220edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}}{{todo|expand}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
Using the patent val, it tempers out | 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 === | ||
{{Harmonics in equal|220}} | {{Harmonics in equal|220}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
220 factors into | 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 == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! 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 | | 0.5302 | ||
| 9.72 | | 9.72 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{ | | {{Monzo| 20 -17 3 }}, {{monzo| 23 6 -14 }} | ||
|{{ | | {{Mapping| 220 349 511 }} | ||
| | | −0.4912 | ||
| 0.4364 | | 0.4364 | ||
| 8.00 | | 8.00 | ||
|- | |- | ||
|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 }} | ||
| | | −0.5538 | ||
| 0.3932 | | 0.3932 | ||
| 7.21 | | 7.21 | ||
|} | |} | ||
=== 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 | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>ratio | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|43\220 | | 43\220 | ||
|234.55 | | 234.55 | ||
| | | 729/640 | ||
|[[Rodan]] | | [[Rodan]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|83\220 | | 83\220 | ||
|452.73 | | 452.73 | ||
|125/81 | | 125/81 | ||
|[[Maja]] | | [[Maja]] (5-limit) | ||
|- | |- | ||
|2 | | 2 | ||
|13\220 | | 13\220 | ||
|70.91 | | 70.91 | ||
|25/24 | | 25/24 | ||
|[[Vishnu]] | | [[Vishnu]] (5-limit) | ||
|- | |- | ||
|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 | |||
| 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 | |||
[[ | |||