Semicanousmic clan: Difference between revisions

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{{Technical data page}}
{{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:
Temperaments discussed elsewhere are:
Line 9: Line 9:
Considered below are syndeute and hemireiwa, in addition to deuteromere, the no-7 subgroup temperament.  
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 ==
== Deuteromere ==
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{{Mapping|legend=2| 1 0 2 1 | 0 1 2 2 | 0 0 -4 -1 }}
{{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 }}
{{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 ===
=== 2.3.5.11.17 subgroup ===
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Comma list: 1089/1088, 14641/14580
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 ===
=== 2.3.5.11.17.19 subgroup ===
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Comma list: 1089/1088, 1216/1215, 1445/1444
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 ==
== Syndeute ==
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{{Mapping|legend=1| 1 0 2 -9 1 | 0 1 2 7 2 | 0 0 -4 1 -1 }}
{{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 }}
{{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 ===
=== 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 }}
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 ===
=== 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 }}
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 ===
=== 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 }}
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 ==
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
[[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|legend=1| 1 0 2 7 1 | 0 2 0 -6 3 | 0 0 4 7 1 }}
: mapping generators: ~2, ~400/231, ~200/189
: 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 }}
{{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 ===
=== 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 }}
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 ===
=== 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 }}
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 ===
=== 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 }}
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:Temperament clans]]
[[Category:Semicanousmic clan| ]] <!-- main article -->
[[Category:Semicanousmic clan| ]] <!-- main article -->
[[Category:Semicanousmic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 10:18, 17 June 2026

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 semicanousmic clan of rank-3 temperaments tempers out the semicanousma (monzo: [-2 -6 -1 0 4⟩, ratio: 14641/14580).

Temperaments discussed elsewhere are:

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

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

Deuteromere

Lattice
Lattice rearranged into neutral thirds

Subgroup: 2.3.5.11

Comma list: 14641/14580

Subgroup-val mapping: [⟨1 0 2 1], ⟨0 1 2 2], ⟨0 0 -4 -1]]

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 702.3342 ¢, ~18/11 = 854.5548 ¢

Optimal ET sequence: 7, 17c, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e

Badness (Smith): 0.142 × 10-3

2.3.5.11.17 subgroup

Subgroup: 2.3.5.11.17

Comma list: 1089/1088, 14641/14580

Subgroup-val mapping: [⟨1 0 2 1 -4], ⟨0 1 2 2 6], ⟨0 0 -4 -1 -2]]

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

Optimal ET sequence: 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e

Badness (Smith): 0.481 × 10-3

2.3.5.11.17.19 subgroup

Subgroup: 2.3.5.11.17.19

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

Subgroup-val mapping: [⟨1 0 2 1 -4 -4], ⟨0 1 2 2 6 7], ⟨0 0 -4 -1 -2 -4]]

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

Optimal ET sequence: 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e

Badness (Smith): 0.403 × 10-3

Syndeute

Subgroup: 2.3.5.7.11

Comma list: 14641/14580, 19712/19683

Mapping: [⟨1 0 2 -9 1], ⟨0 1 2 7 2], ⟨0 0 -4 1 -1]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 702.0812 ¢, ~18/11 = 854.4115 ¢

Optimal ET sequence: 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e

Badness (Smith): 3.26 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1573/1568, 2080/2079, 14641/14580

Mapping: [⟨1 0 2 -9 1 -15], ⟨0 1 2 7 2 10], ⟨0 0 -4 1 -1 4]]

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

Optimal ET sequence: 94, 111, 205, 212, 323, 535ef

Badness (Smith): 3.01 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 561/560, 715/714, 14641/14580

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 = 1200.0000 ¢, ~3/2 = 702.1910 ¢, ~18/11 = 854.4751 ¢

Optimal ET sequence: 94, 111, 205, 212g, 323

Badness (Smith): 1.69 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444

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 = 1200.0000 ¢, ~3/2 = 702.1925 ¢, ~18/11 = 854.4721 ¢

Optimal ET sequence: 94, 111, 205, 212gh, 323

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.

Subgroup: 2.3.5.7.11

Comma list: 14641/14580, 160083/160000

Mapping: [⟨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 = 1200.0000 ¢, ~400/231 = 951.1975 ¢, ~200/189 = 96.5851 ¢

Optimal ET sequence: 87, 111, 125, 198, 212, 323, 410

Badness (Smith): 3.95 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 14641/14580

Mapping: [⟨1 0 2 7 1 1], ⟨0 2 0 -6 3 3], ⟨0 0 4 7 1 4]]

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

Optimal ET sequence: 87, 111, 198, 323, 410

Badness (Smith): 1.89 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 1001/1000, 14641/14580

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 = 1200.0000 ¢, ~26/15 = 951.2022 ¢, ~55/52 = 96.6006 ¢

Optimal ET sequence: 87, 111, 198g, 212g, 299, 323, 410

Badness (Smith): 1.90 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444

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 = 1200.0000 ¢, ~26/15 = 951.1749 ¢, ~55/52 = 96.5813 ¢

Optimal ET sequence: 87, 111, 198gh, 212gh, 299, 323, 410, 622ef

Badness (Smith): 1.73 × 10-3

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