221edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|221}} | |||
== Theory == | |||
221et tempers out 2109375/2097152 (semicomma) and 2541865828329/2500000000000 in the 5-limit; 1029/1024, 19683/19600, and 235298/234375 in the 7-limit, so that it provides the [[optimal patent val]] for the 7-limit [[Gamelismic clan|hemiseven temperament]]. | |||
Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit. | Using the patent val, it tempers out 540/539, 2835/2816, 4375/4356, and 33614/33275 in the 11-limit; 364/363, 625/624, 1701/1690, and 2200/2197 in the 13-limit. | ||
Using the 221ef val, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting the 17-limit hemiseven and the 11-limit [[Semicomma family|triwell]]. | Using the 221ef val, it tempers out 385/384, 441/440, 24057/24010, and 43923/43750 in the 11-limit; 351/350, 676/675, 1287/1280, 1573/1568, and 14641/14625 in the 13-limit; 273/272, 561/560, 715/714, 833/832, 2187/2176, and 10648/10625 in the 17-limit, supporting the 17-limit hemiseven and the 11-limit [[Semicomma family|triwell]]. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|221}} | |||
=== Subsets and supersets === | |||
221 factors into 13 × 17, with its subset edos [[13edo]] and [[17edo]]. | |||
==Regular temperament properties== | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list|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|-350 221}} | |||
|{{val|221 350}} | |||
| 0.4740 | |||
| 0.4742 | |||
| 8.73 | |||
|- | |||
|2.3.5 | |||
|{{monzo|-21 3 7}}, {{monzo|-11 26 -13}} | |||
|{{val|221 350 513}} | |||
| 0.4299 | |||
| 0.3921 | |||
| 7.22 | |||
|- | |||
|2.3.5.7 | |||
|1029/1024, 19683/19600, 235298/234375 | |||
|{{val|221 350 513 620}} | |||
| 0.5282 | |||
| 0.3799 | |||
| 7.00 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
|1 | |||
|50\221 | |||
|271.49 | |||
|75/64 | |||
|[[Orson]] | |||
|- | |||
|1 | |||
|84\221 | |||
|456.11 | |||
|125/96 | |||
|[[Qak]] | |||
|- | |||
|1 | |||
|89\221 | |||
|483.26 | |||
|320/243 | |||
|[[Hemiseven]] | |||
|- | |||
|1 | |||
|93\221 | |||
|504.98 | |||
|104976/78125 | |||
|[[Countermeantone]] | |||
|- | |||
|1 | |||
|103\221 | |||
|559.28 | |||
|864/625 | |||
|[[Tritriple]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||