309edo
| ← 308edo | 309edo | 310edo → |
309 equal divisions of the octave (abbreviated 309edo or 309ed2), also called 309-tone equal temperament (309tet) or 309 equal temperament (309et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 309 equal parts of about 3.88 ¢ each. Each step represents a frequency ratio of 21/309, or the 309th root of 2.
Theory
309edo is inconsistent to the 5-odd-limit, with four possible mappings in the 7-limit:
- ⟨309 490 717 867] (patent val)
- ⟨309 490 718 867] (309c)
- ⟨309 490 717 868] (309d)
- ⟨309 490 718 868] (309cd)
Using the patent val, it tempers out 5120/5103, 117649/116640, 390625/387072, 537824/531441, 823543/819200 and 1953125/1928934 in the 7-limit. Using the 309c val, it tempers out 4375/4374, 153664/151875, 395136/390625, 537824/531441, 2100875/2097152 and 5250987/5242880 in the 7-limit. Using the 309d val, it tempers out 2109375/2097152 in the 5-limit; 4000/3969, 420175/419904, 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. Using the 309cd val, it tempers out 67108864/66430125 in the 5-limit; 3136/3125, 5120/5103, 250047/250000, 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports misty. Note that the 309b val ⟨309 489 717 867] (309b) is enfactored 103edo.
This system is quite accurate in the 2.81.45.63.11.117.17 (or alternatively 2.81.5/9.7/9.11.13/9.17) subgroup, with the same tuning and commas as 1236edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.96 | -1.85 | -1.84 | +0.14 | -1.69 | -0.10 | +1.52 | +0.85 | -0.45 | +0.60 |
| Relative (%) | +0.0 | +24.7 | -47.6 | -47.3 | +3.6 | -43.6 | -2.6 | +39.0 | +21.9 | -11.6 | +15.3 | |
| Steps (reduced) |
309 (0) |
490 (181) |
717 (99) |
867 (249) |
1069 (142) |
1143 (216) |
1263 (27) |
1313 (77) |
1398 (162) |
1501 (265) |
1531 (295) | |
Subsets and supersets
309 factors into prime as 3 × 103, so 309edo has 3edo and 103edo as its subset edos. 1236edo, which quadruples it, gives a good correction to the harmonics 3, 5, 7, and 13.