Semicanousmic clan: Difference between revisions

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These are rank-3 temperaments tempering out the [[semicanousma]], 14641/14580 = {{monzo|-2 -6 -1 0 4}}, but we can begin by looking at the rank-4 temperament.  
{{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 }}.


Temperaments not discussed here include:
Temperaments discussed elsewhere are:
* [[Didymus rank three family #Urania|Urania]]
* ''[[Urania]]'' (+81/80 or 121/120) → [[Didymus rank three family #Urania|Didymus rank-3 family]]
* [[Hemimean family #Triglav|Triglav]]
* ''[[Triglav]]'' (+3025/3024 or 3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
* [[Canou family #Semicanou|Semicanou]]
* ''[[Semicanou]]'' (+9801/9800) → [[Canou family #Semicanou|Canou family]]


= Semicanousmic =
Considered below are syndeute and hemireiwa, in addition to deuteromere, the no-7 subgroup temperament.
Subgroup: 2.3.5.7.11
 
For the rank-4 semicanousmic temperament, see [[Rank-4 temperament #Semicanousmic (14641/14580)]].
 
== Deuteromere ==
[[File:Lattice Semicanousmic.png|thumb|Lattice]]
[[File:Lattice Semicanousmic Rearranged.png|thumb|Lattice rearranged into neutral thirds]]
 
[[Subgroup]]: 2.3.5.11


[[Comma list]]: 14641/14580
[[Comma list]]: 14641/14580


[[Mapping]]: [{{val| 1 0 2 0 1 }}, {{val| 0 1 2 0 2 }}, {{val| 0 0 4 0 1 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=2| 1 0 2 1 | 0 1 2 2 | 0 0 -4 -1 }}
 
: sval mapping generators: ~2, ~3, ~18/11
 
[[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>
 
=== 2.3.5.11.17 subgroup ===
Subgroup: 2.3.5.11.17
 
Comma list: 1089/1088, 14641/14580
 
Sval 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 ET sequence|legend=1| 24, 31g, 55g, 56gg, 63, 70c, 80gg, 87, 111, 118, 205, 528e, 733e }}
 
Badness: 0.481 × 10<sup>-3</sup>
 
=== 2.3.5.11.17.19 subgroup ===
Subgroup: 2.3.5.11.17.19
 
Comma list: 1089/1088, 1216/1215, 1445/1444


[[POTE generator]]s: ~3/2 = 702.2503, ~11/9 = 345.4579, ~7/4 = 968.6866
Sval mapping: {{mapping| 1 0 2 1 -4 -4 | 0 1 2 2 6 7 | 0 0 -4 -1 -2 -4 }}


{{Val list|legend=1| 7, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2987, ~18/11 = 854.5520


[[Badness]]: 0.3506 × 10<sup>-6</sup>
{{Optimal ET sequence|legend=1| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }}


= No-7 semicanousmic =
Badness: 0.403 × 10<sup>-3</sup>
Subgroup: 2.3.5.11


[[Comma list]]: 14641/14580
== Syndeute ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 14641/14580, 19712/19683
 
{{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 ET sequence|legend=1| 7d, 17c, 24, 77c, 87d, 94, 111, 118, 212, 323, 441e }}
 
[[Badness]]: 3.26 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1573/1568, 2080/2079, 14641/14580
 
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 ET sequence|legend=1| 94, 111, 205, 212, 323, 535ef }}
 
Badness: 3.01 × 10<sup>-3</sup>
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 442/441, 561/560, 715/714, 14641/14580
 
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 ET sequence|legend=1| 94, 111, 205, 212g, 323 }}
 
Badness: 1.69 × 10<sup>-3</sup>
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444
 
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 ET sequence|legend=1| 94, 111, 205, 212gh, 323 }}
 
Badness: 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.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 14641/14580, 160083/160000
 
{{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 ET sequence|legend=1| 87, 111, 125, 198, 212, 323, 410 }}
 
[[Badness]]: 3.95 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 14641/14580
 
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 ET sequence|legend=1| 87, 111, 198, 323, 410 }}
 
Badness: 1.89 × 10<sup>-3</sup>
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 676/675, 715/714, 1001/1000, 14641/14580
 
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 ET sequence|legend=1| 87, 111, 198g, 212g, 299, 323, 410 }}
 
Badness: 1.90 × 10<sup>-3</sup>
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444


[[Sval]] [[mapping]]: [{{val| 1 0 2 1 }}, {{val| 0 1 2 2 }}, {{val| 0 0 4 1 }}]
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 }}


[[POTE generator]]s: ~3/2 = 702.2503, ~11/9 = 345.4579
Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.1749, ~55/52 = 96.5813


{{Val list|legend=1| 7, 24, 31, 56, 73, 80, 87, 118, 205, 323, 528e }}
{{Optimal ET sequence|legend=1| 87, 111, 198gh, 212gh, 299, 323, 410, 622ef }}


[[Badness]]: 0.1425 × 10<sup>-3</sup>
Badness: 1.73 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Pages with mostly numerical content]]
[[Category:Semicanousmic]]
[[Category:Semicanousmic clan| ]] <!-- main article -->
[[Category:Semicanousmic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:35, 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 semicanousmic clan of rank-3 temperaments tempers out the semicanousma, 14641/14580 = [-2 -6 -1 0 4⟩.

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 Rank-4 temperament #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]]

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

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

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

Badness: 0.142 × 10-3

2.3.5.11.17 subgroup

Subgroup: 2.3.5.11.17

Comma list: 1089/1088, 14641/14580

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

Badness: 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

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

Badness: 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 = 1\1, ~3/2 = 702.0812, ~18/11 = 854.4115

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

Badness: 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 = 1\1, ~3/2 = 702.1682, ~18/11 = 854.5074

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

Badness: 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 = 1\1, ~3/2 = 702.1910, ~18/11 = 854.4751

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

Badness: 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 = 1\1, ~3/2 = 702.1925, ~18/11 = 854.4721

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

Badness: 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 = 1\1, ~400/231 = 951.1975, ~200/189 = 96.5851

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

Badness: 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 = 1\1, ~26/15 = 951.2316, ~55/52 = 96.6155

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

Badness: 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 = 1\1, ~26/15 = 951.2022, ~55/52 = 96.6006

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

Badness: 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 = 1\1, ~26/15 = 951.1749, ~55/52 = 96.5813

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

Badness: 1.73 × 10-3

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