68edo
The 68 equal temperament, often abbreviated 68-tET, 68-EDO, or 68-ET, is the scale derived by dividing the octave into 68 equally-sized steps. Each step represents a frequency ratio of 17.65 cents; this is half of the step size of 34edo, which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of 17edo, which does well in the 3-limit, but not so well in the 5-limit. The luck continues: 68 is a strong 7-limit system, but does not do as well for in 11-limit; though it's certainly usable for that purpose, it does not represent the 11-limit diamond consistently.
As a 7-limit system it tempers out 2048/2025, 245/243, 4000/3969, 15625/15552, 3136/3125, 6144/6125 and 2401/2400. It supports octacot, shrutar, hemiwürschmidt, hemikleismic, clyde and neptune temperaments, and supplies the optimal patent val for 11-limit hemikleismic. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp.
Intervals
| Degrees | Cents |
|---|---|
| 0 | 0.0000 |
| 1 | 17.6471 |
| 2 | 35.2941 |
| 3 | 52.9411 |
| 4 | 70.5882 |
| 5 | 88.2353 |
| 6 | 105.8824 |
| 7 | 123.5294 |
| 8 | 141.1765 |
| 9 | 158.8235 |
| 10 | 176.4706 |
| 11 | 194.1176 |
| 12 | 211.7647 |
| 13 | 229.4118 |
| 14 | 247.0588 |
| 15 | 264.7059 |
| 16 | 282.3529 |
| 17 | 300.0000 |
| 18 | 317.6471 |
| 19 | 335.2941 |
| 20 | 352.9412 |
| 21 | 370.5882 |
| 22 | 388.2353 |
| 23 | 405.8824 |
| 24 | 423.5294 |
| 25 | 441.1765 |
| 26 | 458.8235 |
| 27 | 476.4706 |
| 28 | 494.1176 |
| 29 | 511.7647 |
| 30 | 529.4118 |
| 31 | 547.0588 |
| 32 | 564.7059 |
| 33 | 582.3529 |
| 34 | 600.0000 |
| 35 | 617.6471 |
| 36 | 635.2941 |
| 37 | 652.9412 |
| 38 | 670.5882 |
| 39 | 688.2353 |
| 40 | 705.8824 |
| 41 | 723.5294 |
| 42 | 741.1765 |
| 43 | 758.8235 |
| 44 | 776.4706 |
| 45 | 794.1176 |
| 46 | 811.7647 |
| 47 | 829.4118 |
| 48 | 847.0588 |
| 49 | 864.7059 |
| 50 | 882.3529 |
| 51 | 900.0000 |
| 52 | 917.6471 |
| 53 | 935.2941 |
| 54 | 952.9412 |
| 55 | 970.5882 |
| 56 | 988.2353 |
| 57 | 1005.8824 |
| 58 | 1023.5294 |
| 59 | 1041.1765 |
| 60 | 1058.8235 |
| 61 | 1076.4706 |
| 62 | 1094.1176 |
| 63 | 1111.7647 |
| 64 | 1129.4118 |
| 65 | 1147.0588 |
| 66 | 1164.7059 |
| 67 | 1182.3529 |
| 68 | 1200.0000 |
Scales
Diatonic scales
Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C5)
Superpyth: 12 12 4 12 12 12 4
Superpyth quarter octave: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1
Flattone: 10 10 9 10 10 10 9
Inverse: 8 8 14 8 8 8 14
Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7