634edo
| ← 633edo | 634edo | 635edo → |
634 equal divisions of the octave (abbreviated 634edo or 634ed2), also called 634-tone equal temperament (634tet) or 634 equal temperament (634et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 634 equal parts of about 1.89 ¢ each. Each step represents a frequency ratio of 21/634, or the 634th root of 2.
Theory
634edo is a good 13-limit and no-17 higher-limit system. The equal temperament tempers out [-53 10 16⟩ (kwazy comma) and [33 -34 9⟩ (countritonic comma) in the 5-limit; 420175/419904 (wizma), 703125/702464 (meter), and 33554432/33480783 (garischisma) in the 7-limit; 9801/9800, 19712/19683, 41503/41472 in the 11-limit; 1716/1715, 2080/2079, 4096/4095, 4225/4224, 14641/14625, and 31250/31213 in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.253 | -0.194 | +0.259 | -0.529 | -0.149 | -0.854 | -0.352 | +0.117 | +0.076 | +0.075 |
| Relative (%) | +0.0 | +13.4 | -10.2 | +13.7 | -28.0 | -7.9 | -45.1 | -18.6 | +6.2 | +4.0 | +4.0 | |
| Steps (reduced) |
634 (0) |
1005 (371) |
1472 (204) |
1780 (512) |
2193 (291) |
2346 (444) |
2591 (55) |
2693 (157) |
2868 (332) |
3080 (544) |
3141 (605) | |
Subsets and supersets
Since 634 factors into 2 × 317, 634edo has 2edo and 317edo as its subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1005 -634⟩ | [⟨634 1005]] | −0.0799 | 0.0799 | 4.22 |
| 2.3.5 | [-53 10 16⟩, [33 -34 9⟩ | [⟨634 1005 1472]] | −0.0254 | 0.1009 | 5.33 |
| 2.3.5.7 | 420175/419904, 703125/702464, 33554432/33480783 | [⟨634 1005 1472 1780]] | −0.0422 | 0.0921 | 4.86 |
| 2.3.5.7.11 | 9801/9800, 19712/19683, 41503/41472, 703125/702464 | [⟨634 1005 1472 1780 2193]] | −0.0031 | 0.1135 | 6.00 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 4096/4095, 14641/14625, 31250/31213 | [⟨634 1005 1472 1780 2193 2346]] | +0.0041 | 0.1048 | 5.54 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 241\634 | 456.15 | 125/96 | Qak |
| 1 | 263\634 | 497.79 | 4/3 | Gary |
| 1 | 311\634 | 588.64 | [-14 15 -4⟩ | Countritonic (5-limit) |
| 2 | 86\634 | 162.78 | 1125/1024 | Kwazy |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct