445edo: Difference between revisions

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== Theory ==

445edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[89edo]], but the approximation to some of the higher harmonics are improved. It is [[consistent]] to the [[7-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] all tuned flat, and it allows an extension to the [[11-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, 35937/35840, [[41503/41472]], 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably [[support]]s [[neptune]].

445edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[89edo]], but the approximation to some of the higher harmonics are improved. It is [[consistent]] to the [[7-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] all tuned flat, and it allows an extension to the [[11-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, 35937/35840, [[41503/41472]], 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably [[support]]s [[Neptune (temperament)|neptune]].

=== Odd harmonics ===

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=== Subsets and supersets ===

Since 445 factors into ~~5 × 89~~, 445edo has [[5edo]] and 89edo as its subsets.

Since 445 factors into {{factorisation|445}}, 445edo has [[5edo]] and 89edo 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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=== 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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| 582.47

| 7/5

| [[Neptune]] (7-limit)

| [[Neptune (temperament)|Neptune]] (7-limit)

|-

| 5

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| 498.88 (18.88)

| 4/3 (81/80)

| [[Pental]] (5-limit)

| [[Pental (temperament)|Pental]] (5-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|445}}
+{{ED intro}}
 == Theory ==
-edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[89edo]], but the approximation to some of the higher harmonics are improved. It is [[consistent]] to the [[7-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] all tuned flat, and it allows an extension to the [[11-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, 35937/35840, [[41503/41472]], 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably [[support]]s [[neptune]].
+edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[89edo]], but the approximation to some of the higher harmonics are improved. It is [[consistent]] to the [[7-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] all tuned flat, and it allows an extension to the [[11-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, 35937/35840, [[41503/41472]], 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably [[support]]s [[Neptune (temperament)|neptune]].
 === Odd harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-Since 445 factors into 5 × 89, 445edo has [[5edo]] and 89edo as its subsets.
+Since 445 factors into {{factorisation|445}}, 445edo has [[5edo]] and 89edo 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 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 69: / Line 71: @@
 | 582.47
 | 7/5
-| [[Neptune]] (7-limit)
+| [[Neptune (temperament)|Neptune]] (7-limit)
 |-
 | 5
@@ Line 75: / Line 77: @@
 | 498.88<br>(18.88)
 | 4/3<br>(81/80)
-| [[Pental]] (5-limit)
+| [[Pental (temperament)|Pental]] (5-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