ | |
| 44 | 600 | ^^G♯, vvA♭ | v4G♯, ^4A♭ | |
| 45 | 613.6 | vvG𝄪, vA♭ | v3G♯, ^5A♭ | |
| 46 | 627.3 | 23/16 | vG𝄪, A♭ | vvG♯, v6A |
| 47 | 640.9 | G𝄪, ^A♭ | vG♯, v5A | |
| 48 | 654.5 | 19/13, 35/24 | ^G𝄪, ^^A♭ | G♯, v4A |
| 49 | 668.2 | ^^G𝄪, vvA | ^G♯, v3A | |
| 50 | 681.8 | vvG♯𝄪, vA | ^^G♯, vvA | |
| 51 | 695.5 | A | ^3G♯, vA | |
| 52 | 709.1 | ^A, ^^B♭♭♭ | A | |
| 53 | 722.7 | ^^A, vvB♭♭ | ^A, v3B♭ | |
| 54 | 736.4 | 26/17 | vvA♯, vB♭♭ | ^^A, vvB♭ |
| 55 | 750 | 37/24 | vA♯, B♭♭ | ^3A, vB♭ |
| 56 | 763.6 | A♯, ^B♭♭ | ^4A, B♭ | |
| 57 | 777.3 | ^A♯, ^^B♭♭ | ^5A, ^B♭ | |
| 58 | 790.9 | ^^A♯, vvB♭ | ^6A, ^^B♭ | |
| 59 | 804.5 | 35/22 | vvA𝄪, vB♭ | v5A♯, ^3B♭ |
| 60 | 818.2 | vA𝄪, B♭ | v4A♯, ^4B♭ | |
| 61 | 831.8 | A𝄪, ^B♭ | v3A♯, ^5B♭ | |
| 62 | 845.5 | 31/19 | ^A𝄪, ^^B♭ | vvA♯, v6B |
| 63 | 859.1 | 23/14, 41/25 | ^^A𝄪, vvB | vA♯, v5B |
| 64 | 872.7 | vvA♯𝄪, vB | A♯, v4B | |
| 65 | 886.4 | 5/3 | B | ^A♯, v3B |
| 66 | 900 | 32/19, 37/22, 42/25 | ^B, ^^C♭♭ | ^^A♯, vvB |
| 67 | 913.6 | 39/23 | ^^B, vvC♭ | ^3A♯, vB |
| 68 | 927.3 | 29/17, 41/24 | vvB♯, vC♭ | B |
| 69 | 940.9 | vB♯, C♭ | ^B, v3C | |
| 70 | 954.5 | B♯, ^C♭ | ^^B, vvC | |
| 71 | 968.2 | 7/4 | ^B♯, ^^C♭ | ^3B, vC |
| 72 | 981.8 | 37/21 | ^^B♯, vvC | C |
| 73 | 995.5 | vvB𝄪, vC | ^C, v3D♭ | |
| 74 | 1009.1 | 34/19 | C | ^^C, vvD♭ |
| 75 | 1022.7 | ^C, ^^D♭♭♭ | ^3C, vD♭ | |
| 76 | 1036.4 | 20/11 | ^^C, vvD♭♭ | ^4C, D♭ |
| 77 | 1050 | 11/6 | vvC♯, vD♭♭ | ^5C, ^D♭ |
| 78 | 1063.6 | 37/20 | vC♯, D♭♭ | ^6C, ^^D♭ |
| 79 | 1077.3 | 41/22 | C♯, ^D♭♭ | v5C♯, ^3D♭ |
| 80 | 1090.9 | ^C♯, ^^D♭♭ | v4C♯, ^4D♭ | |
| 81 | 1104.5 | ^^C♯, vvD♭ | v3C♯, ^5D♭ | |
| 82 | 1118.2 | 21/11, 40/21 | vvC𝄪, vD♭ | vvC♯, v6D |
| 83 | 1131.8 | vC𝄪, D♭ | vC♯, v5D | |
| 84 | 1145.5 | 31/16 | C𝄪, ^D♭ | C♯, v4D |
| 85 | 1159.1 | 41/21 | ^C𝄪, ^^D♭ | ^C♯, v3D |
| 86 | 1172.7 | ^^C𝄪, vvD | ^^C♯, vvD | |
| 87 | 1186.4 | vvC♯𝄪, vD | ^3C♯, vD | |
| 88 | 1200 | 2/1 | D | D |