669edo

← 668edo

669edo

670edo →

Prime factorization	3 × 223
Step size	1.79372¢
Fifth	391\669 (701.345¢)
Semitones (A1:m2)	61:52 (109.4¢ : 93.27¢)
Dual sharp fifth	392\669 (703.139¢)
Dual flat fifth	391\669 (701.345¢)
Dual major 2nd	114\669 (204.484¢) (→38\223)
Consistency limit	7
Distinct consistency limit	7

669 equal divisions of the octave (abbreviated 669edo or 669ed2), also called 669-tone equal temperament (669tet) or 669 equal temperament (669et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 669 equal parts of about 1.79 ¢ each. Each step represents a frequency ratio of 2^1/669, or the 669th root of 2.

669edo is consistent in the 7-odd-limit, although it has significant errors on the 3rd and the 5th harmonics. Besides that, 669c val is a tuning for the sensipent temperament in the 5-limit.

669edo appears better at approximating higher harmonics, with harmonics 37 through 53 all having an error of 20% or less, with a comma basis for the 2.37.41.43.47.53 subgroup being {75809/75776, 1874161/1873232, 151124317/151101728, 9033613312/9032089499, 9795995841727/9788230467584}. Overall, the subgroup which provides satisfactory results for 669edo is 2.7.19.29.37.41.43.47.53.

Odd harmonics

Approximation of odd harmonics in 669edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-0.610	-0.663	-0.216	+0.574	-0.645	+0.728	+0.521	+0.874	+0.245	-0.826	-0.472
	relative (%)	-34	-37	-12	+32	-36	+41	+29	+49	+14	-46	-26
Steps (reduced)		1060 (391)	1553 (215)	1878 (540)	2121 (114)	2314 (307)	2476 (469)	2614 (607)	2735 (59)	2842 (166)	2938 (262)	3026 (350)

Approximation of odd harmonics in 669edo

Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	absolute (¢)	+0.467	-0.035	+0.019	-0.641	+0.539	-0.880	-0.223	+0.118	-0.363	-0.307	-0.089
	relative (%)	+26	-2	+1	-36	+30	-49	-12	+7	-20	-17	-5
Steps (reduced)		3107 (431)	3181 (505)	3250 (574)	3314 (638)	3375 (30)	3431 (86)	3485 (140)	3536 (191)	3584 (239)	3630 (285)	3674 (329)

Approximation of odd harmonics in 669edo

Harmonic		47	49	51	53
Error	absolute (¢)	-0.036	-0.432	+0.264	+0.038
	relative (%)	-2	-24	+15	+2
Steps (reduced)		3716 (371)	3756 (411)	3795 (450)	3832 (487)

Subsets and supersets

Since 669 factors into 3 × 223, 669edo contains 3edo and 223edo as subsets.