Semicanousmic clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
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[[Comma list]]: 14641/14580
[[Comma list]]: 14641/14580


{{Mapping|legend=2| 1 0 2 1 | 0 1 2 2 | 0 0 -4 -1 }}
{{Mapping|legend=2| 1 0 -2 0 | 0 1 2 2 | 0 0 4 1 }}
: mapping generators: ~2, ~3, ~18/11
: mapping generators: ~2, ~3, ~11/9


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.3342{{c}}, ~18/11 = 854.5548{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0495{{c}}, ~3/2 = 702.2793{{c}}, ~11/9 = 345.4721{{c}}
: [[error map]]: {{val| +0.049 +0.374 +0.133 -1.188 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2918{{c}}, ~11/9 = 345.4516{{c}}
: error map: {{val| 0.000 +0.337 +0.077 -1.283 }}


{{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]] (Smith): 0.142 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.18


=== 2.3.5.11.17 subgroup ===
=== 2.3.5.11.17 subgroup ===
Line 33: Line 37:
Comma list: 1089/1088, 14641/14580
Comma list: 1089/1088, 14641/14580


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


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.3427{{c}}, ~18/11 = 854.5587{{c}}
Optimal tunings:
* WE: ~2 = 1200.0291{{c}}, ~3/2 = 702.3333{{c}}, ~11/9 = 345.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3266{{c}}, ~11/9 = 345.4394{{c}}


{{Optimal ET sequence|legend=0| 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 (Smith): 0.481 × 10<sup>-3</sup>
Badness (Sintel): 1.01


=== 2.3.5.11.17.19 subgroup ===
=== 2.3.5.11.17.19 subgroup ===
Line 46: Line 52:
Comma list: 1089/1088, 1216/1215, 1445/1444
Comma list: 1089/1088, 1216/1215, 1445/1444


Subgroup-val 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 0 -6 -8 | 0 1 2 2 6 7 | 0 0 4 1 2 4 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.2987{{c}}, ~18/11 = 854.5520{{c}}
Optimal tunings:
* WE: ~2 = 1200.0556{{c}}, ~3/2 = 702.2904{{c}}, ~11/9 = 345.4565{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2728{{c}}, ~11/9 = 345.4432{{c}}


{{Optimal ET sequence|legend=0| 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 (Smith): 0.403 × 10<sup>-3</sup>
Badness (Sintel): 0.630


== Syndeute ==
== Syndeute ==
Line 59: Line 67:
[[Comma list]]: 14641/14580, 19712/19683
[[Comma list]]: 14641/14580, 19712/19683


{{Mapping|legend=1| 1 0 2 -9 1 | 0 1 2 7 2 | 0 0 -4 1 -1 }}
{{Mapping|legend=1| 1 0 -2 -8 0 | 0 1 2 7 2 | 0 0 4 -1 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.0812{{c}}, ~18/11 = 854.4115{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1265{{c}}, ~3/2 = 702.0849{{c}}, ~11/9 = 345.5759{{c}}
: [[error map]]: {{val| +0.126 +0.256 +0.159 +0.066 -1.319 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0357{{c}}, ~11/9 = 345.5568{{c}}
: error map: {{val| 0.000 +0.081 -0.015 -0.133 -1.690 }}


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


[[Badness]] (Smith): 3.26 × 10<sup>-3</sup>
[[Badness]] (Sintel): 3.91


=== 13-limit ===
=== 13-limit ===
Line 72: Line 84:
Comma list: 1573/1568, 2080/2079, 14641/14580
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 }}
Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 1 2 7 2 10 | 0 0 4 -1 1 -4 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1682{{c}}, ~18/11 = 854.5074{{c}}
Optimal tunings:
* WE: ~2 = 1200.1164{{c}}, ~3/2 = 702.1730{{c}}, ~11/9 = 854.4793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1272{{c}}, ~11/9 = 345.4623{{c}}


{{Optimal ET sequence|legend=0| 94, 111, 205, 212, 323, 535ef }}
{{Optimal ET sequence|legend=0| 94, 111, 205, 212, 323, 535ef }}


Badness (Smith): 3.01 × 10<sup>-3</sup>
Badness (Sintel): 2.82


=== 17-limit ===
=== 17-limit ===
Line 85: Line 99:
Comma list: 442/441, 561/560, 715/714, 14641/14580
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 }}
Mapping: {{mapping| 1 0 -2 -8 0 -11 -6 | 0 1 2 7 2 10 6 | 0 0 4 -1 1 -4 2 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1910{{c}}, ~18/11 = 854.4751{{c}}
Optimal tunings:
* WE: ~2 = 1200.1171{{c}}, ~3/2 = 702.1958{{c}}, ~11/9 = 345.5113{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1497{{c}}, ~11/9 = 345.4943{{c}}


{{Optimal ET sequence|legend=0| 94, 111, 205, 212g, 323 }}
{{Optimal ET sequence|legend=0| 94, 111, 205, 212g, 323 }}


Badness (Smith): 1.69 × 10<sup>-3</sup>
Badness (Sintel): 1.61


=== 19-limit ===
=== 19-limit ===
Line 98: Line 114:
Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444
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 }}
Mapping: {{mapping| 1 0 -2 -8 0 -11 -6 -8 | 0 1 2 7 2 10 6 7 | 0 0 4 -1 1 -4 2 4 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.1925{{c}}, ~18/11 = 854.4721{{c}}
Optimal tunings:
* WE: ~2 = 1200.1120{{c}}, ~3/2 = 702.1996{{c}}, ~11/9 = 345.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1543{{c}}, ~11/9 = 345.5018{{c}}


{{Optimal ET sequence|legend=0| 94, 111, 205, 212gh, 323 }}
{{Optimal ET sequence|legend=0| 94, 111, 205, 212gh, 323 }}


Badness (Smith): 1.27 × 10<sup>-3</sup>
Badness (Sintel): 1.31


== Hemireiwa ==
== Hemireiwa ==
Line 116: Line 134:
: mapping generators: ~2, ~400/231, ~200/189
: mapping generators: ~2, ~400/231, ~200/189


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~400/231 = 951.1975{{c}}, ~200/189 = 96.5851{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0073{{c}}, ~400/231 = 951.1989{{c}}, ~200/189 = 96.5797{{c}}
: [[error map]]: {{val| +0.007 +0.443 +0.020 +0.090 -1.134 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.1950{{c}}, ~200/189 = 96.5816{{c}}
: error map: {{val| 0.000 +0.435 +0.013 +0.076 -1.151 }}


{{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]] (Smith): 3.95 × 10<sup>-3</sup>
[[Badness]] (Sintel): 4.74


=== 13-limit ===
=== 13-limit ===
Line 129: Line 151:
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 = 1200.0000{{c}}, ~26/15 = 951.2316{{c}}, ~55/52 = 96.6155{{c}}
Optimal tuning:
* WE: ~2 = 1199.9814{{c}}, ~26/15 = 951.2264{{c}}, ~55/52 = 96.6277{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.2372{{c}}, ~55/52 = 96.6235{{c}}


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


Badness (Smith): 1.89 × 10<sup>-3</sup>
Badness (Sintel): 1.77


=== 17-limit ===
=== 17-limit ===
Line 142: Line 166:
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 = 1200.0000{{c}}, ~26/15 = 951.2022{{c}}, ~55/52 = 96.6006{{c}}
Optimal tuning:
* WE: ~2 = 1200.0093{{c}}, ~26/15 = 951.2058{{c}}, ~55/52 = 96.5949{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.1999{{c}}, ~55/52 = 96.5966{{c}}


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


Badness (Smith): 1.90 × 10<sup>-3</sup>
Badness (Sintel): 1.81


=== 19-limit ===
=== 19-limit ===
Line 155: Line 181:
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 = 1200.0000{{c}}, ~26/15 = 951.1749{{c}}, ~55/52 = 96.5813{{c}}
Optimal tuning:
* WE: ~2 = 1200.0492{{c}}, ~26/15 = 951.1982{{c}}, ~55/52 = 96.5541{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.1648{{c}}, ~55/52 = 96.5612{{c}}


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


Badness (Smith): 1.73 × 10<sup>-3</sup>
Badness (Sintel): 1.78


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Semicanousmic clan| ]] <!-- main article -->
[[Category:Semicanousmic clan| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 10:38, 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 0], ⟨0 1 2 2], ⟨0 0 4 1]]

mapping generators: ~2, ~3, ~11/9

Optimal tunings:

  • WE: ~2 = 1200.0495 ¢, ~3/2 = 702.2793 ¢, ~11/9 = 345.4721 ¢
error map: ⟨+0.049 +0.374 +0.133 -1.188]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2918 ¢, ~11/9 = 345.4516 ¢
error map: ⟨0.000 +0.337 +0.077 -1.283]

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

Badness (Sintel): 1.18

2.3.5.11.17 subgroup

Subgroup: 2.3.5.11.17

Comma list: 1089/1088, 14641/14580

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

Optimal tunings:

  • WE: ~2 = 1200.0291 ¢, ~3/2 = 702.3333 ¢, ~11/9 = 345.4465 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3266 ¢, ~11/9 = 345.4394 ¢

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

Badness (Sintel): 1.01

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 0 -6 -8], ⟨0 1 2 2 6 7], ⟨0 0 4 1 2 4]]

Optimal tunings:

  • WE: ~2 = 1200.0556 ¢, ~3/2 = 702.2904 ¢, ~11/9 = 345.4565 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.2728 ¢, ~11/9 = 345.4432 ¢

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

Badness (Sintel): 0.630

Syndeute

Subgroup: 2.3.5.7.11

Comma list: 14641/14580, 19712/19683

Mapping: [⟨1 0 -2 -8 0], ⟨0 1 2 7 2], ⟨0 0 4 -1 1]]

Optimal tunings:

  • WE: ~2 = 1200.1265 ¢, ~3/2 = 702.0849 ¢, ~11/9 = 345.5759 ¢
error map: ⟨+0.126 +0.256 +0.159 +0.066 -1.319]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0357 ¢, ~11/9 = 345.5568 ¢
error map: ⟨0.000 +0.081 -0.015 -0.133 -1.690]

Optimal ET sequence: 94, 111, 118, 212, 323, 441e

Badness (Sintel): 3.91

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -2 -8 0 -11], ⟨0 1 2 7 2 10], ⟨0 0 4 -1 1 -4]]

Optimal tunings:

  • WE: ~2 = 1200.1164 ¢, ~3/2 = 702.1730 ¢, ~11/9 = 854.4793 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1272 ¢, ~11/9 = 345.4623 ¢

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

Badness (Sintel): 2.82

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -2 -8 0 -11 -6], ⟨0 1 2 7 2 10 6], ⟨0 0 4 -1 1 -4 2]]

Optimal tunings:

  • WE: ~2 = 1200.1171 ¢, ~3/2 = 702.1958 ¢, ~11/9 = 345.5113 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1497 ¢, ~11/9 = 345.4943 ¢

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

Badness (Sintel): 1.61

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 -8 0 -11 -6 -8], ⟨0 1 2 7 2 10 6 7], ⟨0 0 4 -1 1 -4 2 4]]

Optimal tunings:

  • WE: ~2 = 1200.1120 ¢, ~3/2 = 702.1996 ¢, ~11/9 = 345.5202 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1543 ¢, ~11/9 = 345.5018 ¢

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

Badness (Sintel): 1.31

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 tunings:

  • WE: ~2 = 1200.0073 ¢, ~400/231 = 951.1989 ¢, ~200/189 = 96.5797 ¢
error map: ⟨+0.007 +0.443 +0.020 +0.090 -1.134]
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.1950 ¢, ~200/189 = 96.5816 ¢
error map: ⟨0.000 +0.435 +0.013 +0.076 -1.151]

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

Badness (Sintel): 4.74

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:

  • WE: ~2 = 1199.9814 ¢, ~26/15 = 951.2264 ¢, ~55/52 = 96.6277 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.2372 ¢, ~55/52 = 96.6235 ¢

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

Badness (Sintel): 1.77

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:

  • WE: ~2 = 1200.0093 ¢, ~26/15 = 951.2058 ¢, ~55/52 = 96.5949 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.1999 ¢, ~55/52 = 96.5966 ¢

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

Badness (Sintel): 1.81

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:

  • WE: ~2 = 1200.0492 ¢, ~26/15 = 951.1982 ¢, ~55/52 = 96.5541 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.1648 ¢, ~55/52 = 96.5612 ¢

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

Badness (Sintel): 1.78

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