170edo
| ← 169edo | 170edo | 171edo → |
170 equal divisions of the octave (abbreviated 170edo or 170ed2), also called 170-tone equal temperament (170tet) or 170 equal temperament (170et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 170 equal parts of about 7.06 ¢ each. Each step represents a frequency ratio of 21/170, or the 170th root of 2.
170edo is inconsistent to the 5-odd-limit and higher limits, with four mappings possible for the 7-limit: ⟨170 269 395 477] (patent val), ⟨170 270 395 477] (170b), ⟨170 270 395 478] (170bd), and ⟨170 269 394 477] (170c).
Using the patent val, it tempers out the valentine comma, 1990656/1953125 and 3486784401/3355443200 in the 5-limit; 126/125, 1029/1024, and 215233605/210827008 in the 7-limit, supporting the 7-limit valentine temperament; 540/539, 1944/1925, 2835/2816, and 43923/43904 in the 11-limit; 847/845, 1188/1183, 1287/1280, and 1575/1573 in the 13-limit.
Using the 170c val, it tempers out the python comma, 43046721/41943040 and the sycamore comma, 48828125/47775744 in the 5-limit; 1029/1024, 4375/4374, and 78125/76832 in the 7-limit; 385/384, 441/440, and 8019/8000 in the 11-limit, supporting the 11-limit unidec temperament; 975/968, 1188/1183, 1625/1617, and 3159/3136 in the 13-limit.
The 170b val is enfactored in the 5-limit, with the same tuning as 34edo, tempering out the diaschisma, 2048/2025 and the kleisma, 15625/15552. It tempers out 33614/32805, 50421/50000, and 84035/82944 in the 7-limit; 385/384, 1232/1215, 1331/1323, and 6250/6237 in the 11-limit; 196/195, 364/363, 572/567, and 3146/3125 in the 13-limit.
Using the 170bd val, it tempers out 16875/16807, 51200/50421, and 420175/419904 in the 7-limit; 176/175, 896/891, and 6875/6804 in the 11-limit; 169/168, 640/637, 1001/1000, 2704/2673, and 4235/4212 in the 13-limit. Using the alternative 170bdef val, it tempers out 540/539, 1375/1372, 4375/4356, and 8192/8085 in the 11-limit; 325/324, 364/363, 512/507, 625/624, and 1625/1617 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -3.13 | +1.92 | -1.77 | +0.80 | -0.73 | -0.53 | -1.21 | +0.93 | -1.04 | +2.16 | -0.04 |
| relative (%) | -44 | +27 | -25 | +11 | -10 | -7 | -17 | +13 | -15 | +31 | -1 | |
| Steps (reduced) |
269 (99) |
395 (55) |
477 (137) |
539 (29) |
588 (78) |
629 (119) |
664 (154) |
695 (15) |
722 (42) |
747 (67) |
769 (89) | |
Subsets and supersets
Since 170 factors into 2 × 5 × 17, 170edo has subset edos 2, 5, 10, 17, 34, and 85.