218edo

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← 217edo218edo219edo →
Prime factorization 2 × 109
Step size 5.50459¢
Fifth 128\218 (704.587¢) (→64\109)
Semitones (A1:m2) 24:14 (132.1¢ : 77.06¢)
Dual sharp fifth 128\218 (704.587¢) (→64\109)
Dual flat fifth 127\218 (699.083¢)
Dual major 2nd 37\218 (203.67¢)
Consistency limit 3
Distinct consistency limit 3

218 equal divisions of the octave (abbreviated 218edo or 218ed2), also called 218-tone equal temperament (218tet) or 218 equal temperament (218et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 218 equal parts of about 5.505 ¢ each. Each step represents a frequency ratio of 21/218, or the 218th root of 2.

218edo is inconsistent to the 5-odd-limit, with harmonic 3 falling about halfway between its steps. However, it contains very accurate ratios, such as 7/4, 9/7, 9/8, 10/9, 11/10, 17/16, and 19/16, which are approximated within 0.55-cent deviation (10% the step size). The suggested subgroups are therefore 2.9.7.17.19 and 2.9.5.7.11.17.19.23.

Commas using the 13-limit patent val:

5-limit
20000/19683, 1220703125/1207959552
7-limit
4000/3969, 65625/65536, 245/243, 2401/2400 117649/116640
11-limit
4000/3993, 12005/11979, 16384/16335, 4375/4356, 78125/77616, 896/891, 67228/66825, 1375/1372, 6875/6804, 5632/5625, 385/384, 94325/93312, 15488/15435, 75625/75264, 15488/15309, 3388/3375, 1331/1323, 6655/6561, 65219/64800, 43923/43904, 73205/72576,
13-limit
28672/28561, 86240/85683, 20480/20449, 5600/5577, 16807/16731, 25000/24843, 6125/6084, 86625/86528, 68992/68445, 58080/57967, 96800/95823, 847/845, 41503/41067, 33275/33124, 65219/64896, 29575/29403, 4225/4224, 21632/21609, 676/675, 33124/32805, 9295/9261, 46475/45927, 13013/12960, 28561/28512

Odd harmonics

Approximation of odd harmonics in 218edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) +2.63 -0.99 -0.02 -0.24 -0.86 +1.67 +1.64 -0.37 -0.27 +2.61 -0.75
relative (%) +48 -18 -0 -4 -16 +30 +30 -7 -5 +47 -14
Steps
(reduced)
346
(128)
506
(70)
612
(176)
691
(37)
754
(100)
807
(153)
852
(198)
891
(19)
926
(54)
958
(86)
986
(114)

Subsets and supersets

Since 218 factors into 2 × 109, 218edo contains 2edo and 109edo as its subsets. 436edo, which doubles it, is worth exploring.