4ed5/4: Difference between revisions
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{{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. | '''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. | ||
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| | (5/4)<font style="vertical-align:super;font-size:0.8em;">13</font> = 1220703125/67108864 | | | (5/4)<font style="vertical-align:super;font-size:0.8em;">13</font> = 1220703125/67108864 | ||
|} | |} | ||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 4 | |||
| num = 5 | |||
| denom = 4 | |||
}} | |||
{{Harmonics in equal | |||
| steps = 4 | |||
| num = 5 | |||
| denom = 4 | |||
| start = 12 | |||
| collapsed = 1 | |||
}} | |||
== Related regular temperament == | == Related regular temperament == | ||
4ED5/4 tuning is closely related to the '' | 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. | ||
; [[Luna family #Hemiluna|Hemiluna]] (87 & 323) | ; <font style="font-size:1.15em">[[Luna family #Hemiluna|Hemiluna]] (87 & 323)</font> | ||
'''7-limit'''<br> | '''7-limit'''<br> | ||
Comma list: 48828125/48771072, 67108864/66976875<br> | Comma list: 48828125/48771072, 67108864/66976875<br> | ||
Mapping: [{{val|1 4 2 -5}}, {{val|0 -30 4 97}}]<br> | Mapping: [{{val|1 4 2 -5}}, {{val|0 -30 4 97}}]<br> | ||
POTE generator: ~200/189 = 96.591<br> | POTE generator: ~200/189 = 96.591<br> | ||
{{Optimal ET sequence|legend=1|87, 236, 323, 410, 733}}<br> | |||
Badness: 0.179304<br><br> | Badness: 0.179304<br><br> | ||
'''11-limit'''<br> | '''11-limit'''<br> | ||
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Mapping: [{{val|1 4 2 -5 7}}, {{val|0 -30 4 97 -44}}]<br> | Mapping: [{{val|1 4 2 -5 7}}, {{val|0 -30 4 97 -44}}]<br> | ||
POTE generator: ~200/189 = 96.587<br> | POTE generator: ~200/189 = 96.587<br> | ||
{{Optimal ET sequence|legend=1|87, 236, 323, 410}}<br> | |||
Badness: 0.100077<br><br> | Badness: 0.100077<br><br> | ||
'''13-limit'''<br> | '''13-limit'''<br> | ||
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Mapping: [{{val|1 4 2 -5 7 7}}, {{val|0 -30 4 97 -44 -41}}]<br> | Mapping: [{{val|1 4 2 -5 7 7}}, {{val|0 -30 4 97 -44 -41}}]<br> | ||
POTE generator: ~143/135 = 96.586<br> | POTE generator: ~143/135 = 96.586<br> | ||
{{Optimal ET sequence|legend=1|87, 236, 323, 410}}<br> | |||
Badness: 0.046151<br><br> | Badness: 0.046151<br><br> | ||
[[Category:Major third]] | [[Category:Major third]] | ||
[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||