162edo
| ← 161edo | 162edo | 163edo → |
162 equal divisions of the octave (abbreviated 162edo or 162ed2), also called 162-tone equal temperament (162tet) or 162 equal temperament (162et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 162 equal parts of about 7.41 ¢ each. Each step represents a frequency ratio of 21/162, or the 162nd root of 2.
Using the patent val, the equal temperament tempers out 4000/3969, 10976/10935 and 65625/65536.
The non-patent val ⟨162 257 377] (162c) and its extensions are of considerable interest, as this tempers out 2048/2025. In the 7-limit, ⟨162 257 377 455] tempers out 126/125 and 2048/2025 both, giving a tuning for 7-limit diaschismic. In the 11-limit ⟨162 257 377 455 561] (162ce) tempers out 126/125, 176/175 and 896/891, and so supports 11-limit diaschismic, and in fact has a fifth only 0.01 cents flatter than the POTE tuning. The 13-limit is even closer: ⟨162 257 377 455 561 600] (162cef) tempers out 126/125, 176/175, 196/195, 364/363 giving 13-limit diaschismic, and the fifth of 95\162 is a mere 0.0000383 cents sharp of the 13-limit POTE tuning.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +1.75 | -1.13 | +1.54 | -3.17 | -3.49 | -1.25 | -1.22 | +1.36 | +0.05 | +3.11 |
| relative (%) | +0 | +24 | -15 | +21 | -43 | -47 | -17 | -16 | +18 | +1 | +42 | |
| Steps (reduced) |
162 (0) |
257 (95) |
376 (52) |
455 (131) |
560 (74) |
599 (113) |
662 (14) |
688 (40) |
733 (85) |
787 (139) |
803 (155) | |
Subsets and supersets
Since 162 factors into 2 × 34, 162edo has subset edos 2, 3, 6, 9, 18, 27, 54, and 81.