Semicathartismic clan: Difference between revisions

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The '''semicanousmic clan''' of [[~~Rank-3 temperament|~~rank-3~~]] [[~~temperament]]s [[~~Tempering~~ out|tempers out]] the [[semicanousma]]~~, 14641/14580 =~~ {{monzo| -2 -6 -1 0 4 }}.

The '''semicanousmic clan''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[semicanousma]] ({{monzo|legend=1| -2 -6 -1 0 4 }}, [[ratio]]: 14641/14580).

Temperaments discussed elsewhere are:

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Considered below are syndeute and hemireiwa, in addition to deuteromere, the no-7 subgroup temperament.

For the rank-4 semicanousmic temperament, see [[~~Rank~~-4 ~~temperament~~ #Semicanousmic (14641/14580)]].

For the rank-4 semicanousmic temperament, see [[Catalog of rank-4 temperaments #Semicanousmic (14641/14580)]].

== Deuteromere ==

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: mapping generators: ~2, ~3, ~18/11

~~: sval mapping generators: ~2, ~3, ~18/11~~

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.3342{{c}}, ~18/11 = 854.5548{{c}}

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.3342, ~18/11 = 854.5548

{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}

[[Badness]]: 0.142 × 10-3

[[Badness]] (Smith): 0.142 × 10-3

=== 2.3.5.11.17 subgroup ===

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Comma list: 1089/1088, 14641/14580

~~Sval~~ mapping: {{mapping| 1 0 2 1 -4 | 0 1 2 2 6 | 0 0 -4 -1 -2 }}

Subgroup-val mapping: {{mapping| 1 0 2 1 -4 | 0 1 2 2 6 | 0 0 -4 -1 -2 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.3427, ~18/11 = 854.5587

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.3427{{c}}, ~18/11 = 854.5587{{c}}

{{Optimal ET sequence|legend=1| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}

{{Optimal ET sequence|legend=0| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}

Badness: 0.481 × 10-3

Badness (Smith): 0.481 × 10-3

=== 2.3.5.11.17.19 subgroup ===

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Comma list: 1089/1088, 1216/1215, 1445/1444

~~Sval~~ mapping: {{mapping| 1 0 2 1 -4 -4 | 0 1 2 2 6 7 | 0 0 -4 -1 -2 -4 }}

Subgroup-val mapping: {{mapping| 1 0 2 1 -4 -4 | 0 1 2 2 6 7 | 0 0 -4 -1 -2 -4 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.2987, ~18/11 = 854.5520

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2987{{c}}, ~18/11 = 854.5520{{c}}

{{Optimal ET sequence|legend=1| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}

{{Optimal ET sequence|legend=0| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}

Badness: 0.403 × 10-3

Badness (Smith): 0.403 × 10-3

== Syndeute ==

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~3/2 = 702.0812, ~18/11 = 854.4115

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.0812{{c}}, ~18/11 = 854.4115{{c}}

{{Optimal ET sequence|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}

[[Badness]]: 3.26 × 10-3

[[Badness]] (Smith): 3.26 × 10-3

=== 13-limit ===

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Mapping: {{mapping| 1 0 2 -9 1 -15 | 0 1 2 7 2 10 | 0 0 -4 1 -1 4 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.1682, ~18/11 = 854.5074

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1682{{c}}, ~18/11 = 854.5074{{c}}

Badness: 3.01 × 10-3

Badness (Smith): 3.01 × 10-3

=== 17-limit ===

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Mapping: {{mapping| 1 0 2 -9 1 -15 -4 | 0 1 2 7 2 10 6 | 0 0 -4 1 -1 4 -2 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.1910, ~18/11 = 854.4751

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1910{{c}}, ~18/11 = 854.4751{{c}}

Badness: 1.69 × 10-3

Badness (Smith): 1.69 × 10-3

=== 19-limit ===

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Mapping: {{mapping| 1 0 2 -9 1 -15 -4 -4 | 0 1 2 7 2 10 6 7 | 0 0 -4 1 -1 4 -2 -4 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~3/2 = 702.1925, ~18/11 = 854.4721

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1925{{c}}, ~18/11 = 854.4721{{c}}

Badness: 1.27 × 10-3

Badness (Smith): 1.27 × 10-3

== Hemireiwa ==

Hemireiwa is in a certain sense dual to semicanou. It splits the perfect twelfth into two, whereas semicanou splits the octave.

Hemireiwa is in a certain sense dual to [[semicanou]]. It splits the perfect twelfth into two, whereas semicanou splits the octave.

[[Subgroup]]: 2.3.5.7.11

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: mapping generators: ~2, ~400/231, ~200/189

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~400/231 = 951.1975, ~200/189 = 96.5851

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~400/231 = 951.1975{{c}}, ~200/189 = 96.5851{{c}}

{{Optimal ET sequence|legend=1| 87, 111, 125, 198, 212, 323, 410 }}

[[Badness]]: 3.95 × 10-3

[[Badness]] (Smith): 3.95 × 10-3

=== 13-limit ===

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Mapping: {{mapping| 1 0 2 7 1 1 | 0 2 0 -6 3 3 | 0 0 4 7 1 4 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~26/15 = 951.2316, ~55/52 = 96.6155

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.2316{{c}}, ~55/52 = 96.6155{{c}}

Badness: 1.89 × 10-3

Badness (Smith): 1.89 × 10-3

=== 17-limit ===

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Mapping: {{mapping| 1 0 2 7 1 1 -4 | 0 2 0 -6 3 3 10 | 0 0 4 7 1 4 2 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~26/15 = 951.2022, ~55/52 = 96.6006

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.2022{{c}}, ~55/52 = 96.6006{{c}}

{{Optimal ET sequence|legend=1| 87, 111, 198g, 212g, 299, 323, 410 }}

{{Optimal ET sequence|legend=0| 87, 111, 198g, 212g, 299, 323, 410 }}

Badness: 1.90 × 10-3

Badness (Smith): 1.90 × 10-3

=== 19-limit ===

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Mapping: {{mapping| 1 0 2 7 1 1 -4 -4 | 0 2 0 -6 3 3 10 10 | 0 0 4 7 1 4 2 4 }}

Optimal tuning (CTE): ~2 = ~~1\1~~, ~26/15 = 951.1749, ~55/52 = 96.5813

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.1749{{c}}, ~55/52 = 96.5813{{c}}

{{Optimal ET sequence|legend=1| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}

{{Optimal ET sequence|legend=0| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}

Badness: 1.73 × 10-3

Badness (Smith): 1.73 × 10-3

[[Category:Temperament clans]]

[[Category:Semicanousmic clan| ]]

~~[[Category:Semicanousmic| ]] ~~

[[Category:Rank 3]]

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 {{Technical data page}}
-The '''semicanousmic clan''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[semicanousma]], 14641/14580 = {{monzo| -2 -6 -1 0 4 }}.
+The '''semicanousmic clan''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[semicanousma]] ({{monzo|legend=1| -2 -6 -1 0 4 }}, [[ratio]]: 14641/14580).
 Temperaments discussed elsewhere are:
@@ Line 9: / Line 9: @@
 Considered below are syndeute and hemireiwa, in addition to deuteromere, the no-7 subgroup temperament.
-For the rank-4 semicanousmic temperament, see [[Rank-4 temperament #Semicanousmic (14641/14580)]].
+For the rank-4 semicanousmic temperament, see [[Catalog of rank-4 temperaments #Semicanousmic (14641/14580)]].
 == Deuteromere ==
@@ Line 20: / Line 20: @@
 {{Mapping|legend=2| 1 0 2 1 | 0 1 2 2 | 0 0 -4 -1 }}
+: mapping generators: ~2, ~3, ~18/11
-: sval mapping generators: ~2, ~3, ~18/11
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.3342{{c}}, ~18/11 = 854.5548{{c}}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.3342, ~18/11 = 854.5548
 {{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}
-[[Badness]]: 0.142 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.142 × 10<sup>-3</sup>
 === 2.3.5.11.17 subgroup ===
@@ Line 34: / Line 33: @@
 Comma list: 1089/1088, 14641/14580
-Sval mapping: {{mapping| 1 0 2 1 -4 | 0 1 2 2 6 | 0 0 -4 -1 -2 }}
+Subgroup-val mapping: {{mapping| 1 0 2 1 -4 | 0 1 2 2 6 | 0 0 -4 -1 -2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.3427, ~18/11 = 854.5587
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.3427{{c}}, ~18/11 = 854.5587{{c}}
-{{Optimal ET sequence|legend=1| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}
+{{Optimal ET sequence|legend=0| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}
-Badness: 0.481 × 10<sup>-3</sup>
+Badness (Smith): 0.481 × 10<sup>-3</sup>
 === 2.3.5.11.17.19 subgroup ===
@@ Line 47: / Line 46: @@
 Comma list: 1089/1088, 1216/1215, 1445/1444
-Sval mapping: {{mapping| 1 0 2 1 -4 -4 | 0 1 2 2 6 7 | 0 0 -4 -1 -2 -4 }}
+Subgroup-val mapping: {{mapping| 1 0 2 1 -4 -4 | 0 1 2 2 6 7 | 0 0 -4 -1 -2 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2987, ~18/11 = 854.5520
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2987{{c}}, ~18/11 = 854.5520{{c}}
-{{Optimal ET sequence|legend=1| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}
+{{Optimal ET sequence|legend=0| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}
-Badness: 0.403 × 10<sup>-3</sup>
+Badness (Smith): 0.403 × 10<sup>-3</sup>
 == Syndeute ==
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 {{Mapping|legend=1| 1 0 2 -9 1 | 0 1 2 7 2 | 0 0 -4 1 -1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.0812, ~18/11 = 854.4115
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.0812{{c}}, ~18/11 = 854.4115{{c}}
 {{Optimal ET sequence|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}
-[[Badness]]: 3.26 × 10<sup>-3</sup>
+[[Badness]] (Smith): 3.26 × 10<sup>-3</sup>
 === 13-limit ===
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 Mapping: {{mapping| 1 0 2 -9 1 -15 | 0 1 2 7 2 10 | 0 0 -4 1 -1 4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1682, ~18/11 = 854.5074
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1682{{c}}, ~18/11 = 854.5074{{c}}
-{{Optimal ET sequence|legend=1| 94, 111, 205, 212, 323, 535ef }}
+{{Optimal ET sequence|legend=0| 94, 111, 205, 212, 323, 535ef }}
-Badness: 3.01 × 10<sup>-3</sup>
+Badness (Smith): 3.01 × 10<sup>-3</sup>
 === 17-limit ===
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 Mapping: {{mapping| 1 0 2 -9 1 -15 -4 | 0 1 2 7 2 10 6 | 0 0 -4 1 -1 4 -2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1910, ~18/11 = 854.4751
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1910{{c}}, ~18/11 = 854.4751{{c}}
-{{Optimal ET sequence|legend=1| 94, 111, 205, 212g, 323 }}
+{{Optimal ET sequence|legend=0| 94, 111, 205, 212g, 323 }}
-Badness: 1.69 × 10<sup>-3</sup>
+Badness (Smith): 1.69 × 10<sup>-3</sup>
 === 19-limit ===
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 Mapping: {{mapping| 1 0 2 -9 1 -15 -4 -4 | 0 1 2 7 2 10 6 7 | 0 0 -4 1 -1 4 -2 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1925, ~18/11 = 854.4721
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1925{{c}}, ~18/11 = 854.4721{{c}}
-{{Optimal ET sequence|legend=1| 94, 111, 205, 212gh, 323 }}
+{{Optimal ET sequence|legend=0| 94, 111, 205, 212gh, 323 }}
-Badness: 1.27 × 10<sup>-3</sup>
+Badness (Smith): 1.27 × 10<sup>-3</sup>
 == Hemireiwa ==
-Hemireiwa is in a certain sense dual to semicanou. It splits the perfect twelfth into two, whereas semicanou splits the octave.
+Hemireiwa is in a certain sense dual to [[semicanou]]. It splits the perfect twelfth into two, whereas semicanou splits the octave.
 [[Subgroup]]: 2.3.5.7.11
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 {{Mapping|legend=1| 1 0 2 7 1 | 0 2 0 -6 3 | 0 0 4 7 1 }}
 : mapping generators: ~2, ~400/231, ~200/189
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~400/231 = 951.1975, ~200/189 = 96.5851
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~400/231 = 951.1975{{c}}, ~200/189 = 96.5851{{c}}
 {{Optimal ET sequence|legend=1| 87, 111, 125, 198, 212, 323, 410 }}
-[[Badness]]: 3.95 × 10<sup>-3</sup>
+[[Badness]] (Smith): 3.95 × 10<sup>-3</sup>
 === 13-limit ===
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 Mapping: {{mapping| 1 0 2 7 1 1 | 0 2 0 -6 3 3 | 0 0 4 7 1 4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.2316, ~55/52 = 96.6155
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.2316{{c}}, ~55/52 = 96.6155{{c}}
-{{Optimal ET sequence|legend=1| 87, 111, 198, 323, 410 }}
+{{Optimal ET sequence|legend=0| 87, 111, 198, 323, 410 }}
-Badness: 1.89 × 10<sup>-3</sup>
+Badness (Smith): 1.89 × 10<sup>-3</sup>
 === 17-limit ===
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 Mapping: {{mapping| 1 0 2 7 1 1 -4 | 0 2 0 -6 3 3 10 | 0 0 4 7 1 4 2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.2022, ~55/52 = 96.6006
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.2022{{c}}, ~55/52 = 96.6006{{c}}
-{{Optimal ET sequence|legend=1| 87, 111, 198g, 212g, 299, 323, 410 }}
+{{Optimal ET sequence|legend=0| 87, 111, 198g, 212g, 299, 323, 410 }}
-Badness: 1.90 × 10<sup>-3</sup>
+Badness (Smith): 1.90 × 10<sup>-3</sup>
 === 19-limit ===
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 Mapping: {{mapping| 1 0 2 7 1 1 -4 -4 | 0 2 0 -6 3 3 10 10 | 0 0 4 7 1 4 2 4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.1749, ~55/52 = 96.5813
+Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.1749{{c}}, ~55/52 = 96.5813{{c}}
-{{Optimal ET sequence|legend=1| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}
+{{Optimal ET sequence|legend=0| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}
-Badness: 1.73 × 10<sup>-3</sup>
+Badness (Smith): 1.73 × 10<sup>-3</sup>
 [[Category:Temperament clans]]
 [[Category:Semicanousmic clan| ]] <!-- main article -->
-[[Category:Semicanousmic| ]] <!-- key article -->
 [[Category:Rank 3]]