Septischismic clan: Difference between revisions

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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 '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783).

== Gary ==

The head of this clan is gary, which is generated by a [[3/2|perfect fifth]]. Two [[2187/2048|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).

[[Subgroup]]: 2.3.7

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: mapping generators: ~2, ~3

~~: sval mapping generators: ~2, ~3~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.2079{{c}}

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

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

[[Badness]]: 0.0135

[[Badness]] (Smith): 0.0135

=== Overview to extensions ===

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Comma list: 19712/19683, 41503/41472

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

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

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

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2292{{c}}

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

{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}

Badness: 0.00731

Badness (Smith): 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 &~~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.

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 {{nowrap| 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 Pythagorean 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

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.317{{c}}

[[Badness]]: 0.105632

[[Badness]] (Smith): 0.105632

=== 11-limit ===

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Mapping: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }}

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

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.303{{c}}

Badness: 0.050966

Badness (Smith): 0.050966

=== 13-limit ===

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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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.306{{c}}

Badness: 0.036951

Badness (Smith): 0.036951

=== 17-limit ===

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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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.307{{c}}

Badness: 0.029495

Badness (Smith): 0.029495

=== 19-limit ===

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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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.308{{c}}

Badness: 0.021811

Badness (Smith): 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.

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

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000{{c}}, ~91125/57344 = 801.0947{{c}}

[[Badness]]: 0.292

[[Badness]] (Smith): 0.292

=== 2.3.5.7.17 subgroup ===

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

~~Sval~~ mapping: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}

Subgroup-val mapping: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}

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

Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~27/17 = 801.0908{{c}}

Badness: 0.0743

Badness (Smith): 0.0743

=== 2.3.5.7.17.19 subgroup ===

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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 tuning (POTE): ~25/21 = 300.0000{{c}}, ~27/17 = 801.0945{{c}}

Badness: 0.0378

Badness (Smith): 0.0378

[[Category:Temperament clans]]

[[Category:Garischismic clan| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
 {{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 '''garischismic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[garischisma]] ({{monzo|legend=1| 25 -14 0 -1 }}, [[ratio]]: 33554432/33480783).
 == Gary ==
+The head of this clan is gary, which is generated by a [[3/2|perfect fifth]]. Two [[2187/2048|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).
 [[Subgroup]]: 2.3.7
@@ Line 8: / Line 9: @@
 {{Mapping|legend=2| 1 0 25 | 0 1 -14 }}
+: mapping generators: ~2, ~3
-: sval mapping generators: ~2, ~3
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.2079{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2079
 {{Optimal ET sequence|legend=1| 12, 29, 41, 94, 135, 364, 499, 634, 3035bd, 3669bd, 4303bd, 4937bbdd, 5571bbdd }}
-[[Badness]]: 0.0135
+[[Badness]] (Smith): 0.0135
 === Overview to extensions ===
@@ Line 41: / Line 41: @@
 Comma list: 19712/19683, 41503/41472
-Sval mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}
+Subgroup-val mapping: {{mapping| 1 0 25 -33 | 0 1 -14 23 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2292
+Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2292{{c}}
-{{Optimal ET sequence|legend=1| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
+{{Optimal ET sequence|legend=0| 12e, 41, 94, 135, 716, 851, 986, 1121, 1256 }}
-Badness: 0.00731
+Badness (Smith): 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.
+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 {{nowrap| 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 Pythagorean 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
@@ Line 60: / Line 60: @@
 {{Mapping|legend=1| 1 0 80 25 | 0 1 -49 -14 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.317
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.317{{c}}
 {{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 258, 475 }}
-[[Badness]]: 0.105632
+[[Badness]] (Smith): 0.105632
 === 11-limit ===
@@ Line 73: / Line 73: @@
 Mapping: {{mapping| 1 0 80 25 -33 | 0 1 -49 -14 23 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303
+Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.303{{c}}
-{{Optimal ET sequence|legend=1| 41, 135c, 176, 217 }}
+{{Optimal ET sequence|legend=0| 41, 135c, 176, 217 }}
-Badness: 0.050966
+Badness (Smith): 0.050966
 === 13-limit ===
@@ Line 86: / Line 86: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.306{{c}}
-{{Optimal ET sequence|legend=1| 41, 176, 217 }}
+{{Optimal ET sequence|legend=0| 41, 176, 217 }}
-Badness: 0.036951
+Badness (Smith): 0.036951
 === 17-limit ===
@@ Line 99: / Line 99: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.307{{c}}
-{{Optimal ET sequence|legend=1| 41, 176, 217 }}
+{{Optimal ET sequence|legend=0| 41, 176, 217 }}
-Badness: 0.029495
+Badness (Smith): 0.029495
 === 19-limit ===
@@ Line 112: / Line 112: @@
 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 tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 702.308{{c}}
-{{Optimal ET sequence|legend=1| 41, 176, 217 }}
+{{Optimal ET sequence|legend=0| 41, 176, 217 }}
-Badness: 0.021811
+Badness (Smith): 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.
+World calendar tempers out the [[dimcomp comma]] and the garischisma, and can be described as the {{nowrap| 12 & 364 }} temperament. The name derives from a [[wikipedia: World Calendar|certain calendar layout]] by the same name.
 [[Subgroup]]: 2.3.5.7
@@ Line 126: / Line 126: @@
 {{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 tuning]] ([[POTE]]): ~25/21 = 300.0000{{c}}, ~91125/57344 = 801.0947{{c}}
 {{Optimal ET sequence|legend=1| 12, …, 352, 364 }}
-[[Badness]]: 0.292
+[[Badness]] (Smith): 0.292
 === 2.3.5.7.17 subgroup ===
@@ Line 140: / Line 139: @@
 Comma list: 2025/2023, 24576/24565, 390625/388962
-Sval mapping: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}
+Subgroup-val mapping: {{mapping| 4 1 -44 86 3 | 0 2 -13 -28 5 }}
-Optimal tuning (POTE): ~25/21 = 1\4, ~27/17 = 801.0908
+Optimal tuning (POTE): ~25/21 = 300.0000{{c}}, ~27/17 = 801.0908{{c}}
-{{Optimal ET sequence|legend=1| 12, …, 352, 364 }}
+{{Optimal ET sequence|legend=0| 12, …, 352, 364 }}
-Badness: 0.0743
+Badness (Smith): 0.0743
 === 2.3.5.7.17.19 subgroup ===
@@ Line 155: / Line 154: @@
 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 tuning (POTE): ~25/21 = 300.0000{{c}}, ~27/17 = 801.0945{{c}}
-{{Optimal ET sequence|legend=1| 12, …, 352, 364 }}
+{{Optimal ET sequence|legend=0| 12, …, 352, 364 }}
-Badness: 0.0378
+Badness (Smith): 0.0378
 [[Category:Temperament clans]]
 [[Category:Garischismic clan| ]] <!-- main article -->
 [[Category:Rank 2]]