165edo
| ← 164edo | 165edo | 166edo → |
165 equal divisions of the octave (abbreviated 165edo or 165ed2), also called 165-tone equal temperament (165tet) or 165 equal temperament (165et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 165 equal parts of about 7.27 ¢ each. Each step represents a frequency ratio of 21/165, or the 165th root of 2.
165edo is inconsistent to the 5-odd-limit and higher limits, with two mappings possible for the 5-limit: ⟨165 262 383] (patent val) and ⟨165 261 383] (165b).
Using the patent val (with a sharp fifth), it tempers out 1638400/1594323 (immunity comma) and [-27 -2 13⟩ (ditonma) in the 5-limit; 4000/3969, 65625/65536, and 84035/82944 in the 7-limit; 385/384, 2401/2376, 3388/3375, and 6655/6561 in the 11-limit; 196/195, 364/363, 676/675, 3185/3168, and 3200/3159 in the 13-limit.
Using the 165b val (with a flat fifth), it tempers out 34171875/33554432 (ampersand) and 129140163/125000000 in the 5-limit; 225/224, 1029/1024, and 100442349/97656250 in the 7-limit; 1944/1925, 2187/2156, 4000/3993, and 12005/11979 in the 11-limit; 144/143, 351/350, 625/624, 847/845, and 9261/9152 in the 13-limit. Using the 165bf val, it tempers out 364/363, 975/968, 1001/1000, 1701/1690, and 1716/1715 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +3.50 | -0.86 | -1.55 | -0.27 | +1.41 | +3.11 | +2.64 | -3.14 | +0.67 | +1.95 | -2.82 |
| relative (%) | +48 | -12 | -21 | -4 | +19 | +43 | +36 | -43 | +9 | +27 | -39 | |
| Steps (reduced) |
262 (97) |
383 (53) |
463 (133) |
523 (28) |
571 (76) |
611 (116) |
645 (150) |
674 (14) |
701 (41) |
725 (65) |
746 (86) | |
Subsets and supersets
Since 165 factors into 3 × 5 × 11, 165edo has subset edos 3, 5, 11, 15, 33, and 55. 330edo, which doubles it, provides good correction for the approximation of harmonic 3.