638edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro | {{EDO intro}} | ||
== Theory == | == Theory == | ||
The equal temperament [[tempering out|tempers out]] the minortone comma, {{monzo| -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. | The equal temperament [[tempering out|tempers out]] the minortone comma, {{monzo| -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 {{nowrap|224 & 414}} temperament. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 638 factors as | Since 638 factors as {{factorisation|638}}, 638edo has subset edos {{EDOs| 2, 11, 22, 29, 58, and 319 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| [[Quatracot]] | | [[Quatracot]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Quatracot]] | [[Category:Quatracot]] | ||
Revision as of 18:51, 15 January 2025
| ← 637edo | 638edo | 639edo → |
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 (%) | -20.6 | -39.0 | -9.2 | -41.2 | -11.7 | +11.9 | +40.4 | +19.9 | -17.8 | -29.9 | -3.3 | |
| 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 distinct