4ed5/4: Difference between revisions

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'''4ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into four parts of 96.5784 [[cent|cents]] each, corresponding to 12.4251 [[~~edo~~]]. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit, which is supported by [[87edo]], [[323edo]], [[410edo]], and [[733edo]] among others.

'''4ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into four parts of 96.5784 [[cent|cents]] each, corresponding to 12.4251 [[EDO]]. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit (''hemiluna'' temperament), which is supported by [[87edo|87EDO]], [[323edo|323EDO]], [[410edo|410EDO]], and [[733edo|733EDO]] among others.

==Intervals==

== Intervals ==

{| class="wikitable"

|-

Line 221:

Line 222:

|}

==~~Related regular temperament~~==

== Harmonics ==

===~~7-limit 87&323~~===

{{Harmonics in equal

~~Commas: 48828125/48771072, 67108864/66976875~~

| steps = 4

| num = 5

| denom = 4

}}

{{Harmonics in equal

| steps = 4

| num = 5

| denom = 4

| start = 12

| collapsed = 1

}}

~~POTE generator: ~200~~/~~189 = 96~~.~~591~~

== Related regular temperament ==

4ED5/4 tuning is closely related to the ''[[Luna family #Hemiluna|Hemiluna]]'' temperament (87&323), which tempers out 48828125/48771072 (''neptunisma'', laruruleyo) and 67108864/66976875 (''decovulture'' comma, sasabirugugu) in the 7-limit. It splits a hemithird interval into two equal parts.

~~Map: [<1 4 2 -5|, <0 -30 4 97|]~~

~~EDOs~~: 87, 323, 410, 733

; [[Luna family #Hemiluna|Hemiluna]] (87 & 323)

'''7-limit'''

Badness: 0.~~1793~~

Comma list: 48828125/48771072, 67108864/66976875

Mapping: [{{val|1 4 2 -5}}, {{val|0 -30 4 97}}]

~~===~~11-limit ~~87&323===~~

POTE generator: ~200/189 = 96.591

~~Commas~~: 5632/5625, 14641/14580, 131072/130977

{{Optimal ET sequence|legend=1|87, 236, 323, 410, 733}}

Badness: 0.179304

~~POTE generator: ~200/189 = 96.587~~

'''11-limit'''

Comma list: 5632/5625, 14641/14580, 131072/130977

~~Map~~: [~~<~~1 4 2 -5 7|~~, <~~0 -30 4 97 -44|]

Mapping: [{{val|1 4 2 -5 7}}, {{val|0 -30 4 97 -44}}]

POTE generator: ~200/189 = 96.587

~~EDOs:~~ 87, 323, 410~~, 733e~~

{{Optimal ET sequence|legend=1|87, 236, 323, 410}}

Badness: 0.100077

Badness: 0.~~1001~~

'''13-limit'''

Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580

~~===~~13-limit ~~87&323===~~

Mapping: [{{val|1 4 2 -5 7 7}}, {{val|0 -30 4 97 -44 -41}}]

~~Commas~~: 676/675, 1001/1000, 4096/4095, 14641/14580

POTE generator: ~143/135 = 96.586

{{Optimal ET sequence|legend=1|87, 236, 323, 410}}

~~POTE generator: ~143/135 = 96.586~~

Badness: 0.046151

~~Map~~: [~~<~~1 4 2 -5 7 7|~~, <~~0 -30 4 97 -44 -41|]

~~EDOs:~~ 87, 323, 410~~, 733e~~

Badness: 0.~~0462~~

[[Category:Major third]]

[[Category:Equal-step tuning]]

~~[[Category:Edonoi]]~~

@@ Line 1: / Line 1: @@
-'''4ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into four parts of 96.5784 [[cent|cents]] each, corresponding to 12.4251 [[edo]]. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit, which is supported by [[87edo]], [[323edo]], [[410edo]], and [[733edo]] among others.
+{{Infobox ET}}
+'''4ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into four parts of 96.5784 [[cent|cents]] each, corresponding to 12.4251 [[EDO]]. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit (''hemiluna'' temperament), which is supported by [[87edo|87EDO]], [[323edo|323EDO]], [[410edo|410EDO]], and [[733edo|733EDO]] among others.
-==Intervals==
+== Intervals ==
 {| class="wikitable"
 |-
@@ Line 221: / Line 222: @@
 |}
-==Related regular temperament==
+== Harmonics ==
-===7-limit 87&amp;323===
+{{Harmonics in equal
-Commas: 48828125/48771072, 67108864/66976875
+| steps = 4
+| num = 5
+| denom = 4
+}}
+{{Harmonics in equal
+| steps = 4
+| num = 5
+| denom = 4
+| start = 12
+| collapsed = 1
+}}
-POTE generator: ~200/189 = 96.591
+== Related regular temperament ==
+ED5/4 tuning is closely related to the ''[[Luna family #Hemiluna|Hemiluna]]'' temperament (87&amp;323), which tempers out 48828125/48771072 (''neptunisma'', laruruleyo) and 67108864/66976875 (''decovulture'' comma, sasabirugugu) in the 7-limit. It splits a hemithird interval into two equal parts.
-Map: [&lt;1 4 2 -5|, &lt;0 -30 4 97|]
-EDOs: 87, 323, 410, 733
+; <font style="font-size:1.15em">[[Luna family #Hemiluna|Hemiluna]] (87 &amp; 323)</font>
+'''7-limit'''<br>
-Badness: 0.1793
+Comma list: 48828125/48771072, 67108864/66976875<br>
+Mapping: [{{val|1 4 2 -5}}, {{val|0 -30 4 97}}]<br>
-===11-limit 87&amp;323===
+POTE generator: ~200/189 = 96.591<br>
-Commas: 5632/5625, 14641/14580, 131072/130977
+{{Optimal ET sequence|legend=1|87, 236, 323, 410, 733}}<br>
+Badness: 0.179304<br><br>
-POTE generator: ~200/189 = 96.587
+'''11-limit'''<br>
+Comma list: 5632/5625, 14641/14580, 131072/130977<br>
-Map: [&lt;1 4 2 -5 7|, &lt;0 -30 4 97 -44|]
+Mapping: [{{val|1 4 2 -5 7}}, {{val|0 -30 4 97 -44}}]<br>
+POTE generator: ~200/189 = 96.587<br>
-EDOs: 87, 323, 410, 733e
+{{Optimal ET sequence|legend=1|87, 236, 323, 410}}<br>
+Badness: 0.100077<br><br>
-Badness: 0.1001
+'''13-limit'''<br>
+Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580<br>
-===13-limit 87&amp;323===
+Mapping: [{{val|1 4 2 -5 7 7}}, {{val|0 -30 4 97 -44 -41}}]<br>
-Commas: 676/675, 1001/1000, 4096/4095, 14641/14580
+POTE generator: ~143/135 = 96.586<br>
+{{Optimal ET sequence|legend=1|87, 236, 323, 410}}<br>
-POTE generator: ~143/135 = 96.586
+Badness: 0.046151<br><br>
-Map: [&lt;1 4 2 -5 7 7|, &lt;0 -30 4 97 -44 -41|]
-EDOs: 87, 323, 410, 733e
-Badness: 0.0462
 [[Category:Major third]]
 [[Category:Equal-step tuning]]
-[[Category:Edonoi]]