208edo

← 207edo

208edo

209edo →

Prime factorization	24 × 13
Step size	5.76923¢
Fifth	122\208 (703.846¢) (→61\104)
Semitones (A1:m2)	22:14 (126.9¢ : 80.77¢)
Consistency limit	7
Distinct consistency limit	7

208 equal divisions of the octave (abbreviated 208edo or 208ed2), also called 208-tone equal temperament (208tet) or 208 equal temperament (208et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 208 equal parts of about 5.77 ¢ each. Each step represents a frequency ratio of 2^1/208, or the 208th root of 2.

Theory

208edo is closely related to 104edo, but the mappings for harmonic 5 differ. As an equal temperament, it tempers out 15625/15552, the kleisma, and is the optimal patent val for the kleismic temperament metakleismic, and 7-, 11- and 13-limit rank-3 tolerant temperament. It is also the optimal patent val for the rank-4 11-limit temperament tempering out 896/891, the pentacircle temperament. Other commas it tempers out include 2200/2187 in the 11-limit and 325/324, 352/351, 364/363 and 625/624 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 208edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+1.89	+0.22	+0.40	-1.99	+2.53	+1.78	+2.12	-1.11	+2.49	+2.30	+0.57
	relative (%)	+33	+4	+7	-34	+44	+31	+37	-19	+43	+40	+10
Steps (reduced)		330 (122)	483 (67)	584 (168)	659 (35)	720 (96)	770 (146)	813 (189)	850 (18)	884 (52)	914 (82)	941 (109)

Subsets and supersets

Since 208 factors into 2⁴ × 13, 208edo has subset edos 2, 4, 8, 16, 13, 26, 52, and 104.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3.5	15625/15552, [57 -33 -2⟩	[⟨208 330 483]]	-0.4301	0.5409	9.38
2.3.5.7	2401/2400, 15625/15552, 179200/177147	[⟨208 330 483 584]]	-0.3586	0.4845	8.40
2.3.5.7.11	896/891, 2200/2187, 2401/2400, 3025/3024	[⟨208 330 483 584 720]]	-0.4330	0.4582	7.94
2.3.5.7.11.13	325/324, 352/351, 364/363, 676/675, 2401/2400	[⟨208 330 483 584 720 770]]	-0.4410	0.4187	7.26

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	47\208	251.15	1024/875	Quasiorwell
1	55\208	317.31	6/5	Metakleismic
4	55\208 (3\208)	317.31 (17.31)	6/5 (126/125)	Quadritikleismic (7-limit)