10009edo
| ← 10008edo | 10009edo | 10010edo → |
10009 equal divisions of the octave (abbreviated 10009edo or 10009ed2), also called 10009-tone equal temperament (10009tet) or 10009 equal temperament (10009et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 10009 equal parts of about 0.12 ¢ each. Each step represents a frequency ratio of 21/10009, or the 10009th root of 2.
Theory
10009edo is consistent to the 9-odd-limit. It can be used in the 2.3.5.7.13.19.29.31.41.47 subgroup, tempering out 60025/60021, 138240/138229, 140625/140608, 482125/482112, 4751360/4750893, 739375/739328, 5137600/5137263, 19552/19551 and 103936/103935. Using the 2.3.7.13.23.31 subgroup, it tempers out 8464/8463.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0132 | -0.0214 | +0.0221 | -0.0541 | +0.0358 | -0.0498 | +0.0592 | -0.0398 | +0.0561 | +0.0538 |
| Relative (%) | +0.0 | +11.0 | -17.8 | +18.5 | -45.1 | +29.9 | -41.6 | +49.4 | -33.2 | +46.8 | +44.9 | |
| Steps (reduced) |
10009 (0) |
15864 (5855) |
23240 (3222) |
28099 (8081) |
34625 (4598) |
37038 (7011) |
40911 (875) |
42518 (2482) |
45276 (5240) |
48624 (8588) |
49587 (9551) | |
Subsets and supersets
10009edo is the 1231st prime edo. 20018edo, which doubles it, gives a good correction to the harmonic 11.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [15864 -10009⟩ | [⟨10009 15864]] | -0.0042 | 0.0042 | 3.50 |
| 2.3.5 | [56 -91 38⟩, [-304 79 77⟩ | [⟨10009 15864 23240]] | +0.0003 | 0.0072 | 6.01 |
| 2.3.5.7 | [-2 -3 15 -10⟩, [-48 0 11 8⟩, [5 -44 17 9⟩ | [⟨10009 15864 23240 28099]] | -0.0018 | 0.0071 | 5.92 |