221edo: Difference between revisions

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 {{Infobox ET}}
-'''221edo''' is the [[EDO|equal division of the octave]] into 221 parts of 5.4299 [[cent]]s each.
+{{ED intro}}
-It 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]].
+== Theory ==
+edo 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 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 [[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 ===
 {{Harmonics in equal|221}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+=== Subsets and supersets ===
+Since 221 factors into 13 × 17, 221edo has [[13edo]] and [[17edo]] as its subsets.
+== 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| -350 221 }}
+| {{mapping| 221 350 }}
+| +0.4740
+| 0.4742
+| 8.73
+|-
+| 2.3.5
+| {{monzo| -21 3 7 }}, {{monzo| -11 26 -13 }}
+| {{mapping| 221 350 513 }}
+| +0.4299
+| 0.3921
+| 7.22
+|-
+| 2.3.5.7
+| 1029/1024, 19683/19600, 235298/234375
+| {{mapping| 221 350 513 620 }}
+| +0.5282
+| 0.3799
+| 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 ===
+{| 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
+| 50\221
+| 271.49
+| 75/64
+| [[Orson]]
+|-
+| 1
+| 57\221
+| 309.50
+| 448/375
+| [[Triwell]] (221e)
+|-
+| 1
+| 84\221
+| 456.11
+| 125/96
+| [[Qak]]
+|-
+| 1
+| 89\221
+| 483.26
+| 320/243
+| [[Hemiseven]] (221ef)
+|-
+| 1
+| 93\221
+| 504.98
+| 104976/78125
+| [[Countermeantone]]
+|-
+| 1
+| 103\221
+| 559.28
+| 864/625
+| [[Tritriple]] (221e)
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct