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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|221}}
{{ED intro}}
 
== Theory ==
== 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]].  
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 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, 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 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 ===
=== Odd harmonics ===
{{Harmonics in equal|221}}
{{Harmonics in equal|221}}
=== Subsets and supersets ===
=== Subsets and supersets ===
221 factors into 13 × 17, with its subset edos [[13edo]] and [[17edo]].
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" |[[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
! rowspan="2" | [[Subgroup]]
|{{monzo|-350 221}}
! rowspan="2" | [[Comma list]]
|{{val|221 350}}
! rowspan="2" | [[Mapping]]
| 0.4740
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -350 221 }}
| {{mapping| 221 350 }}
| +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 }}
|{{val|221 350 513}}
| {{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
|{{val|221 350 513 620}}
| {{mapping| 221 350 513 620 }}
| 0.5282
| +0.5282
| 0.3799
| 0.3799
| 7.00
| 7.00
|-
| 2.3.5.7.11
| 385/384, 441/440, 19683/19600, 235298/234375
| {{mapping| 221 350 513 620 764 }} (221e)
| +0.5904
| 0.3618
| 6.66
|}
|}
=== 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
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|50\221
| 50\221
|271.49
| 271.49
|75/64
| 75/64
|[[Orson]]
| [[Orson]]
|-
| 1
| 57\221
| 309.50
| 448/375
| [[Triwell]] (221e)
|-
|-
|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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
Retrieved from "https://en.xen.wiki/w/221edo"