669edo

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← 668edo669edo670edo →
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 669-tone equal temperament (669tet), 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 of 669edo represents a frequency ratio of 21/669, 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.