638edo: Difference between revisions
m +category |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" Tags: Mobile edit Mobile web edit |
||
| (7 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
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. | == 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 {{nowrap|224 & 414}} temperament. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 8: | Line 9: | ||
=== 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 == | ||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -1011 638 }} | |||
| {{mapping| 638 1011 }} | |||
| +0.1223 | |||
| 0.1223 | |||
| 6.50 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| -51 19 9 }}, {{monzo| -16 35 -17 }} | |||
| {{mapping| 638 1011 1481 }} | |||
| +0.1869 | |||
| 0.1353 | |||
| 7.19 | |||
|- | |||
| 2.3.5.7 | |||
| 4375/4374, 2100875/2097152, {{monzo| -11 5 11 -8 }} | |||
| {{mapping| 638 1011 1481 1791 }} | |||
| +0.1556 | |||
| 0.1291 | |||
| 6.86 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 4375/4374, 825000/823543, 1265625/1261568 | |||
| {{mapping| 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 | |||
| {{mapping| 638 1011 1481 1791 2207 2361 }} | |||
| +0.1043 | |||
| 0.1330 | |||
| 7.07 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />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]] | |||
|} | |||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
[[Category:Quatracot]] | |||