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}}
* [[Vulture family #Vulture|Vulture]]
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).
* [[Amity family #Paramity|Paramity]]
* [[Breedsmic temperaments #Newt|Newt]]
* [[Schismatic family #Garibaldi|Garibaldi]]
* [[Landscape microtemperaments #Sextile|Sextile]]
* [[Canousmic temperaments #Satin|Satin]]
* [[Stearnsmic clan #Garistearn|Garistearn]]
* [[Hemimage temperaments #Cotoneum|Cotoneum]]
* [[Hemimean clan #Quintoneum|Quintoneum]]


== No-five garischismic ==
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.
Subgroup: 2.3.7
 
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:
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+225/224)
* ''[[Newt]]'' → [[Breedsmic temperaments #Newt|Breedsmic temperaments]] (+2401/2400)
* ''[[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 ==
[[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


[[Category:Regular temperament theory]]
=== 2.3.7.11 subgroup ===
[[Category:Temperament clan]]
Subgroup: 2.3.7.11
[[Category:Garischismic]]
 
Comma list: 19712/19683, 41503/41472
 
Sval mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292
 
{{Optimal ET sequence|legend=1| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
 
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 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
 
[[Comma list]]: 390625/388962, 33554432/33480783
 
{{Mapping|legend=1| 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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 12, …, 352, 364 }}
 
Badness: 0.0378
 
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Garischismic clan| ]] <!-- main article -->
[[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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