User:Francium/8951edo
| ← 8950edo | 8951edo | 8952edo → |
(semiconvergent)
8951 equal divisions of the octave (abbreviated 8951edo or 8951ed2), also called 8951-tone equal temperament (8951tet) or 8951 equal temperament (8951et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8951 equal parts of about 0.134 ¢ each. Each step represents a frequency ratio of 21/8951, or the 8951st root of 2.
Theory
8951edo is consistent to the 9-odd-limit, although the error of its harmonic 5 is very high. It has an almost exact harmonic 3 with a relative error of 0.1 percent. 8951edo is strong in the 2.3.17.19.31 subgroup, tempering out 90876411/90870848, 49615462203392/49613455241829, 17594077438737/17592186044416 and 129639437691032/129635157244779.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0001 | +0.0565 | +0.0491 | -0.0499 | +0.0488 | +0.0161 | -0.0267 | -0.0540 | +0.0284 | -0.0015 |
| Relative (%) | +0.0 | +0.1 | +42.2 | +36.6 | -37.2 | +36.4 | +12.0 | -19.9 | -40.3 | +21.2 | -1.1 | |
| Steps (reduced) |
8951 (0) |
14187 (5236) |
20784 (2882) |
25129 (7227) |
30965 (4112) |
33123 (6270) |
36587 (783) |
38023 (2219) |
40490 (4686) |
43484 (7680) |
44345 (8541) | |
Subsets and supersets
8951edo is the 1113th prime edo. 17902edo, which doubles it, gives a good correction to its harmonic 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [14187 -8951⟩ | [⟨8951 14187]] | −0.00003 | 0.00003 | 0.02 |
| 2.3.5 | [183 -51 -44⟩, [-102 142 -53⟩ | [⟨8951 14187 20784]] | −0.0081 | 0.0115 | 8.58 |
| 2.3.5.7 | [-12 29 -11 -3⟩, [47 -18 -14 5⟩, [42 3 -2 -15⟩ | [⟨8951 14187 20784 25129]] | −0.0105 | 0.0107 | 7.98 |