221edo: Difference between revisions
No edit summary |
Rework |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|221}} | {{EDO intro|221}} | ||
== Theory == | == Theory == | ||
221edo has a flat tendency, with [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] all tuned flat. The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| -11 26 -13 }} 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 [[hemiseven]] temperament. | |||
Using the 221ef val, which does the best into the 17-limit, 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 17-limit hemiseven and 11-limit [[triwell]]. | |||
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. | ||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|221}} | {{Harmonics in equal|221}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
221 factors into 13 × 17, | Since 221 factors into 13 × 17, 221edo has [[13edo]] and [[17edo]] as its subsets. | ||
==Regular temperament properties== | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-350 221}} | | {{monzo| -350 221 }} | ||
|{{ | | {{mapping| 221 350 }} | ||
| 0.4740 | | 0.4740 | ||
| 0.4742 | | 0.4742 | ||
| 8.73 | | 8.73 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|-21 3 7}}, {{monzo|-11 26 -13}} | | {{monzo| -21 3 7 }}, {{monzo| -11 26 -13 }} | ||
|{{ | | {{mapping| 221 350 513 }} | ||
| 0.4299 | | 0.4299 | ||
| 0.3921 | | 0.3921 | ||
| 7.22 | | 7.22 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|1029/1024, 19683/19600, 235298/234375 | | 1029/1024, 19683/19600, 235298/234375 | ||
|{{ | | {{mapping| 221 350 513 620 }} | ||
| 0.5282 | | 0.5282 | ||
| 0.3799 | | 0.3799 | ||
| 7.00 | | 7.00 | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|50\221 | | 50\221 | ||
|271.49 | | 271.49 | ||
|75/64 | | 75/64 | ||
|[[Orson]] | | [[Orson]] | ||
|- | |- | ||
|1 | | 1 | ||
|84\221 | | 84\221 | ||
|456.11 | | 456.11 | ||
|125/96 | | 125/96 | ||
|[[Qak]] | | [[Qak]] | ||
|- | |- | ||
|1 | | 1 | ||
|89\221 | | 89\221 | ||
|483.26 | | 483.26 | ||
|320/243 | | 320/243 | ||
|[[Hemiseven]] | | [[Hemiseven]] (221ef) | ||
|- | |- | ||
|1 | | 1 | ||
|93\221 | | 93\221 | ||
|504.98 | | 504.98 | ||
|104976/78125 | | 104976/78125 | ||
|[[Countermeantone]] | | [[Countermeantone]] | ||
|- | |- | ||
|1 | | 1 | ||
|103\221 | | 103\221 | ||
|559.28 | | 559.28 | ||
|864/625 | | 864/625 | ||
|[[Tritriple]] | | [[Tritriple]] (221e) | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[ | |||