User:Francium/3271edo
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Prime factorization
3271 (prime)
Step size
0.36686 ¢
Fifth
1913\3271 (701.804 ¢)
Semitones (A1:m2)
307:248 (112.6 ¢ : 90.98 ¢)
Dual sharp fifth
1914\3271 (702.171 ¢)
Dual flat fifth
1913\3271 (701.804 ¢)
Dual major 2nd
556\3271 (203.974 ¢)
Consistency limit
5
Distinct consistency limit
5
| ← 3270edo | 3271edo | 3272edo → |
3271 equal divisions of the octave (abbreviated 3271edo or 3271ed2), also called 3271-tone equal temperament (3271tet) or 3271 equal temperament (3271et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3271 equal parts of about 0.367 ¢ each. Each step represents a frequency ratio of 21/3271, or the 3271st root of 2.
Theory
3271edo is consistent to the 5-limit and its harmonic 3 is about halfway its steps. It is strong in the 2.9.5.7.11.13.17.19.31 subgroup, tempering out 12376/12375, 14400/14399, 6175/6174, 123201/123200, 228096/228095, 1549184/1549125, 27625/27621 and 110825/110808.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.151 | -0.010 | +0.052 | +0.064 | +0.073 | -0.051 | -0.161 | -0.033 | +0.011 | -0.099 | +0.157 |
| Relative (%) | -41.2 | -2.7 | +14.2 | +17.5 | +19.9 | -13.8 | -43.9 | -9.1 | +2.9 | -27.0 | +42.9 | |
| Steps (reduced) |
5184 (1913) |
7595 (1053) |
9183 (2641) |
10369 (556) |
11316 (1503) |
12104 (2291) |
12779 (2966) |
13370 (286) |
13895 (811) |
14367 (1283) |
14797 (1713) | |
Subsets and supersets
3271edo is the 462nd prime edo. 6542edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [10369 -3271⟩ | [⟨3271 10369]] | −0.0101 | 0.0101 | 2.75 |
| 2.9.5 | [144 -11 -47⟩, [95 -60 41⟩ | [⟨3271 10369 7595]] | −0.0054 | 0.0107 | 2.92 |
| 2.9.5.7 | 282475249000/282429536481, 70368744177664/70338939985125, 19073486328125/19062721117944 | [⟨3271 10369 7595 9183]] | −0.0087 | 0.0109 | 2.97 |
| 2.9.5.7.11 | 56953125/56942116, 184549376/184528125, 3294225/3294172, 29541015625/29530050606 | [⟨3271 10369 7595 9183 11316]] | −0.0111 | 0.0109 | 2.97 |
| 2.9.5.7.11.13 | 123201/123200, 6656/6655, 8859375/8859136, 823680/823543, 43061200/43046721 | [⟨3271 10369 7595 9183 11316 12104]] | −0.0070 | 0.0136 | 3.71 |
| 2.9.5.7.11.13.17 | 12376/12375, 14400/14399, 123201/123200, 4685824/4685625, 361250/361179, 81331250/81310473 | [⟨3271 10369 7595 9183 11316 12104 13370]] | −0.0048 | 0.0137 | 3.73 |
| 2.9.5.7.11.13.17.19 | 12376/12375, 14400/14399, 6175/6174, 123201/123200, 228096/228095, 1549184/1549125, 8978125/8975448 | [⟨3271 10369 7595 9183 11316 12104 13370 13895]] | −0.0045 | 0.0128 | 3.49 |