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
Line 29: Line 39:
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 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
Line 59: Line 138:
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
Line 72: Line 151:
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]]

Latest revision as of 00:36, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The garischismic clan of temperaments tempers out the garischisma (monzo: [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 septimal major second (8/7). Equivalently stated, the harmonic seventh (7/4) is found at the double-diminished octave (C–Cbb).

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.

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.

Temperaments discussed elsewhere are:

Considered below are cotoneum and world calendar.

Gary

Subgroup: 2.3.7

Comma list: 33554432/33480783

Sval mapping: [⟨1 0 25], ⟨0 1 -14]]

sval mapping generators: ~2, ~3

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

Optimal ET sequence: 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd

Badness: 0.0135

2.3.7.11 subgroup

Subgroup: 2.3.7.11

Comma list: 19712/19683, 41503/41472

Sval mapping: [⟨1 0 25 -33], ⟨0 1 -14 23]]

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

Optimal ET sequence: 12e, 41, 94, 135, 716, 851, 986, 1121, 1256

Badness: 0.00731

Cotoneum

Main article: 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 176-, 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: [⟨1 0 80 25], ⟨0 1 -49 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.317

Optimal ET sequence: 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: [⟨1 0 80 25 -33], ⟨0 1 -49 -14 23]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 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: [⟨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: 41, 176, 217

Badness: 0.021811

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 certain calendar layout by the same name.

Subgroup: 2.3.5.7

Comma list: 390625/388962, 33554432/33480783

Mapping: [⟨4 1 -44 86], ⟨0 2 -13 -28]]

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

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

Optimal ET sequence: 12, …, 352, 364

Badness: 0.292

2.3.5.7.17 subgroup

Subgroup: 2.3.5.7.17

Comma list: 2025/2023, 24576/24565, 390625/388962

Sval mapping: [⟨4 1 -44 86 3], ⟨0 2 -13 -28 5]]

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

Optimal ET sequence: 12, …, 352, 364

Badness: 0.0743

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

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

Sval mapping: [⟨4 1 -44 86 3 25], ⟨0 2 -13 -28 5 -3]]

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

Optimal ET sequence: 12, …, 352, 364

Badness: 0.0378

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