1861edo
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Prime factorization
1861 (prime)
Step size
0.644815¢
Fifth
1089\1861 (702.203¢)
Semitones (A1:m2)
179:138 (115.4¢ : 88.98¢)
Dual sharp fifth
1089\1861 (702.203¢)
Dual flat fifth
1088\1861 (701.558¢)
Dual major 2nd
316\1861 (203.761¢)
Consistency limit
5
Distinct consistency limit
5
| ← 1860edo | 1861edo | 1862edo → |
1861 equal divisions of the octave (abbreviated 1861edo or 1861ed2), also called 1861-tone equal temperament (1861tet) or 1861 equal temperament (1861et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1861 equal parts of about 0.645 ¢ each. Each step represents a frequency ratio of 21/1861, or the 1861st root of 2.
Theory
1861edo is only consistent to the 5-odd-limit, and the harmonic 3 is about halfway between its steps. It has a reasonable approximation of the 2.9.5.21.11 subgroup.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.248 | -0.070 | -0.314 | -0.149 | -0.001 | +0.311 | +0.178 | +0.149 | -0.253 | -0.066 | -0.225 |
| relative (%) | +38 | -11 | -49 | -23 | -0 | +48 | +28 | +23 | -39 | -10 | -35 | |
| Steps (reduced) |
2950 (1089) |
4321 (599) |
5224 (1502) |
5899 (316) |
6438 (855) |
6887 (1304) |
7271 (1688) |
7607 (163) |
7905 (461) |
8174 (730) |
8418 (974) | |
Subsets and supersets
1861edo is the 284th prime edo. 3722edo, which doubles it, provides a good correction to harmonics 3, 7, and 13.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-5899 1861⟩ | [⟨1861 5899]] | +0.0234 | 0.0234 | 3.63 |