Garischismic clan: Difference between revisions

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The '''garischismic clan''' of temperaments tempers out the [[garischisma]], {{monzo| 25 -14 0 -1 }} = 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~~).

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).

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.

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. ~~Trident~~ 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.

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)

* [[Cotoneum]] → [[Hemimage temperaments #Cotoneum|Hemimage temperaments]] (+10976/10935)

* ''[[Newt]]'' → [[Breedsmic temperaments #Newt|Breedsmic temperaments]] (+2401/2400)

* ''[[Sextile]]'' → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]] (+250047/250000)

* ''[[Satin]]'' → [[Canousmic temperaments #Satin|Canousmic temperaments]] (+2100875/2097152)

* ''[[~~Trident~~]]'' → [[~~Tricot~~ family #~~Trident~~|~~Tricot~~ family]] (+6144/6125)

* ''[[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 ==

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[[Comma list]]: 33554432/33480783

: sval mapping generators: ~2, ~3

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

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Comma list: 19712/19683, 41503/41472

Sval mapping: {{mapping| 1 2 -~~3 13~~ | 0 -1 14 -23 }}

Sval mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}

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

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Badness: 0.00731

== 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

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

[[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

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

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

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

Badness: 0.021811

== World calendar ==

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[[Category:Temperament clans]]

[[Category:Pages with mostly numerical content]]

[[Category:Garischismic clan| ]]

[[Category:Rank 2]]

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-The '''garischismic clan''' of temperaments tempers out the [[garischisma]], {{monzo| 25 -14 0 -1 }} = 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).
+{{Technical data page}}
+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).
-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.
+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. Trident 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.
+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)
-* [[Cotoneum]] → [[Hemimage temperaments #Cotoneum|Hemimage temperaments]] (+10976/10935)
 * ''[[Newt]]'' → [[Breedsmic temperaments #Newt|Breedsmic temperaments]] (+2401/2400)
 * ''[[Sextile]]'' → [[Landscape microtemperaments #Sextile|Landscape microtemperaments]] (+250047/250000)
 * ''[[Satin]]'' → [[Canousmic temperaments #Satin|Canousmic temperaments]] (+2100875/2097152)
-* ''[[Trident]]'' → [[Tricot family #Trident|Tricot family]] (+6144/6125)
+* ''[[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 ==
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 [[Comma list]]: 33554432/33480783
-{{Mapping|legend=2| 1 2 -3 | 0 -1 14 }}
+{{Mapping|legend=2| 1 0 25 | 0 1 -14 }}
+: sval mapping generators: ~2, ~3
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2079
@@ Line 34: / Line 39: @@
 Comma list: 19712/19683, 41503/41472
-Sval mapping: {{mapping| 1 2 -3 13 | 0 -1 14 -23 }}
+Sval mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}
 Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292
@@ Line 41: / Line 46: @@
 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 &amp; 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 ==
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 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Garischismic clan| ]] <!-- main article -->
 [[Category:Rank 2]]