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Garischismic clan: Difference between revisions

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This clan of temperaments tempers out the [[garischisma]], {{monzo| 25 -14 0 -1 }} = 33554432/33480783, and includes these:
{{Technical data page}}
* [[Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
The '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783). The head of this clan is gary, which is generated by a perfect fifth. Two apotomes i.e. 14 fifths octave reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–Cbb).
* ''[[Newt]]'', {2401/2400, 33554432/33480783} → [[Breedsmic temperaments #Newt|Breedsmic temperaments]]
 
* ''[[Quintagar]]'', {3136/3125, 33554432/33480783} → [[Quindromeda family #Quintagar|Quindromeda family]]
The second comma of the comma list determines which full 7-limit family member we are looking at. Garibaldi adds the [[schisma]], or equivalently [[225/224]] and finds 5/4 at the diminished fourth. Cotoneum adds [[10976/10935]] and finds 5/4 at the septuple-diminished octave. These are generated by the fifth as is gary.
* ''[[Vulture]]'', {4375/4374, 33554432/33480783} → [[Vulture family #Vulture|Vulture family]]
 
* ''[[Trident]]'', {6144/6125, 14348907/14336000} → [[Tricot family #Trident|Tricot family]]
Newt adds [[2401/2400]], slicing the fifth in two. Sextile adds [[250047/250000]] with a 1/3-octave period. Alphatrident adds [[6144/6125]], slicing the twelfth in three. Satin adds [[2100875/2097152]], slicing the fourth in three. Vulture adds [[4375/4374]], slicing the twelfth in four. World calendar adds [[390625/388962]] with a 1/4-octave period as well as a bisect generator. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five.
* [[Cotoneum]], {10976/10935, 823543/819200} → [[Hemimage temperaments #Cotoneum|Hemimage temperaments]]
 
* ''[[Paramity]]'', {65625/65536, 1600000/1594323} → [[Amity family #Paramity|Amity family]]
Temperaments discussed elsewhere are:
* ''[[Garistearn]]'', {118098/117649, 33554432/33480783} → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+225/224)
* ''[[Sextile]]'', {250047/250000, 33554432/33480783} → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]]
* ''[[Newt]]'' → [[Breedsmic temperaments #Newt|Breedsmic temperaments]] (+2401/2400)
* ''[[Satin]]'', {2100875/2097152, 4802000/4782969} → [[Canousmic temperaments #Satin|Canousmic temperaments]]
* ''[[Sextile]]'' → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]] (+250047/250000)
* ''[[Satin]]'' → [[Canousmic temperaments #Satin|Canousmic temperaments]] (+2100875/2097152)
* ''[[Alphatrident]]'' → [[Alphatricot family #Alphatrident|Alphatricot family]] (+6144/6125)
* ''[[Vulture]]'' → [[Vulture family #Vulture|Vulture family]] (+4375/4374)
* ''[[Quintagar]]'' → [[Quindromeda family #Quintagar|Quindromeda family]] (+3136/3125)
* ''[[Paramity]]'' → [[Amity family #Paramity|Amity family]] (+65625/65536)
* ''[[Garistearn]]'' → [[Stearnsmic clan #Garistearn|Stearnsmic clan]] (+118098/117649)
 
Considered below are cotoneum and world calendar.


== Gary ==
== Gary ==
Subgroup: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: 33554432/33480783
[[Comma list]]: 33554432/33480783


[[Sval]] [[mapping]]: [{{val| 1 2 -3 }}, {{val| 0 -1 14 }}]
{{Mapping|legend=2| 1 0 25 | 0 1 -14 }}
 
: sval mapping generators: ~2, ~3


[[POTE generator]]: ~3/2 = 702.2079
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2079


{{Val list|legend=1| 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd }}


[[Badness]]: 0.0135
[[Badness]]: 0.0135
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Comma list: 19712/19683, 41503/41472
Comma list: 19712/19683, 41503/41472


Sval mapping: [{{val| 1 2 -3 13 }}, {{val| 0 -1 14 -23 }}]
Sval mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}


POTE generator: ~3/2 = 702.2292
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292


Vals: {{Val list| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
{{Optimal ET sequence|legend=1| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}


Badness: 0.00731
Badness: 0.00731
== Cotoneum ==
{{Main| Cotoneum }}
The cotoneum temperament tempers out 10976/10935 ([[hemimage comma]]), and 823543/819200 ([[quince comma]]) in addition to the garischisma. This temperament can be described as 41 & 217, and is supported by [[176edo|176-]], [[217edo|217-]], and [[258edo]]. 5/4 is found at the septuple diminished octave (C-Cbbbbbbb) or equivalently at the perfect fourth minus four Pyth. commas. It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 10976/10935, 823543/819200
{{Mapping|legend=1| 1 0 80 25 | 0 1 -49 -14 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.317
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 258, 475 }}
[[Badness]]: 0.105632
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 10976/10935, 16384/16335
Mapping: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217 }}
Badness: 0.050966
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 3584/3575, 10976/10935
Mapping: {{mapping| 1 0 80 25 -33 -93 | 0 1 -49 -14 23 61 }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.306
{{Optimal ET sequence|legend=1| 41, 176, 217 }}
Badness: 0.036951
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
Mapping: {{mapping| 1 0 80 25 -33 -93 -137 | 0 1 -49 -14 23 61 89 }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.307
{{Optimal ET sequence|legend=1| 41, 176, 217 }}
Badness: 0.029495
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
Mapping: {{mapping| 1 0 80 25 -33 -93 -137 74 | 0 1 -49 -14 23 61 89 -44 }}
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.308
{{Optimal ET sequence|legend=1| 41, 176, 217 }}
Badness: 0.021811


== World calendar ==
== World calendar ==
''World calendar'' tempers out the [[dimcomp comma]] and the garischisma, and can be described as the 12 & 364 temperament. The name derives from a [[wikipedia:World Calendar|certain calendar layout]] by the same name.  
World calendar tempers out the [[dimcomp comma]] and the garischisma, and can be described as the 12 & 364 temperament. The name derives from a [[wikipedia: World Calendar|certain calendar layout]] by the same name.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 390625/388962, 33554432/33480783
[[Comma list]]: 390625/388962, 33554432/33480783


[[Mapping]]: [{{val| 4 1 -44 86 }}, {{val| 0 2 -13 -28 }}]
{{Mapping|legend=1| 4 1 -44 86 | 0 2 -13 -28 }}


Mapping generators: ~25/21, ~91125/57344
: mapping generators: ~25/21, ~91125/57344


[[POTE generator]]: ~91125/57344 = 801.0947
[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~91125/57344 = 801.0947


{{Val list|legend=1| 12, …, 352, 364 }}
{{Optimal ET sequence|legend=1| 12, …, 352, 364 }}


[[Badness]]: 0.292
[[Badness]]: 0.292
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Comma list: 2025/2023, 24576/24565, 390625/388962
Comma list: 2025/2023, 24576/24565, 390625/388962


Sval mapping: [{{val| 4 1 -44 86 3 }}, {{val| 0 2 -13 -28 5 }}]
Sval mapping: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}


POTE generator: ~27/17 = 801.0908
Optimal tuning (POTE): ~25/21 = 1\4, ~27/17 = 801.0908


Optimal GPV sequence: {{Val list| 12, …, 352, 364 }}
{{Optimal ET sequence|legend=1| 12, …, 352, 364 }}


Badness: 0.0743
Badness: 0.0743
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Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000
Comma list: 1216/1215, 2025/2023, 8075/8064, 48013/48000


Sval mapping: [{{val| 4 1 -44 86 3 25 }}, {{val| 0 2 -13 -28 5 -3 }}]
Sval mapping: {{mapping| 4 1 -44 86 3 25 | 0 2 -13 -28 5 -3 }}


POTE generator: ~27/17 = 801.0945
Optimal tuning (POTE): ~25/21 = 1\4, ~27/17 = 801.0945


Optimal GPV sequence: {{Val list| 12, …, 352, 364 }}
{{Optimal ET sequence|legend=1| 12, …, 352, 364 }}


Badness: 0.0378
Badness: 0.0378


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Garischismic clan| ]] <!-- main article -->
[[Category:Garischismic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Garischismic_clan"