5632/5625: Difference between revisions

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'''5632/5625''', the '''vishdelisma''' or '''vishdelic comma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.5.11 subgroup|2.3.5.11-subgroup]]) [[comma]] with a size of about 2.15 [[cent]]s. It is the amount by which a stack of four [[5/4]] major thirds falls short of [[11/9]] plus an [[octave]]. Equivalently, it is the amount by which [[11/8]] overshoots a stack of two [[75/64]] classical augmented seconds.

In terms of other commas, it is the difference between [[176/175]] and [[225/224]]. In the [[13-limit]], it equals ([[676/675]])/([[4225/4224]]) and factors as ([[1001/1000]])⋅([[4096/4095]]). In the [[19-limit]] it factors as ([[1216/1215]])⋅([[2376/2375]]).

== Temperaments ==

[[Tempering out]] this comma in the full 11-limit gives the rank-4 [[vishdelismic]] temperament, or in the 2.3.5.11 subgroup, the rank-3 '''vishdelic''' temperament. It is tempered out in 11-limit extensions of temperaments such as [[vishnu]], [[grendel]], [[quasiorwell]], [[hemithirds]], and [[decoid]].

~~=== Vishdelic ===~~

See [[Vishdelismic clan]] for the [[clan of temperaments]] where it is tempered out.

[[~~Subgroup~~]]~~: 2.3.5.11~~

~~[[Comma list]]: 5632/5625~~

~~{{Mapping|legend=1| 1 0 0 -9 | 0 1 0 2 | 0 0 1 4 }}~~

~~: mapping generators: ~2, ~3, ~5~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.8976{{c}}, ~3/2 = 702.1145{{c}}, ~5/4 = 386.7638{{c}}

~~: [[error map]]: {{val| -0.102 +0.057 +0.245 -0.136 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1309{{c}}, ~5/4 = 386.7770{{c}}

~~: error map: {{val| 0.000 +0.176 +0.463 +0.052 }}~~

~~{{Optimal ET sequence|legend=1| 31, 34, 53, 65, 87, 118, 152, 270, 335, 605, 940, 1092, 1362c, 1697c, 2302c }}~~

[[~~Badness~~]] ~~(Sintel): 0~~.~~183~~

[[Category:Vishdelismic]]

[[Category:Commas named by combining multiple temperament names]]

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 }}
 '''5632/5625''', the '''vishdelisma''' or '''vishdelic comma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.5.11 subgroup|2.3.5.11-subgroup]]) [[comma]] with a size of about 2.15 [[cent]]s. It is the amount by which a stack of four [[5/4]] major thirds falls short of [[11/9]] plus an [[octave]]. Equivalently, it is the amount by which [[11/8]] overshoots a stack of two [[75/64]] classical augmented seconds.
+In terms of other commas, it is the difference between [[176/175]] and [[225/224]]. In the [[13-limit]], it equals ([[676/675]])/([[4225/4224]]) and factors as ([[1001/1000]])⋅([[4096/4095]]). In the [[19-limit]] it factors as ([[1216/1215]])⋅([[2376/2375]]).
 == Temperaments ==
 [[Tempering out]] this comma in the full 11-limit gives the rank-4 [[vishdelismic]] temperament, or in the 2.3.5.11 subgroup, the rank-3 '''vishdelic''' temperament. It is tempered out in 11-limit extensions of temperaments such as [[vishnu]], [[grendel]], [[quasiorwell]], [[hemithirds]], and [[decoid]].
-=== Vishdelic ===
+See [[Vishdelismic clan]] for the [[clan of temperaments]] where it is tempered out.
-[[Subgroup]]: 2.3.5.11
-[[Comma list]]: 5632/5625
-{{Mapping|legend=1| 1 0 0 -9 | 0 1 0 2 | 0 0 1 4 }}
-: mapping generators: ~2, ~3, ~5
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.8976{{c}}, ~3/2 = 702.1145{{c}}, ~5/4 = 386.7638{{c}}
-: [[error map]]: {{val| -0.102 +0.057 +0.245 -0.136 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1309{{c}}, ~5/4 = 386.7770{{c}}
-: error map: {{val| 0.000 +0.176 +0.463 +0.052 }}
-{{Optimal ET sequence|legend=1| 31, 34, 53, 65, 87, 118, 152, 270, 335, 605, 940, 1092, 1362c, 1697c, 2302c }}
-[[Badness]] (Sintel): 0.183
 [[Category:Vishdelismic]]
 [[Category:Commas named by combining multiple temperament names]]