170edo

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← 169edo170edo171edo →
Prime factorization 2 × 5 × 17
Step size 7.05882¢ 
Fifth 99\170 (698.824¢)
Semitones (A1:m2) 13:15 (91.76¢ : 105.9¢)
Dual sharp fifth 100\170 (705.882¢) (→10\17)
Dual flat fifth 99\170 (698.824¢)
Dual major 2nd 29\170 (204.706¢)
Consistency limit 3
Distinct consistency limit 3

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

Approximation of odd harmonics in 170edo
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 (%) -44.4 +27.2 -25.0 +11.3 -10.3 -7.5 -17.1 +13.1 -14.8 +30.6 -0.6
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.