117edo

← 116edo

117edo

118edo →

Prime factorization	32 × 13
Step size	10.2564¢
Fifth	68\117 (697.436¢)
Semitones (A1:m2)	8:11 (82.05¢ : 112.8¢)
Dual sharp fifth	69\117 (707.692¢) (→23\39)
Dual flat fifth	68\117 (697.436¢)
Dual major 2nd	20\117 (205.128¢)
Consistency limit	3
Distinct consistency limit	3

117 equal divisions of the octave (abbreviated 117edo or 117ed2), also called 117-tone equal temperament (117tet) or 117 equal temperament (117et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 117 equal parts of about 10.256 ¢ each. Each step represents a frequency ratio of 2^1/117, or the 117th root of 2.

Theory

117edo is inconsistent to the 5-odd-limit and higher odd limits, with four mappings possible for the 11-limit: ⟨117 185 272 328 405] (patent val), ⟨117 186 272 329 405] (117bd), ⟨117 185 271 328 404] (117ce), and ⟨117 185 272 329 405] (117d).

Commas

Using the patent val, it tempers out the syntonic comma (81/80) and [69 -1 -29⟩ in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting the 7-limit mohajira temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the rank-3 terpsichore temperament; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.

Using the 117bd val, it tempers out the kleisma (15625/15552) and 17179869184/16142520375 in the 5-limit; 245/243, 3136/3125, and 51200/50421 in the 7-limit; 176/175, 1232/1215, 1375/1372, and 2560/2541 in the 11-limit; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-limit.

Using the 117ce val, it tempers out the magic comma (3125/3072) and 282429536481/268435456000 in the 5-limit; 2401/2400, 3645/3584, and 4375/4374 in the 7-limit; 243/242, 441/440, and 1815/1792 in the 11-limit; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-limit.

Using the 117d val, it tempers out 126/125, 225/224, and 14495514624/13841287201 in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit grosstone temperament.

Harmonics

Approximation of odd harmonics in 117edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-4.52	+3.43	-4.72	+1.22	+2.53	+0.50	-1.09	-2.39	-0.08	+1.01	-2.63
	relative (%)	-44	+33	-46	+12	+25	+5	-11	-23	-1	+10	-26
Steps (reduced)		185 (68)	272 (38)	328 (94)	371 (20)	405 (54)	433 (82)	457 (106)	478 (10)	497 (29)	514 (46)	529 (61)