210edo
| ← 209edo | 210edo | 211edo → |
210 equal divisions of the octave (abbreviated 210edo or 210ed2), also called 210-tone equal temperament (210tet) or 210 equal temperament (210et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 210 equal parts of about 5.71 ¢ each. Each step represents a frequency ratio of 21/210, or the 210th root of 2.
Theory
Since 210 = 3 × 70, 210edo shares its fifth with 70edo. It is consistent to the 9-odd-limit, but there is a sharp tendency in the lower harmonics. The equal temperament tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit.
Using the 210e val, which does the best, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.90 | +2.26 | +2.60 | +1.80 | -2.75 | -0.53 | -2.55 | -2.10 | -0.37 | -2.21 | +0.30 |
| Relative (%) | +15.8 | +39.5 | +45.5 | +31.6 | -48.1 | -9.2 | -44.7 | -36.7 | -6.5 | -38.7 | +5.2 | |
| Steps (reduced) |
333 (123) |
488 (68) |
590 (170) |
666 (36) |
726 (96) |
777 (147) |
820 (190) |
858 (18) |
892 (52) |
922 (82) |
950 (110) | |
Subsets and supersets
Since 210 factors into 2 × 3 × 5 × 7, 210edo has subset edos 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [26 -12 -3⟩, [19 10 -15⟩ | [⟨210 333 488]] | −0.5138 | 0.3987 | 6.98 |
| 2.3.5.7 | 3136/3125, 5120/5103, 118098/117649 | [⟨210 333 488 590]] | −0.6170 | 0.3888 | 6.80 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 3 | 123\210 (17\210) |
702.86 (97.14) |
3/2 (18/17) |
Misty (210gh) |
| 5 | 13\210 | 74.29 | 25/24 | Countdown (210e) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct