71edo: Difference between revisions
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== Theory == | == Theory == | ||
{{Harmonics in equal|71}} | {{Harmonics in equal|71}} | ||
It tempers out 20480/19683 and [[393216/390625]] in the [[5-limit]], 875/864, 4000/3969 and 1029/1024 in the [[7-limit]], 245/242 and [[100/99]] in the [[11-limit]], and 91/90 in the [[13-limit]]. In the 13-limit it supplies the optimal [[patent val]] for the 29&71 and 34&37 temperaments. | |||
It tempers out 20480/19683 and [[393216/390625]] in the [[5-limit]], 875/864, 4000/3969 and 1029/1024 in the [[7-limit]], 245/242 and [[100/99]] in the [[11-limit]], and 91/90 in the [[13-limit]]. In the 13-limit it supplies the optimal [[patent val]] for the 29&71 and 34&37 temperaments. | |||
== Intervals == | == Intervals == | ||
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== Music == | == Music == | ||
* [https://www.youtube.com/watch?v=_FPTUlO6jNI Dancing in the Mosh Pit] by [[User:Francium|Francium]] | * [https://www.youtube.com/watch?v=_FPTUlO6jNI Dancing in the Mosh Pit] by [[User:Francium|Francium]] | ||
Revision as of 14:41, 27 November 2023
| ← 70edo | 71edo | 72edo → |
The 71 equal temperament or 71-EDO divides the octave into 71 equal parts of 16.901 cents each.
71edo is the 20th prime EDO.
71edo is a dual-fifth system, with the flat fifth (which is near the fifths of 26edo and 45edo) supporting flattone temperament, and the sharp fifth (which is near 22edo's fifth) supporting superpyth and archy. Unlike small dual-fifth systems such as 18edo, both fifths are close approximations of 3/2.
Theory
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.90 | +2.42 | -5.45 | -1.09 | +6.43 | +4.54 | -6.58 | -3.55 | +6.71 | +2.46 | -2.92 |
| Relative (%) | +46.8 | +14.3 | -32.2 | -6.5 | +38.0 | +26.9 | -38.9 | -21.0 | +39.7 | +14.5 | -17.3 | |
| Steps (reduced) |
113 (42) |
165 (23) |
199 (57) |
225 (12) |
246 (33) |
263 (50) |
277 (64) |
290 (6) |
302 (18) |
312 (28) |
321 (37) | |
It tempers out 20480/19683 and 393216/390625 in the 5-limit, 875/864, 4000/3969 and 1029/1024 in the 7-limit, 245/242 and 100/99 in the 11-limit, and 91/90 in the 13-limit. In the 13-limit it supplies the optimal patent val for the 29&71 and 34&37 temperaments.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 41\71) |
Ups and downs notation (Dual sharp fifth 42\71) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 16.9 | ^D, vE♭♭♭ | ^D, vvE♭ | |
| 2 | 33.8 | vD♯, E♭♭♭ | ^^D, vE♭ | |
| 3 | 50.7 | 35/34 | D♯, ^E♭♭♭ | ^3D, E♭ |
| 4 | 67.6 | 25/24, 26/25 | ^D♯, vE♭♭ | ^4D, ^E♭ |
| 5 | 84.5 | 21/20 | vD𝄪, E♭♭ | ^5D, ^^E♭ |
| 6 | 101.4 | 17/16 | D𝄪, ^E♭♭ | v4D♯, ^3E♭ |
| 7 | 118.3 | 31/29 | ^D𝄪, vE♭ | v3D♯, ^4E♭ |
| 8 | 135.2 | 13/12 | vD♯𝄪, E♭ | vvD♯, v5E |
| 9 | 152.1 | 12/11, 35/32 | D♯𝄪, ^E♭ | vD♯, v4E |
| 10 | 169 | 32/29 | ^D♯𝄪, vE | D♯, v3E |
| 11 | 185.9 | 29/26 | E | ^D♯, vvE |
| 12 | 202.8 | ^E, vF♭♭ | ^^D♯, vE | |
| 13 | 219.7 | 25/22 | vE♯, F♭♭ | E |
| 14 | 236.6 | E♯, ^F♭♭ | ^E, vvF | |
| 15 | 253.5 | 22/19, 29/25, 37/32 | ^E♯, vF♭ | ^^E, vF |
| 16 | 270.4 | vE𝄪, F♭ | F | |
| 17 | 287.3 | 13/11 | E𝄪, ^F♭ | ^F, vvG♭ |
| 18 | 304.2 | 25/21, 31/26, 37/31 | ^E𝄪, vF | ^^F, vG♭ |
| 19 | 321.1 | F | ^3F, G♭ | |
| 20 | 338 | 17/14, 28/23 | ^F, vG♭♭♭ | ^4F, ^G♭ |
| 21 | 354.9 | vF♯, G♭♭♭ | ^5F, ^^G♭ | |
| 22 | 371.8 | 26/21, 31/25 | F♯, ^G♭♭♭ | v4F♯, ^3G♭ |
| 23 | 388.7 | 5/4 | ^F♯, vG♭♭ | v3F♯, ^4G♭ |
| 24 | 405.6 | 19/15, 24/19 | vF𝄪, G♭♭ | vvF♯, v5G |
| 25 | 422.5 | 37/29 | F𝄪, ^G♭♭ | vF♯, v4G |
| 26 | 439.4 | 31/24 | ^F𝄪, vG♭ | F♯, v3G |
| 27 | 456.3 | 13/10 | vF♯𝄪, G♭ | ^F♯, vvG |
| 28 | 473.2 | 21/16, 25/19 | F♯𝄪, ^G♭ | ^^F♯, vG |
| 29 | 490.1 | ^F♯𝄪, vG | G | |
| 30 | 507 | G | ^G, vvA♭ | |
| 31 | 523.9 | 23/17 | ^G, vA♭♭♭ | ^^G, vA♭ |
| 32 | 540.8 | 26/19 | vG♯, A♭♭♭ | ^3G, A♭ |
| 33 | 557.7 | 29/21 | G♯, ^A♭♭♭ | ^4G, ^A♭ |
| 34 | 574.6 | 32/23 | ^G♯, vA♭♭ | ^5G, ^^A♭ |
| 35 | 591.5 | 31/22 | vG𝄪, A♭♭ | v4G♯, ^3A♭ |
| 36 | 608.5 | 37/26 | G𝄪, ^A♭♭ | v3G♯, ^4A♭ |
| 37 | 625.4 | 23/16 | ^G𝄪, vA♭ | vvG♯, v5A |
| 38 | 642.3 | 29/20 | vG♯𝄪, A♭ | vG♯, v4A |
| 39 | 659.2 | 19/13 | G♯𝄪, ^A♭ | G♯, v3A |
| 40 | 676.1 | 31/21, 34/23, 37/25 | ^G♯𝄪, vA | ^G♯, vvA |
| 41 | 693 | A | ^^G♯, vA | |
| 42 | 709.9 | ^A, vB♭♭♭ | A | |
| 43 | 726.8 | 32/21, 35/23 | vA♯, B♭♭♭ | ^A, vvB♭ |
| 44 | 743.7 | 20/13 | A♯, ^B♭♭♭ | ^^A, vB♭ |
| 45 | 760.6 | 31/20 | ^A♯, vB♭♭ | ^3A, B♭ |
| 46 | 777.5 | vA𝄪, B♭♭ | ^4A, ^B♭ | |
| 47 | 794.4 | 19/12, 30/19 | A𝄪, ^B♭♭ | ^5A, ^^B♭ |
| 48 | 811.3 | 8/5 | ^A𝄪, vB♭ | v4A♯, ^3B♭ |
| 49 | 828.2 | 21/13 | vA♯𝄪, B♭ | v3A♯, ^4B♭ |
| 50 | 845.1 | 31/19 | A♯𝄪, ^B♭ | vvA♯, v5B |
| 51 | 862 | 23/14, 28/17 | ^A♯𝄪, vB | vA♯, v4B |
| 52 | 878.9 | B | A♯, v3B | |
| 53 | 895.8 | ^B, vC♭♭ | ^A♯, vvB | |
| 54 | 912.7 | 22/13 | vB♯, C♭♭ | ^^A♯, vB |
| 55 | 929.6 | B♯, ^C♭♭ | B | |
| 56 | 946.5 | 19/11 | ^B♯, vC♭ | ^B, vvC |
| 57 | 963.4 | vB𝄪, C♭ | ^^B, vC | |
| 58 | 980.3 | 37/21 | B𝄪, ^C♭ | C |
| 59 | 997.2 | ^B𝄪, vC | ^C, vvD♭ | |
| 60 | 1014.1 | C | ^^C, vD♭ | |
| 61 | 1031 | 29/16 | ^C, vD♭♭♭ | ^3C, D♭ |
| 62 | 1047.9 | 11/6 | vC♯, D♭♭♭ | ^4C, ^D♭ |
| 63 | 1064.8 | 24/13, 37/20 | C♯, ^D♭♭♭ | ^5C, ^^D♭ |
| 64 | 1081.7 | ^C♯, vD♭♭ | v4C♯, ^3D♭ | |
| 65 | 1098.6 | 32/17 | vC𝄪, D♭♭ | v3C♯, ^4D♭ |
| 66 | 1115.5 | C𝄪, ^D♭♭ | vvC♯, v5D | |
| 67 | 1132.4 | 25/13 | ^C𝄪, vD♭ | vC♯, v4D |
| 68 | 1149.3 | vC♯𝄪, D♭ | C♯, v3D | |
| 69 | 1166.2 | C♯𝄪, ^D♭ | ^C♯, vvD | |
| 70 | 1183.1 | ^C♯𝄪, vD | ^^C♯, vD | |
| 71 | 1200 | 2/1 | D | D |