167edo
167edo is the equal division of the octave into 167 parts of 7.18562874251 cents each. It tempers out the würschmidt comma, 393216/390625 and 10737418240/10460353203 in the 5-limit; 2401/2400, 3136/3125, and 179200/177147 in the 7-limit; 896/891, 2200/2187, and 3388/3375 in the 11-limit; 325/324, 352/351, 364/363, 1001/1000, and 1716/1715 in the 13-limit, providing the optimal patent val for 11- and 13-limit polypyth temperament; 256/255, 442/441, 595/594, 715/714, and 936/935 in the 17-limit. It also supports 11-limit unthirds temperament.
| ← 166edo | 167edo | 168edo → |
167edo also has a very close approximation to the golden magic scale.
167edo is the 39th prime EDO.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +2.24 | +1.71 | +1.23 | +1.98 | +0.19 | +2.83 | -2.90 | -3.12 | -2.03 | -2.52 | +0.15 | +2.08 |
| Relative (%) | +0.0 | +31.1 | +23.8 | +17.2 | +27.5 | +2.7 | +39.4 | -40.4 | -43.5 | -28.3 | -35.1 | +2.1 | +28.9 | |
| Steps (reduced) |
167 (0) |
265 (98) |
388 (54) |
469 (135) |
578 (77) |
618 (117) |
683 (15) |
709 (41) |
755 (87) |
811 (143) |
827 (159) |
870 (35) |
895 (60) | |
| Harmonic | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.34 | +2.76 | +3.14 | -2.88 | -3.11 | -0.27 | -0.06 | +2.15 | +1.93 | +2.65 | -3.22 | -1.33 |
| Relative (%) | -18.6 | +38.4 | +43.7 | -40.1 | -43.3 | -3.7 | -0.8 | +29.9 | +26.9 | +36.8 | -44.7 | -18.5 | |
| Steps (reduced) |
906 (71) |
928 (93) |
957 (122) |
982 (147) |
990 (155) |
1013 (11) |
1027 (25) |
1034 (32) |
1053 (51) |
1065 (63) |
1081 (79) |
1102 (100) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [265 -167⟩ | ⟨167 265] | -0.7056 | 0.7052 | 9.81 |
| 2.3.5 | [17 1 -8⟩, [14 -22 9⟩ | ⟨167 265 388] | -0.7158 | 0.5759 | 8.01 |
| 2.3.5.7 | 6144/6125, 3136/3125, 179200/177147 | ⟨167 265 388 469] | -0.6467 | 0.5129 | 7.14 |
| 2.3.5.7.11 | 896/891, 2200/2187, 6144/6125, 6250/6237 | ⟨167 265 388 469 578] | -0.6315 | 0.4598 | 6.40 |
| 2.3.5.7.11.13 | 325/324, 352/351, 896/891, 1001/1000, 6656/6615 | ⟨167 265 388 469 578 618] | -0.5349 | 0.4721 | 6.57 |
| 2.3.5.7.11.13.17 | 325/324, 352/351, 896/891, 256/255, 1001/1000, 1225/1224 | ⟨167 265 388 469 578 618 683] | -0.5573 | 0.4405 | 6.13 |