220edo: Difference between revisions

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-{{Infobox ET}}
+{{Infobox ET}}{{todo|expand}}
-'''220edo''' is the [[EDO|equal division of the octave]] into 220 parts of 5.4545 [[cent]]s each. It is consistent to the 7-odd-limit.
+{{ED intro}}
-Using the patent val, it tempers out 131072000/129140163 (rodan comma) and 6115295232/6103515625 (vishnuzma) in the 5-limit; 6144/6125, 10976/10935, and 390625/388962 in the 7-limit; 1331/1323, 1375/1372, 2200/2187, and 16384/16335 in the 11-limit; 325/324, 352/351, 1001/1000, 1573/1568, and 2704/2695 in the 13-limit.
+== Theory ==
+Using the patent val, it tempers out {{monzo| 20 -17 3 }} ([[rodan comma]]) and {{monzo| 23 6 -14 }} ([[vishnuzma]]) in the 5-limit; [[6144/6125]], [[10976/10935]], and [[390625/388962]] in the 7-limit; 1331/1323, 1375/1372, [[2200/2187]], and [[16384/16335]] in the 11-limit; [[325/324]], [[352/351]], [[1001/1000]], [[1573/1568]], and 2704/2695 in the 13-limit.
+=== Odd harmonics ===
 {{Harmonics in equal|220}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+=== Subsets and supersets ===
+Since 220 factors into {{factorisation|220}}, 220edo has subset edos {{EDOs| 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 }}.
+== 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| 349 -220 }}
+| {{Mapping| 220 349 }}
+| −0.5304
+| 0.5302
+| 9.72
+|-
+| 2.3.5
+| {{Monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }}
+| {{Mapping| 220 349 511 }}
+| −0.4912
+| 0.4364
+| 8.00
+|-
+| 2.3.5.7
+| 6144/6125, 10976/10935, 390625/388962
+| {{Mapping| 220 349 511 618 }}
+| −0.5538
+| 0.3932
+| 7.21
+|}
+=== 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
+| 43\220
+| 234.55
+| 729/640
+| [[Rodan]] (5-limit)
+|-
+| 1
+| 83\220
+| 452.73
+| 125/81
+| [[Maja]] (5-limit)
+|-
+| 2
+| 13\220
+| 70.91
+| 25/24
+| [[Vishnu]] (5-limit)
+|-
+| 11
+| 91\220<br>(9\220)
+| 496.36<br>(49.09)
+| 4/3<br>(36/35)
+| [[Hendecatonic (temperament)|Hendecatonic]]
+|-
+| 20
+| 91\220<br>(3\220)
+| 496.36<br>(16.36)
+| 4/3<br>(126/125)
+| [[Degrees]]
+|}
+<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