User:Francium/5113edo
| ← 5112edo | 5113edo | 5114edo → |
5113 equal divisions of the octave (abbreviated 5113edo or 5113ed2), also called 5113-tone equal temperament (5113tet) or 5113 equal temperament (5113et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 5113 equal parts of about 0.235 ¢ each. Each step represents a frequency ratio of 21/5113, or the 5113th root of 2.
Theory
5113edo is consistent to the 11-limit, tempering out 21437500/21434787, 47265625/47258883, 184549376/184528125 and 246071287/246037500 in the 11-limit; 123201/123200, 196625/196608, 1664000/1663893, 5175625/5174928 and 1063348/1063125 in the 13-limit; and 12376/12375, 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893 and 7109375/7108992 in the 17-limit. It is strong in the 2.3.5.7.11.17.23.29.31 subgroup, tempering out 21505/21504, 126225/126224, 30625/30624, 150920/150903, 15625/15624, 53361/53360, 750200/750141 and 69632/69629.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.020 | -0.004 | -0.001 | -0.017 | -0.082 | -0.046 | +0.081 | +0.007 | +0.034 | +0.046 |
| Relative (%) | +0.0 | +8.7 | -1.8 | -0.6 | -7.4 | -34.8 | -19.8 | +34.7 | +2.8 | +14.3 | +19.4 | |
| Steps (reduced) |
5113 (0) |
8104 (2991) |
11872 (1646) |
14354 (4128) |
17688 (2349) |
18920 (3581) |
20899 (447) |
21720 (1268) |
23129 (2677) |
24839 (4387) |
25331 (4879) | |
Subsets and supersets
5113edo is the 684th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [8104 -5113⟩ | [⟨5113 8104]] | −0.0064 | 0.0064 | 2.73 |
| 2.3.5 | [56 -91 38⟩, [144 -22 -47⟩ | [⟨5113 8104 11872]] | −0.0037 | 0.0065 | 2.77 |
| 2.3.5.7 | 24414062500/24407490807, 13841287201/13839609375, 281484423828125/281474976710656 | [⟨5113 8104 11872 14354]] | −0.0026 | 0.0059 | 2.51 |
| 2.3.5.7.11 | 21437500/21434787, 47265625/47258883, 184549376/184528125, 246071287/246037500 | [⟨5113 8104 11872 14354 17688]] | −0.0011 | 0.0061 | 2.60 |
| 2.3.5.7.11.13 | 123201/123200, 196625/196608, 1664000/1663893, 5175625/5174928, 1063348/1063125 | [⟨5113 8104 11872 14354 17688 18920]] | +0.0028 | 0.0103 | 4.39 |
| 2.3.5.7.11.13.17 | 12376/12375, 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893, 7109375/7108992 | [⟨5113 8104 11872 14354 17688 18920 20899]] | +0.0040 | 0.0100 | 4.26 |