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220edo: Difference between revisions

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{{Infobox ET}}{{todo|expand}#
{{Infobox ET}}{{todo|expand}}
{{EDO intro|220}}
{{ED intro}}


== Theory ==
== 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.  
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 ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 220 factors into 2<sup>2</sup> × 5 × 11, 220edo has subset edos {{EDOs| 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 }}
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 ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 349 -220 }}
| {{Monzo| 349 -220 }}
| {{mapping| 220 349 }}
| {{Mapping| 220 349 }}
| &minus;0.5304
| −0.5304
| 0.5302
| 0.5302
| 9.72
| 9.72
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }}
| {{Monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }}
| {{mapping| 220 349 511 }}
| {{Mapping| 220 349 511 }}
| &minus;0.4912
| −0.4912
| 0.4364
| 0.4364
| 8.00
| 8.00
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| 2.3.5.7
| 2.3.5.7
| 6144/6125, 10976/10935, 390625/388962
| 6144/6125, 10976/10935, 390625/388962
| {{mapping| 220 349 511 618 }}
| {{Mapping| 220 349 511 618 }}
| &minus;0.5538
| −0.5538
| 0.3932
| 0.3932
| 7.21
| 7.21
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 11
| 11
| 91\220<br />(9\220)
| 91\220<br>(9\220)
| 496.36<br />(49.09)
| 496.36<br>(49.09)
| 4/3<br />(36/35)
| 4/3<br>(36/35)
| [[Hendecatonic]]
| [[Hendecatonic (temperament)|Hendecatonic]]
|-
|-
| 20
| 20
| 91\220<br />(3\220)
| 91\220<br>(3\220)
| 496.36<br />(16.36)
| 496.36<br>(16.36)
| 4/3<br />(126/125)
| 4/3<br>(126/125)
| [[Degrees]]
| [[Degrees]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<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
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