451edo: Difference between revisions

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== Theory ==

~~451 = 11 × 41, and~~ 451edo shares its [[3/2|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]].

451edo shares its [[3/2|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]].

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 451 factors into ~~11 × 41~~, 451edo has [[11edo]] and [[41edo]] as its subsets.

Since 451 factors into {{factorisation|451}}, 451edo has [[11edo]] and [[41edo]] as its subsets.

== 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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| 390625000/387420489, {{monzo| -59 5 22 }}

| {{mapping| 451 715 1047 }}

| -0.0294

| −0.0294

| 0.2144

| 8.06

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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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| [[Tertiaseptal]]

|}

<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

[[Category:Quartonic]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|451}}
+{{ED intro}}
 == Theory ==
-= 11 × 41, and 451edo shares its [[3/2|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]].
+edo shares its [[3/2|fifth]] with [[41edo]]. Unlike 41, however, 451 is only [[consistent]] to the [[7-odd-limit]], though it has a reasonable approximation up to the [[13-limit]] using the [[patent val]]. The equal temperament [[tempering out|tempers out]] 390625000/387420489 ([[quartonic comma]]) in the 5-limit; [[2401/2400]], [[65625/65536]], [[703125/702464]], [[2100875/2097152]], in the 7-limit; [[6250/6237]], 42592/42525, 42875/42768, 43923/43904 in the 11-limit; and [[625/624]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], 4459/4455, and 17303/17280 in the 13-limit. It [[support]]s [[tertiaseptal]], [[tertiseptisix]], and [[hemermacomp]], providing the [[optimal patent val]] for 5-limit [[quartonic]].
 === Prime harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-Since 451 factors into 11 × 41, 451edo has [[11edo]] and [[41edo]] as its subsets.
+Since 451 factors into {{factorisation|451}}, 451edo has [[11edo]] and [[41edo]] as its subsets.
 == 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 25: / Line 26: @@
 | 390625000/387420489, {{monzo| -59 5 22 }}
 | {{mapping| 451 715 1047 }}
-| -0.0294
+| &minus;0.0294
 | 0.2144
 | 8.06
@@ Line 53: / Line 54: @@
 === 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 72: / Line 74: @@
 | [[Tertiaseptal]]
 |}
-<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
 [[Category:Quartonic]]