71edo
| ← 70edo | 71edo | 72edo → |
71 equal divisions of the octave (abbreviated 71edo or 71ed2), also called 71-tone equal temperament (71tet) or 71 equal temperament (71et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 71 equal parts of about 16.9 ¢ each. Each step represents a frequency ratio of 21/71, or the 71st root of 2.
71edo is the 20th prime EDO. It 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 (%) | +47 | +14 | -32 | -6 | +38 | +27 | -39 | -21 | +40 | +15 | -17 | |
| 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 | Ups and downs notation (dual flat fifth 41\71) |
Ups and downs notation (dual sharp fifth 42\71) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 16.9014 | ↑D, ↓E♭♭♭ | ↑D, ↓↓E♭ | |
| 2 | 33.8028 | ↑↑D, E♭♭♭ | ↑↑D, ↓E♭ | |
| 3 | 50.7042 | D♯, ↓↓E♭♭ | ↑3D, E♭ | 65/63 |
| 4 | 67.6056 | ↑D♯, ↓E♭♭ | ↑4D, ↓9E | 25/24, 26/25, 80/77 |
| 5 | 84.507 | ↑↑D♯, E♭♭ | ↑5D, ↓8E | 21/20, 22/21 |
| 6 | 101.408 | D𝄪, ↓↓E♭ | ↑6D, ↓7E | 55/52 |
| 7 | 118.31 | ↑D𝄪, ↓E♭ | ↑7D, ↓6E | |
| 8 | 135.211 | ↑↑D𝄪, E♭ | ↑8D, ↓5E | 13/12 |
| 9 | 152.113 | D♯𝄪, ↓↓E | ↑9D, ↓4E | 12/11, 35/32 |
| 10 | 169.014 | ↑D♯𝄪, ↓E | D♯, ↓3E | 11/10 |
| 11 | 185.915 | E | ↑D♯, ↓↓E | |
| 12 | 202.817 | ↑E, ↓F♭♭ | ↑↑D♯, ↓E | 44/39 |
| 13 | 219.718 | ↑↑E, F♭♭ | E | 25/22 |
| 14 | 236.62 | E♯, ↓↓F♭ | ↑E, ↓↓F | 8/7, 55/48, 63/55 |
| 15 | 253.521 | ↑E♯, ↓F♭ | ↑↑E, ↓F | 15/13 |
| 16 | 270.423 | ↑↑E♯, F♭ | F | |
| 17 | 287.324 | E𝄪, ↓↓F | ↑F, ↓↓G♭ | 13/11, 77/65 |
| 18 | 304.225 | ↑E𝄪, ↓F | ↑↑F, ↓G♭ | 25/21 |
| 19 | 321.127 | F | ↑3F, G♭ | 6/5, 77/64 |
| 20 | 338.028 | ↑F, ↓G♭♭♭ | ↑4F, ↓9G | 63/52 |
| 21 | 354.93 | ↑↑F, G♭♭♭ | ↑5F, ↓8G | 16/13 |
| 22 | 371.831 | F♯, ↓↓G♭♭ | ↑6F, ↓7G | 26/21 |
| 23 | 388.732 | ↑F♯, ↓G♭♭ | ↑7F, ↓6G | 5/4 |
| 24 | 405.634 | ↑↑F♯, G♭♭ | ↑8F, ↓5G | 63/50 |
| 25 | 422.535 | F𝄪, ↓↓G♭ | ↑9F, ↓4G | 32/25 |
| 26 | 439.437 | ↑F𝄪, ↓G♭ | F♯, ↓3G | |
| 27 | 456.338 | ↑↑F𝄪, G♭ | ↑F♯, ↓↓G | 13/10 |
| 28 | 473.239 | F♯𝄪, ↓↓G | ↑↑F♯, ↓G | 21/16, 55/42 |
| 29 | 490.141 | ↑F♯𝄪, ↓G | G | |
| 30 | 507.042 | G | ↑G, ↓↓A♭ | |
| 31 | 523.944 | ↑G, ↓A♭♭♭ | ↑↑G, ↓A♭ | 65/48 |
| 32 | 540.845 | ↑↑G, A♭♭♭ | ↑3G, A♭ | 15/11 |
| 33 | 557.746 | G♯, ↓↓A♭♭ | ↑4G, ↓9A | 11/8 |
| 34 | 574.648 | ↑G♯, ↓A♭♭ | ↑5G, ↓8A | |
| 35 | 591.549 | ↑↑G♯, A♭♭ | ↑6G, ↓7A | 55/39 |
| 36 | 608.451 | G𝄪, ↓↓A♭ | ↑7G, ↓6A | 78/55 |
| 37 | 625.352 | ↑G𝄪, ↓A♭ | ↑8G, ↓5A | 63/44 |
| 38 | 642.254 | ↑↑G𝄪, A♭ | ↑9G, ↓4A | 16/11 |
| 39 | 659.155 | G♯𝄪, ↓↓A | G♯, ↓3A | 22/15 |
| 40 | 676.056 | ↑G♯𝄪, ↓A | ↑G♯, ↓↓A | 65/44, 77/52 |
| 41 | 692.958 | A | ↑↑G♯, ↓A | |
| 42 | 709.859 | ↑A, ↓B♭♭♭ | A | |
| 43 | 726.761 | ↑↑A, B♭♭♭ | ↑A, ↓↓B♭ | 32/21 |
| 44 | 743.662 | A♯, ↓↓B♭♭ | ↑↑A, ↓B♭ | 20/13, 77/50 |
| 45 | 760.563 | ↑A♯, ↓B♭♭ | ↑3A, B♭ | 65/42 |
| 46 | 777.465 | ↑↑A♯, B♭♭ | ↑4A, ↓9B | 25/16 |
| 47 | 794.366 | A𝄪, ↓↓B♭ | ↑5A, ↓8B | |
| 48 | 811.268 | ↑A𝄪, ↓B♭ | ↑6A, ↓7B | 8/5 |
| 49 | 828.169 | ↑↑A𝄪, B♭ | ↑7A, ↓6B | 21/13 |
| 50 | 845.07 | A♯𝄪, ↓↓B | ↑8A, ↓5B | 13/8 |
| 51 | 861.972 | ↑A♯𝄪, ↓B | ↑9A, ↓4B | |
| 52 | 878.873 | B | A♯, ↓3B | 5/3 |
| 53 | 895.775 | ↑B, ↓C♭♭ | ↑A♯, ↓↓B | 42/25 |
| 54 | 912.676 | ↑↑B, C♭♭ | ↑↑A♯, ↓B | 22/13 |
| 55 | 929.577 | B♯, ↓↓C♭ | B | 75/44 |
| 56 | 946.479 | ↑B♯, ↓C♭ | ↑B, ↓↓C | 26/15 |
| 57 | 963.38 | ↑↑B♯, C♭ | ↑↑B, ↓C | 7/4 |
| 58 | 980.282 | B𝄪, ↓↓C | C | 44/25 |
| 59 | 997.183 | ↑B𝄪, ↓C | ↑C, ↓↓D♭ | 39/22 |
| 60 | 1014.08 | C | ↑↑C, ↓D♭ | |
| 61 | 1030.99 | ↑C, ↓D♭♭♭ | ↑3C, D♭ | 20/11 |
| 62 | 1047.89 | ↑↑C, D♭♭♭ | ↑4C, ↓9D | 11/6, 64/35 |
| 63 | 1064.79 | C♯, ↓↓D♭♭ | ↑5C, ↓8D | 24/13 |
| 64 | 1081.69 | ↑C♯, ↓D♭♭ | ↑6C, ↓7D | |
| 65 | 1098.59 | ↑↑C♯, D♭♭ | ↑7C, ↓6D | |
| 66 | 1115.49 | C𝄪, ↓↓D♭ | ↑8C, ↓5D | 21/11, 40/21 |
| 67 | 1132.39 | ↑C𝄪, ↓D♭ | ↑9C, ↓4D | 25/13, 48/25, 77/40 |
| 68 | 1149.3 | ↑↑C𝄪, D♭ | C♯, ↓3D | |
| 69 | 1166.2 | C♯𝄪, ↓↓D | ↑C♯, ↓↓D | |
| 70 | 1183.1 | ↑C♯𝄪, ↓D | ↑↑C♯, ↓D | |
| 71 | 1200 | D | D | 2/1 |
Music
- Dancing in the Mosh Pit (2023)