638edo
| ← 637edo | 638edo | 639edo → |
638 equal divisions of the octave (abbreviated 638edo or 638ed2), also called 638-tone equal temperament (638tet) or 638 equal temperament (638et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 638 equal parts of about 1.88 ¢ each. Each step represents a frequency ratio of 21/638, or the 638th root of 2.
Theory
The equal temperament tempers out the minortone comma, [-16 35 -17⟩, in the 5-limit, 4375/4374 in the 7-limit, 3025/3024, 9801/9800, and 43923/43904, in the 11-limit; and 625/624, 729/728, 1575/1573, 2200/2197 and 4225/4224 in the 13-limit. It supplies the optimal patent val for quatracot, the 224 & 414 temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -0.388 | -0.734 | -0.174 | -0.775 | -0.221 | +0.225 | +0.759 | +0.374 | -0.334 | -0.561 | -0.061 |
| relative (%) | -21 | -39 | -9 | -41 | -12 | +12 | +40 | +20 | -18 | -30 | -3 | |
| Steps (reduced) |
1011 (373) |
1481 (205) |
1791 (515) |
2022 (108) |
2207 (293) |
2361 (447) |
2493 (579) |
2608 (56) |
2710 (158) |
2802 (250) |
2886 (334) | |
Subsets and supersets
Since 638 factors as 2 × 11 × 29, 638edo has subset edos 2, 11, 22, 29, 58, and 319.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1011 638⟩ | [⟨638 1011]] | +0.1223 | 0.1223 | 6.50 |
| 2.3.5 | [-51 19 9⟩, [-16 35 -17⟩ | [⟨638 1011 1481]] | +0.1869 | 0.1353 | 7.19 |
| 2.3.5.7 | 4375/4374, 2100875/2097152, [-11 5 11 -8⟩ | [⟨638 1011 1481 1791]] | +0.1556 | 0.1291 | 6.86 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 825000/823543, 1265625/1261568 | [⟨638 1011 1481 1791 2207]] | +0.1373 | 0.1212 | 6.44 |
| 2.3.5.7.11.13 | 625/624, 729/728, 1575/1573, 2200/2197, 823680/823543 | [⟨638 1011 1481 1791 2207 2361]] | +0.1043 | 0.1330 | 7.07 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 97\638 | 182.45 | 10/9 | Mitonic |
| 1 | 313\638 | 588.71 | 7/5 | Untriton |
| 2 | 94\638 | 176.80 | 448/405 | Quatracot |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct