220edo: Difference between revisions

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== 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 ===

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=== Subsets and supersets ===

Since 220 factors into ~~22 × 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 ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{~~monzo~~| 349 -220 }}

| {{Monzo| 349 -220 }}

| {{~~mapping~~| 220 349 }}

| {{Mapping| 220 349 }}

| -0.5304

| −0.5304

| 0.5302

| 9.72

|-

| 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 }}

| -0.4912

| −0.4912

| 0.4364

| 8.00

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| 2.3.5.7

| 6144/6125, 10976/10935, 390625/388962

| {{~~mapping~~| 220 349 511 618 }}

| {{Mapping| 220 349 511 618 }}

| -0.5538

| −0.5538

| 0.3932

| 7.21

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=== Rank-2 temperaments ===

{| 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 per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 496.36 (49.09)

| 4/3 (36/35)

| [[Hendecatonic]]

| [[Hendecatonic (temperament)|Hendecatonic]]

|-

| 20

| 91\220 (3\220)

| 496.36 (16.36)

| 4/3 (126/125)

| [[Degrees]]

|}

<nowiki>*~~</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

@@ Line 1: / Line 1: @@
-{{Infobox ET}}
+{{Infobox ET}}{{todo|expand}}
-{{EDO intro|220}}
+{{ED intro}}
 == 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 ===
@@ Line 9: / Line 9: @@
 === 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 ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
-| {{monzo| 349 -220 }}
+| {{Monzo| 349 -220 }}
-| {{mapping| 220 349 }}
+| {{Mapping| 220 349 }}
-| -0.5304
+| −0.5304
 | 0.5302
 | 9.72
 |-
 | 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 }}
-| -0.4912
+| −0.4912
 | 0.4364
 | 8.00
@@ Line 38: / Line 39: @@
 | 2.3.5.7
 | 6144/6125, 10976/10935, 390625/388962
-| {{mapping| 220 349 511 618 }}
+| {{Mapping| 220 349 511 618 }}
-| -0.5538
+| −0.5538
 | 0.3932
 | 7.21
@@ Line 46: / Line 47: @@
 === Rank-2 temperaments ===
 {| 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*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 75: / Line 77: @@
 | 496.36<br>(49.09)
 | 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>*</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