209edo
| ← 208edo | 209edo | 210edo → |
209 equal divisions of the octave (abbreviated 209edo or 209ed2), also called 209-tone equal temperament (209tet) or 209 equal temperament (209et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 209 equal parts of about 5.74 ¢ each. Each step represents a frequency ratio of 21/209, or the 209th root of 2.
Theory
209edo is only consistent to the 5-odd-limit. The equal temperament tempers out 129140163/128000000 (graviton) and 1220703125/1207959552 (ditonma) in the 5-limit. Using the patent val, it tempers out 225/224, 2125764/2100875, and 2500000/2470629 in the 7-limit; 243/242, 3025/3024, 4000/3993, and 16896/16807 in the 11-limit; 351/350, 625/624, 1573/1568, 1625/1617, and 15379/15360 in the 13-limit, so that it provides the optimal patent val for the 13-limit marvo temperament. It also supports the 13-limit spectacle temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -1.48 | -1.62 | +1.51 | +2.79 | -0.12 | -2.25 | +2.64 | -1.61 | +1.05 | +0.03 | -2.44 |
| relative (%) | -26 | -28 | +26 | +49 | -2 | -39 | +46 | -28 | +18 | +1 | -42 | |
| Steps (reduced) |
331 (122) |
485 (67) |
587 (169) |
663 (36) |
723 (96) |
773 (146) |
817 (190) |
854 (18) |
888 (52) |
918 (82) |
945 (109) | |
Subsets and supersets
Since 209 factors into 11 × 19, 209edo contains 11edo and 19edo as its subsets. 627edo, which triples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-331 209⟩ | [⟨209 331]] | 0.4658 | 0.4660 | 8.12 |
| 2.3.5 | [-13 17 -6⟩, [-27 -2 13⟩ | [⟨209 331 485]] | 0.5439 | 0.3962 | 6.90 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 71\209 | 407.66 | 15625/12288 | Ditonic |
| 1 | 90\209 | 516.75 | 27/20 | Larry / marvo (209) / zarvo (209d) |
| 19 | 122\209 (1\209) |
700.48 (5.74) |
3/2 (225/224) |
Enneadecal (209d) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct