167edo
| ← 166edo | 167edo | 168edo → |
167 equal divisions of the octave (abbreviated 167edo or 167ed2), also called 167-tone equal temperament (167tet) or 167 equal temperament (167et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 167 equal parts of about 7.19 ¢ each. Each step represents a frequency ratio of 21/167, or the 167th root of 2.
Theory
167et tempers out the würschmidt comma, 393216/390625, and the leapday comma, [31 -21 1⟩, in the 5-limit; 2401/2400, 3136/3125, 6144/6125, and 179200/177147 in the 7-limit; 896/891, 2200/2187, 3025/3024, 3388/3375, and 4000/3993 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 the 11-limit unthirds temperament.
167edo also has a very close approximation to the golden magic scale.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 |
| 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 | |
| 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) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.08 | -1.34 | +2.76 | +3.14 | -2.88 | -3.11 | -0.27 | -0.06 | +2.15 | +1.93 | +2.65 | -3.22 |
| Relative (%) | +28.9 | -18.6 | +38.4 | +43.7 | -40.1 | -43.3 | -3.7 | -0.8 | +29.9 | +26.9 | +36.8 | -44.7 | |
| Steps (reduced) |
895 (60) |
906 (71) |
928 (93) |
957 (122) |
982 (147) |
990 (155) |
1013 (11) |
1027 (25) |
1034 (32) |
1053 (51) |
1065 (63) |
1081 (79) | |
Subsets and supersets
167edo is the 39th prime edo.
Intervals
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 | 393216/390625, [31 -21 1⟩ | [⟨167 265 388]] | −0.7158 | 0.5759 | 8.01 |
| 2.3.5.7 | 2401/2400, 3136/3125, 179200/177147 | [⟨167 265 388 469]] | −0.6467 | 0.5129 | 7.14 |
| 2.3.5.7.11 | 896/891, 2200/2187, 2401/2400, 3136/3125 | [⟨167 265 388 469 578]] | −0.6315 | 0.4598 | 6.40 |
| 2.3.5.7.11.13 | 325/324, 352/351, 364/363, 1001/1000, 1716/1715 | [⟨167 265 388 469 578 618]] | −0.5349 | 0.4721 | 6.57 |
| 2.3.5.7.11.13.17 | 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000 | [⟨167 265 388 469 578 618 683]] | −0.5573 | 0.4405 | 6.13 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 27\167 | 194.01 | 28/25 | Hemiwürschmidt |
| 1 | 44\167 | 316.17 | 6/5 | Counterhanson |
| 1 | 54\167 | 388.02 | 5/4 | Würschmidt |
| 1 | 58\167 | 416.77 | 14/11 | Unthirds (11-limit) |
| 1 | 63\167 | 452.69 | 125/96 | Majo |
| 1 | 69\167 | 495.81 | 4/3 | Polypyth |
| 1 | 70\167 | 502.99 | 147/110 | Quadrawürschmidt |
| 1 | 78\167 | 560.48 | 242/175 | Whoops |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
Music
- "way too random partying" from Helayo EP (2023) – Spotify | Bandcamp | YouTube – helayo in 167edo tuning
- "moving on" from hope in dark times (2024) – Spotify | Bandcamp | YouTube
- "ordering the universal theme on wish" from End of Sartorius Membranes (2024) – Spotify | Bandcamp | YouTube
- "Funky Man's Love" from Abbreviations Gone Wrong (2024) – Spotify | Bandcamp | YouTube