218edo

← 217edo

218edo

219edo →

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 5.5 ¢ each. Each step represents a frequency ratio of 2^1/218, 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.