888edo

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← 887edo888edo889edo →
Prime factorization 23 × 3 × 37
Step size 1.35135¢
Fifth 519\888 (701.351¢) (→173\296)
Semitones (A1:m2) 81:69 (109.5¢ : 93.24¢)
Dual sharp fifth 520\888 (702.703¢) (→65\111)
Dual flat fifth 519\888 (701.351¢) (→173\296)
Dual major 2nd 151\888 (204.054¢)
Consistency limit 3
Distinct consistency limit 3

888 equal divisions of the octave (abbreviated 888edo), or 888-tone equal temperament (888tet), 888 equal temperament (888et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 888 equal parts of about 1.35 ¢ each. Each step of 888edo represents a frequency ratio of 21/888, or the 888th root of 2.

888edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise it is excellent in approximating harmonics 5, 7, 9, 11, and 13, making it suitable for a 2.9.5.7.11.13 subgroup interpretation. The equal temperament tempers out 4096/4095, 6656/6655, 9801/9800, 10648/10647, 105644/105625, 151263/151200, and 250047/250000 in the above subgroup.

Odd harmonics

Approximation of odd harmonics in 888edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -0.604 +0.173 +0.093 +0.144 +0.033 +0.013 -0.431 +0.450 -0.216 -0.511 +0.104
relative (%) -45 +13 +7 +11 +2 +1 -32 +33 -16 -38 +8
Steps
(reduced)
1407
(519)
2062
(286)
2493
(717)
2815
(151)
3072
(408)
3286
(622)
3469
(805)
3630
(78)
3772
(220)
3900
(348)
4017
(465)

Subsets and supersets

Since 888 factors into 23 × 3 × 37, 888edo has subset edos 2, 3, 4, 6, 8, 12, 24, 37, 74, 111, 148, 222, 296, and 444. 1776edo, which doubles it, provides a good correction for harmonic 3.