76edo
| ← 75edo | 76edo | 77edo → |
76 equal divisions of the octave (abbreviated 76edo or 76ed2), also called 76-tone equal temperament (76tet) or 76 equal temperament (76et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 76 equal parts of about 15.789 ¢ each. Each step represents a frequency ratio of 21/76, or the 76th root of 2.
Theory
| 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 (%) | -46 | -47 | -36 | +9 | +8 | -23 | +8 | +35 | +16 | +18 | +21 | |
| Steps (reduced) |
120 (44) |
176 (24) |
213 (61) |
241 (13) |
263 (35) |
281 (53) |
297 (69) |
311 (7) |
323 (19) |
334 (30) |
344 (40) | |
This tuning's 5-limit patent val is contorted in the 5-limit, reflecting the fact that 76 = 4 * 19. In the 7-limit it tempers out 2401/2400 as well as 81/80, and so supports squares temperament. In the 11-limit, it tempers out 245/242 and 385/384, and supports 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 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 non-patent vals, 76edo provides an excellent tuning for teff temperament, a low complexity, medium accuracy, and high limit (17 or 19) temperament.
Intervals
| Steps | Cents | Ups and downs notation (dual flat fifth 44\76) |
Ups and downs notation (dual sharp fifth 45\76) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 15.7895 | ^D, v3Ebb | ^D, vvEb | |
| 2 | 31.5789 | ^^D, vvEbb | ^^D, vEb | 49/48, 50/49, 56/55, 66/65 |
| 3 | 47.3684 | ^3D, vEbb | ^3D, Eb | 33/32, 36/35, 40/39 |
| 4 | 63.1579 | D#, Ebb | ^4D, v10E | 80/77 |
| 5 | 78.9474 | ^D#, v3Eb | ^5D, v9E | 21/20 |
| 6 | 94.7368 | ^^D#, vvEb | ^6D, v8E | 55/52 |
| 7 | 110.526 | ^3D#, vEb | ^7D, v7E | |
| 8 | 126.316 | Dx, Eb | ^8D, v6E | 14/13 |
| 9 | 142.105 | ^Dx, v3E | ^9D, v5E | 13/12 |
| 10 | 157.895 | ^^Dx, vvE | ^10D, v4E | |
| 11 | 173.684 | ^3Dx, vE | D#, v3E | 72/65 |
| 12 | 189.474 | E | ^D#, vvE | 39/35 |
| 13 | 205.263 | ^E, v3Fb | ^^D#, vE | 55/49 |
| 14 | 221.053 | ^^E, vvFb | E | |
| 15 | 236.842 | ^3E, vFb | ^E, vvF | 8/7, 55/48 |
| 16 | 252.632 | E#, Fb | ^^E, vF | 65/56 |
| 17 | 268.421 | ^E#, v3F | F | 7/6, 64/55 |
| 18 | 284.211 | ^^E#, vvF | ^F, vvGb | 13/11, 33/28 |
| 19 | 300 | ^3E#, vF | ^^F, vGb | 25/21 |
| 20 | 315.789 | F | ^3F, Gb | 6/5, 77/64 |
| 21 | 331.579 | ^F, v3Gbb | ^4F, v10G | 40/33 |
| 22 | 347.368 | ^^F, vvGbb | ^5F, v9G | 49/40, 60/49 |
| 23 | 363.158 | ^3F, vGbb | ^6F, v8G | 16/13 |
| 24 | 378.947 | F#, Gbb | ^7F, v7G | |
| 25 | 394.737 | ^F#, v3Gb | ^8F, v6G | 49/39, 63/50 |
| 26 | 410.526 | ^^F#, vvGb | ^9F, v5G | 33/26 |
| 27 | 426.316 | ^3F#, vGb | ^10F, v4G | 50/39 |
| 28 | 442.105 | Fx, Gb | F#, v3G | |
| 29 | 457.895 | ^Fx, v3G | ^F#, vvG | 13/10 |
| 30 | 473.684 | ^^Fx, vvG | ^^F#, vG | |
| 31 | 489.474 | ^3Fx, vG | G | 65/49 |
| 32 | 505.263 | G | ^G, vvAb | |
| 33 | 521.053 | ^G, v3Abb | ^^G, vAb | 65/48, 66/49 |
| 34 | 536.842 | ^^G, vvAbb | ^3G, Ab | 49/36 |
| 35 | 552.632 | ^3G, vAbb | ^4G, v10A | 11/8, 48/35 |
| 36 | 568.421 | G#, Abb | ^5G, v9A | 25/18, 39/28 |
| 37 | 584.211 | ^G#, v3Ab | ^6G, v8A | 7/5 |
| 38 | 600 | ^^G#, vvAb | ^7G, v7A | 55/39, 78/55 |
| 39 | 615.789 | ^3G#, vAb | ^8G, v6A | 10/7 |
| 40 | 631.579 | Gx, Ab | ^9G, v5A | 36/25, 56/39 |
| 41 | 647.368 | ^Gx, v3A | ^10G, v4A | 16/11, 35/24 |
| 42 | 663.158 | ^^Gx, vvA | G#, v3A | 72/49 |
| 43 | 678.947 | ^3Gx, vA | ^G#, vvA | 49/33, 77/52 |
| 44 | 694.737 | A | ^^G#, vA | |
| 45 | 710.526 | ^A, v3Bbb | A | |
| 46 | 726.316 | ^^A, vvBbb | ^A, vvBb | |
| 47 | 742.105 | ^3A, vBbb | ^^A, vBb | 20/13 |
| 48 | 757.895 | A#, Bbb | ^3A, Bb | 65/42 |
| 49 | 773.684 | ^A#, v3Bb | ^4A, v10B | 39/25 |
| 50 | 789.474 | ^^A#, vvBb | ^5A, v9B | 52/33 |
| 51 | 805.263 | ^3A#, vBb | ^6A, v8B | 78/49 |
| 52 | 821.053 | Ax, Bb | ^7A, v7B | 77/48 |
| 53 | 836.842 | ^Ax, v3B | ^8A, v6B | 13/8 |
| 54 | 852.632 | ^^Ax, vvB | ^9A, v5B | 49/30, 80/49 |
| 55 | 868.421 | ^3Ax, vB | ^10A, v4B | 33/20 |
| 56 | 884.211 | B | A#, v3B | 5/3 |
| 57 | 900 | ^B, v3Cb | ^A#, vvB | 42/25 |
| 58 | 915.789 | ^^B, vvCb | ^^A#, vB | 22/13, 56/33 |
| 59 | 931.579 | ^3B, vCb | B | 12/7, 55/32 |
| 60 | 947.368 | B#, Cb | ^B, vvC | |
| 61 | 963.158 | ^B#, v3C | ^^B, vC | 7/4 |
| 62 | 978.947 | ^^B#, vvC | C | |
| 63 | 994.737 | ^3B#, vC | ^C, vvDb | |
| 64 | 1010.53 | C | ^^C, vDb | 70/39 |
| 65 | 1026.32 | ^C, v3Dbb | ^3C, Db | 65/36 |
| 66 | 1042.11 | ^^C, vvDbb | ^4C, v10D | |
| 67 | 1057.89 | ^3C, vDbb | ^5C, v9D | 24/13 |
| 68 | 1073.68 | C#, Dbb | ^6C, v8D | 13/7 |
| 69 | 1089.47 | ^C#, v3Db | ^7C, v7D | |
| 70 | 1105.26 | ^^C#, vvDb | ^8C, v6D | |
| 71 | 1121.05 | ^3C#, vDb | ^9C, v5D | 40/21 |
| 72 | 1136.84 | Cx, Db | ^10C, v4D | 77/40 |
| 73 | 1152.63 | ^Cx, v3D | C#, v3D | 35/18, 39/20, 64/33 |
| 74 | 1168.42 | ^^Cx, vvD | ^C#, vvD | 49/25, 55/28, 65/33 |
| 75 | 1184.21 | ^3Cx, vD | ^^C#, vD | |
| 76 | 1200 | D | D | 2/1 |