208edo
| ← 207edo | 208edo | 209edo → |
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 21/208, 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
| 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 (%) | +32.8 | +3.9 | +7.0 | -34.4 | +43.8 | +30.9 | +36.7 | -19.2 | +43.1 | +39.8 | +9.9 | |
| 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
| 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) |