User:Francium/1663edo
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Prime factorization
1663 (prime)
Step size
0.721587 ¢
Fifth
973\1663 (702.105 ¢)
Semitones (A1:m2)
159:124 (114.7 ¢ : 89.48 ¢)
Consistency limit
3
Distinct consistency limit
3
| ← 1662edo | 1663edo | 1664edo → |
1663 equal divisions of the octave (abbreviated 1663edo or 1663ed2), also called 1663-tone equal temperament (1663tet) or 1663 equal temperament (1663et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1663 equal parts of about 0.722 ¢ each. Each step represents a frequency ratio of 21/1663, or the 1663rd root of 2.
Theory
1663edo is only consistent to the 3-limit and the errors of its lower harmonics are quite large, except for its harmonic 11/1 with an relative error of only 3.5 percent.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.150 | -0.264 | +0.266 | +0.299 | -0.025 | +0.122 | -0.115 | -0.325 | -0.219 | -0.306 | +0.228 |
| Relative (%) | +20.7 | -36.6 | +36.9 | +41.5 | -3.5 | +16.9 | -15.9 | -45.1 | -30.3 | -42.4 | +31.6 | |
| Steps (reduced) |
2636 (973) |
3861 (535) |
4669 (1343) |
5272 (283) |
5753 (764) |
6154 (1165) |
6497 (1508) |
6797 (145) |
7064 (412) |
7304 (652) |
7523 (871) | |
Subsets and supersets
1663edo is the 261st prime edo. 3326edo, which doubles it, gives a good correction to its harmonic 5.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2636 -1663⟩ | [⟨1663 2636]] | −0.0472 | 0.0472 | 6.54 |