76edo
| ← 75edo | 76edo | 77edo → |
Theory
76edo's patent val is contorted in the 5-limit, reflecting the fact that 76 = 4 × 19. In the 7-limit it tempers out 2401/2400 in addition to 81/80, and so supports the squares temperament. In the 11-limit, it tempers out 245/242 and 385/384, and supports pombe, the 24 & 26 temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the 44\76 = 11\19 fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the hornbostel temperament's POTE fifth, whereas its sharp fifth, 45\76, makes for an excellent superpyth fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.
Using the 76dgh val, 76edo provides an excellent tuning for teff temperament, a low-complexity, medium-accuracy, and high-limit (17- or 19-limit) temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -7.22 | -7.37 | -5.67 | +1.35 | +1.31 | -3.69 | +1.20 | +5.57 | +2.49 | +2.90 | +3.30 |
| Relative (%) | -45.7 | -46.7 | -35.9 | +8.6 | +8.3 | -23.3 | +7.6 | +35.3 | +15.8 | +18.4 | +20.9 | |
| Steps (reduced) |
120 (44) |
176 (24) |
213 (61) |
241 (13) |
263 (35) |
281 (53) |
297 (69) |
311 (7) |
323 (19) |
334 (30) |
344 (40) | |
Subsets and supersets
Since 76 factors into 22 × 19, 76edo has subset edos 2, 4, 19, and 38. 152edo, which doubles it, is a zeta peak edo.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 44\76) |
Ups and downs notation (Dual sharp fifth 45\76) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 15.8 | ^D, ^E♭♭♭ | ^D, vvE♭ | |
| 2 | 31.6 | ^^D, vvE♭♭ | ^^D, vE♭ | |
| 3 | 47.4 | 36/35, 38/37 | vD♯, vE♭♭ | ^3D, E♭ |
| 4 | 63.2 | 29/28 | D♯, E♭♭ | ^4D, ^E♭ |
| 5 | 78.9 | 23/22 | ^D♯, ^E♭♭ | ^5D, ^^E♭ |
| 6 | 94.7 | ^^D♯, vvE♭ | v5D♯, ^3E♭ | |
| 7 | 110.5 | vD𝄪, vE♭ | v4D♯, ^4E♭ | |
| 8 | 126.3 | 14/13 | D𝄪, E♭ | v3D♯, ^5E♭ |
| 9 | 142.1 | ^D𝄪, ^E♭ | vvD♯, v5E | |
| 10 | 157.9 | 34/31 | ^^D𝄪, vvE | vD♯, v4E |
| 11 | 173.7 | 32/29 | vD♯𝄪, vE | D♯, v3E |
| 12 | 189.5 | 19/17, 29/26, 39/35 | E | ^D♯, vvE |
| 13 | 205.3 | ^E, ^F♭♭ | ^^D♯, vE | |
| 14 | 221.1 | 33/29 | ^^E, vvF♭ | E |
| 15 | 236.8 | vE♯, vF♭ | ^E, vvF | |
| 16 | 252.6 | 22/19, 37/32 | E♯, F♭ | ^^E, vF |
| 17 | 268.4 | 7/6 | ^E♯, ^F♭ | F |
| 18 | 284.2 | 33/28 | ^^E♯, vvF | ^F, vvG♭ |
| 19 | 300 | 19/16, 25/21 | vE𝄪, vF | ^^F, vG♭ |
| 20 | 315.8 | 6/5 | F | ^3F, G♭ |
| 21 | 331.6 | 23/19 | ^F, ^G♭♭♭ | ^4F, ^G♭ |
| 22 | 347.4 | ^^F, vvG♭♭ | ^5F, ^^G♭ | |
| 23 | 363.2 | vF♯, vG♭♭ | v5F♯, ^3G♭ | |
| 24 | 378.9 | F♯, G♭♭ | v4F♯, ^4G♭ | |
| 25 | 394.7 | ^F♯, ^G♭♭ | v3F♯, ^5G♭ | |
| 26 | 410.5 | 33/26 | ^^F♯, vvG♭ | vvF♯, v5G |
| 27 | 426.3 | vF𝄪, vG♭ | vF♯, v4G | |
| 28 | 442.1 | F𝄪, G♭ | F♯, v3G | |
| 29 | 457.9 | ^F𝄪, ^G♭ | ^F♯, vvG | |
| 30 | 473.7 | ^^F𝄪, vvG | ^^F♯, vG | |
| 31 | 489.5 | vF♯𝄪, vG | G | |
| 32 | 505.3 | G | ^G, vvA♭ | |
| 33 | 521.1 | 23/17 | ^G, ^A♭♭♭ | ^^G, vA♭ |
| 34 | 536.8 | ^^G, vvA♭♭ | ^3G, A♭ | |
| 35 | 552.6 | 11/8 | vG♯, vA♭♭ | ^4G, ^A♭ |
| 36 | 568.4 | 25/18, 32/23 | G♯, A♭♭ | ^5G, ^^A♭ |
| 37 | 584.2 | 7/5 | ^G♯, ^A♭♭ | v5G♯, ^3A♭ |
| 38 | 600 | ^^G♯, vvA♭ | v4G♯, ^4A♭ | |
| 39 | 615.8 | 10/7 | vG𝄪, vA♭ | v3G♯, ^5A♭ |
| 40 | 631.6 | 23/16, 36/25 | G𝄪, A♭ | vvG♯, v5A |
| 41 | 647.4 | 16/11 | ^G𝄪, ^A♭ | vG♯, v4A |
| 42 | 663.2 | ^^G𝄪, vvA | G♯, v3A | |
| 43 | 678.9 | 34/23 | vG♯𝄪, vA | ^G♯, vvA |
| 44 | 694.7 | A | ^^G♯, vA | |
| 45 | 710.5 | ^A, ^B♭♭♭ | A | |
| 46 | 726.3 | ^^A, vvB♭♭ | ^A, vvB♭ | |
| 47 | 742.1 | vA♯, vB♭♭ | ^^A, vB♭ | |
| 48 | 757.9 | A♯, B♭♭ | ^3A, B♭ | |
| 49 | 773.7 | ^A♯, ^B♭♭ | ^4A, ^B♭ | |
| 50 | 789.5 | ^^A♯, vvB♭ | ^5A, ^^B♭ | |
| 51 | 805.3 | vA𝄪, vB♭ | v5A♯, ^3B♭ | |
| 52 | 821.1 | 37/23 | A𝄪, B♭ | v4A♯, ^4B♭ |
| 53 | 836.8 | ^A𝄪, ^B♭ | v3A♯, ^5B♭ | |
| 54 | 852.6 | ^^A𝄪, vvB | vvA♯, v5B | |
| 55 | 868.4 | 33/20, 38/23 | vA♯𝄪, vB | vA♯, v4B |
| 56 | 884.2 | 5/3 | B | A♯, v3B |
| 57 | 900 | 32/19, 37/22 | ^B, ^C♭♭ | ^A♯, vvB |
| 58 | 915.8 | ^^B, vvC♭ | ^^A♯, vB | |
| 59 | 931.6 | 12/7 | vB♯, vC♭ | B |
| 60 | 947.4 | 19/11 | B♯, C♭ | ^B, vvC |
| 61 | 963.2 | ^B♯, ^C♭ | ^^B, vC | |
| 62 | 978.9 | ^^B♯, vvC | C | |
| 63 | 994.7 | vB𝄪, vC | ^C, vvD♭ | |
| 64 | 1010.5 | 34/19 | C | ^^C, vD♭ |
| 65 | 1026.3 | 29/16 | ^C, ^D♭♭♭ | ^3C, D♭ |
| 66 | 1042.1 | 31/17 | ^^C, vvD♭♭ | ^4C, ^D♭ |
| 67 | 1057.9 | vC♯, vD♭♭ | ^5C, ^^D♭ | |
| 68 | 1073.7 | 13/7 | C♯, D♭♭ | v5C♯, ^3D♭ |
| 69 | 1089.5 | ^C♯, ^D♭♭ | v4C♯, ^4D♭ | |
| 70 | 1105.3 | ^^C♯, vvD♭ | v3C♯, ^5D♭ | |
| 71 | 1121.1 | vC𝄪, vD♭ | vvC♯, v5D | |
| 72 | 1136.8 | C𝄪, D♭ | vC♯, v4D | |
| 73 | 1152.6 | 35/18, 37/19 | ^C𝄪, ^D♭ | C♯, v3D |
| 74 | 1168.4 | ^^C𝄪, vvD | ^C♯, vvD | |
| 75 | 1184.2 | vC♯𝄪, vD | ^^C♯, vD | |
| 76 | 1200 | 2/1 | D | D |