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.
Theory
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. Unlike small dual-fifth systems such as 18edo, both fifths are close approximations of 3/2.
Using the patent val, the equal temperament tempers out 20480/19683 and 393216/390625 in the 5-limit, 875/864, 1029/1024 and 4000/3969 in the 7-limit, 100/99 and 245/242 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.
Odd harmonics
| 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) | |
Subsets and supersets
71edo is the 20th prime edo, following 67edo and before 73edo. 142edo, which doubles it, provides correction for the harmonic 3.
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 |
Notation
Sagittal notation
Best fifth notation
Evo flavor
Revo flavor
Evo-SZ flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
Second-best fifth notation
This notation uses the same sagittal sequence as EDOs 50, 57, and 64.
Evo flavor
Revo flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
Music
- Dancing in the Mosh Pit (2023)
- Spiraling - Randomness (2025)