412edo: Difference between revisions

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== Theory ==

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=== Subsets and supersets ===

412 factors into ~~22 × 103~~, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.

412 factors into {{factorisation|412}}, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.

== 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]] (%)

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| {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }}

| {{mapping| 412 653 957 }}

| -0.1501

| −0.1501

| 0.2182

| 7.49

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| 6144/6125, 2460375/2458624, 49009212/48828125

| {{mapping| 412 653 957 1157 }}

| -0.2085

| −0.2085

| 0.2143

| 7.36

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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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| [[Septisuperfourth]] (7-limit)

|}

<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}}
-{{EDO intro|412}}
+{{ED intro}}
 == Theory ==
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-factors into 2<sup>2</sup> × 103, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.
+factors into {{factorisation|412}}, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11.
 == 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]] (%)
@@ Line 32: / Line 33: @@
 | {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }}
 | {{mapping| 412 653 957 }}
-| -0.1501
+| −0.1501
 | 0.2182
 | 7.49
@@ Line 39: / Line 40: @@
 | 6144/6125, 2460375/2458624, 49009212/48828125
 | {{mapping| 412 653 957 1157 }}
-| -0.2085
+| −0.2085
 | 0.2143
 | 7.36
@@ 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
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 77: / Line 79: @@
 | [[Septisuperfourth]] (7-limit)
 |}
-<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