Rastmic rank-3 clan: Difference between revisions

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== Spectacle ==

Spectacle, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17700.html Yahoo! Tuning Group | ''Spectacle temperament?'']</ref>, can be described as the {{nowrap| 31 & 34d & 41 }} temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.

It is [[associated temperament|associated]] with the [[marvo]] temperamment.

[[Subgroup]]: 2.3.5.7.11

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: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11

{{Optimal ET sequence|legend=1| 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}

{{Optimal ET sequence|legend=1| 24d, 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}

[[Badness]] (Sintel): 0.599

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Scales: [[spectacle31]]

~~[[Associated temperament]]: [[marvo]]~~

=== 13-limit ===

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== Mirwomo ==

~~The mirwomo~~ temperament ~~tunes ~36/35 to exactly half of an apotome ~2187/2048~~, ~~and tunes~~ ~~~128~~/~~105 to exactly half of a perfect fifth~~ ~3/2.

Mirwomo tempers out 385/384 and may be described as the {{nowrap| 24 & 31 & 41 }} temperament, equating the undecimal quartertone ~33/32 with the septimal quartertone ~36/35.

=== 7-limit ===

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== Mandos ==

Mandos tempers out 176/175 and may be described as the {{nowrap| 24 & 27e & 31 }} temperament.

[[Subgroup]]: 2.3.5.7.11

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== Cuckoo ==

Cuckoo tempers out 126/125 and may be described as the {{nowrap| 24d & 27e & 31 }} temperament.

[[Subgroup]]: 2.3.5.7.11

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== Parahemif ==

Parahemif tempers out 896/891 and may be described as the {{nowrap| 24 & 34d & 41 }} temperament. It is related to [[hemif]], the no-5 rank-2 temperament that tempers out the same list of commas. As such, it finds the interval class of 7 at +13 generator steps, as a semi-augmented sixth (C–At). In the 13-limit, it tempers out 144/143, 352/351, 364/363 among others, and finds ~16/13 at the same neutral third as ~11/9.

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

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== Urania ==

Urania tempers out 81/80, the syntonic comma. It is essentially [[mohaha]] with an independent generator for prime 7.

[[Subgroup]]: 2.3.5.7.11

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== Mirage ==

Mirage is [[miracle]] with an independent generator for prime 13.

[[Subgroup]]: 2.3.5.7.11.13

@@ Line 11: / Line 11: @@
 == Spectacle ==
 Spectacle, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17700.html Yahoo! Tuning Group | ''Spectacle temperament?'']</ref>, can be described as the {{nowrap| 31 & 34d & 41 }} temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.
+It is [[associated temperament|associated]] with the [[marvo]] temperamment.
 [[Subgroup]]: 2.3.5.7.11
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 : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
-{{Optimal ET sequence|legend=1| 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}
+{{Optimal ET sequence|legend=1| 24d, 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}
 [[Badness]] (Sintel): 0.599
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 Scales: [[spectacle31]]
-[[Associated temperament]]: [[marvo]]
 === 13-limit ===
@@ Line 58: / Line 58: @@
 == Mirwomo ==
-The mirwomo temperament tunes ~36/35 to exactly half of an apotome ~2187/2048, and tunes ~128/105 to exactly half of a perfect fifth ~3/2.
+Mirwomo tempers out 385/384 and may be described as the {{nowrap| 24 & 31 & 41 }} temperament, equating the undecimal quartertone ~33/32 with the septimal quartertone ~36/35.
 === 7-limit ===
@@ Line 97: / Line 97: @@
 == Mandos ==
+Mandos tempers out 176/175 and may be described as the {{nowrap| 24 & 27e & 31 }} temperament.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 131: / Line 133: @@
 == Cuckoo ==
+Cuckoo tempers out 126/125 and may be described as the {{nowrap| 24d & 27e & 31 }} temperament.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 165: / Line 169: @@
 == Parahemif ==
+Parahemif tempers out 896/891 and may be described as the {{nowrap| 24 & 34d & 41 }} temperament. It is related to [[hemif]], the no-5 rank-2 temperament that tempers out the same list of commas. As such, it finds the interval class of 7 at +13 generator steps, as a semi-augmented sixth (C–At). In the 13-limit, it tempers out 144/143, 352/351, 364/363 among others, and finds ~16/13 at the same neutral third as ~11/9.
 === 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 219: / Line 225: @@
 == Urania ==
+Urania tempers out 81/80, the syntonic comma. It is essentially [[mohaha]] with an independent generator for prime 7.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 248: / Line 256: @@
 == Mirage ==
+Mirage is [[miracle]] with an independent generator for prime 13.
 [[Subgroup]]: 2.3.5.7.11.13