68edo
| ← 67edo | 68edo | 69edo → |
68 equal divisions of the octave (abbreviated 68edo or 68ed2), also called 68-tone equal temperament (68tet) or 68 equal temperament (68et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 68 equal parts of about 17.6 ¢ each. Each step represents a frequency ratio of 21/68, or the 68th root of 2.
Theory
68edo's step 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.
The 3rd degree of 68edo can be used as a generator for stretched 23edo, which also acts as the quartkeenlig temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents). It also works as a 22L 1s MOS of the quartkeenlig temperament.
The 5th degree of 68edo can be used as a generator for 88cET.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +3.93 | +1.92 | +1.76 | -4.26 | +6.53 | +0.93 | +2.49 | +7.02 | -6.05 | +2.02 |
| relative (%) | +0 | +22 | +11 | +10 | -24 | +37 | +5 | +14 | +40 | -34 | +11 | |
| Steps (reduced) |
68 (0) |
108 (40) |
158 (22) |
191 (55) |
235 (31) |
252 (48) |
278 (6) |
289 (17) |
308 (36) |
330 (58) |
337 (65) | |
Intervals
| Degrees | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0.00 | 1/1 |
| 1 | 17.65 | 64/63, 126/125, 225/224 |
| 2 | 35.29 | 81/80, 49/48, 50/49 |
| 3 | 52.94 | 28/27, 36/35, 33/32 |
| 4 | 70.59 | 25/24, 22/21 |
| 5 | 88.24 | 21/20, 19/18, 20/19 |
| 6 | 105.88 | 16/15, 17/16, 18/17 |
| 7 | 123.53 | 15/14, 14/13 |
| 8 | 141.18 | 13/12 |
| 9 | 158.82 | 12/11, 11/10 |
| 10 | 176.47 | 10/9 |
| 11 | 194.12 | 28/25, 19/17 |
| 12 | 211.76 | 9/8 |
| 13 | 229.41 | 8/7 |
| 14 | 247.06 | 15/13 |
| 15 | 264.71 | 7/6 |
| 16 | 282.35 | 20/17 |
| 17 | 300.00 | 13/11, 19/16 |
| 18 | 317.65 | 6/5 |
| 19 | 335.29 | 11/9, 40/33, 17/14 |
| 20 | 352.94 | 16/13, 39/32 |
| 21 | 370.59 | 27/22, 26/21, 21/17 |
| 22 | 388.24 | 5/4 |
| 23 | 405.88 | 24/19, 19/15 |
| 24 | 423.53 | 14/11 |
| 25 | 441.18 | 9/7 |
| 26 | 458.82 | 13/10, 17/13 |
| 27 | 476.47 | 21/16 |
| 28 | 494.12 | 4/3 |
| 29 | 511.76 | 75/56 |
| 30 | 529.41 | 27/20, 19/14 |
| 31 | 547.06 | 11/8, 15/11 |
| 32 | 564.71 | 25/18, 18/13, 26/19 |
| 33 | 582.35 | 7/5 |
| 34 | 600.00 | 17/12, 24/17 |
| 35 | 617.65 | 10/7 |
| 36 | 635.29 | 36/25, 13/9, 19/13 |
| 37 | 652.94 | 16/11, 22/15 |
| 38 | 670.59 | 40/27, 28/19 |
| 39 | 688.24 | 112/75 |
| 40 | 705.88 | 3/2 |
| 41 | 723.53 | 32/21 |
| 42 | 741.18 | 16/13, 26/17 |
| 43 | 758.82 | 14/9 |
| 44 | 776.47 | 11/7 |
| 45 | 794.12 | 19/12, 30/19 |
| 46 | 811.76 | 8/5 |
| 47 | 829.41 | 44/27, 21/13, 34/21 |
| 48 | 847.06 | 13/8, 64/39 |
| 49 | 864.71 | 18/11, 33/20, 28/17 |
| 50 | 882.35 | 5/3 |
| 51 | 900.00 | 22/13, 32/19 |
| 52 | 917.65 | 17/10 |
| 53 | 935.29 | 12/7 |
| 54 | 952.94 | 26/15 |
| 55 | 970.59 | 7/4 |
| 56 | 988.24 | 16/9 |
| 57 | 1005.88 | 25/14, 34/19 |
| 58 | 1023.53 | 9/5 |
| 59 | 1041.18 | 11/6, 20/11 |
| 60 | 1058.82 | 24/13 |
| 61 | 1076.47 | 28/15, 13/7 |
| 62 | 1094.12 | 15/8, 32/17, 17/9 |
| 63 | 1111.76 | 40/21, 36/19, 19/10 |
| 64 | 1129.41 | 48/25, 21/11 |
| 65 | 1147.06 | 27/14, 35/18, 64/33 |
| 66 | 1164.71 | 160/81, 96/49, 49/25 |
| 67 | 1182.35 | 63/32, 125/64, 448/225 |
| 68 | 1200.00 | 2/1 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 245/243, 2048/2025, 2401/2400 | [⟨68 108 158 191]] | -0.983 | 0.915 | 5.19 |
| 2.3.5.7.11 | 121/120, 176/175, 245/243, 1375/1372 | [⟨68 108 158 191 235]] | -0.541 | 1.206 | 6.84 |
| 2.3.5.7.11.13 | 121/120, 176/175, 196/195, 245/243, 275/273 | [⟨68 108 158 191 235 252]] | -0.745 | 1.191 | 6.75 |
| 2.3.5.7.11.13.17 | 121/120, 136/135, 154/153, 176/175, 196/195, 275/273 | [⟨68 108 158 191 235 252 278]] | -0.671 | 1.118 | 6.34 |
| 2.3.5.7.11.13.17.19 | 121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273 | [⟨68 108 158 191 235 252 278 289]] | -0.661 | 1.046 | 5.93 |
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
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
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
Other
Quartkeenlig (Stretched 23edo): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2