Semicathartismic clan: Difference between revisions
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The ''' | {{Technical data page}} | ||
The '''semicathartismic clan''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[semicathartisma]] ({{monzo|legend=1| -2 -6 -1 0 4 }}, [[ratio]]: 14641/14580). | |||
For the rank-4 semicathartismic temperament, see [[Catalog of rank-4 temperaments #Semicathartismic (14641/14580)]]. | |||
== Deuteromere == | |||
[[File:Lattice Semicanousmic.png|thumb|Lattice for deutermere.]] | |||
[[File:Lattice Semicanousmic Rearranged.png|thumb|Ditto, but re-arranged into neutral thirds]] | |||
[[Subgroup]]: 2.3.5.11 | |||
[[Comma list]]: 14641/14580 | |||
{{Mapping|legend=2| 1 0 -2 0 | 0 1 2 2 | 0 0 4 1 }} | |||
: mapping generators: ~2, ~3, ~11/9 | |||
[[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 }} | ||
[[Badness]] (Sintel): 1.18 | |||
[[ | === Overview to extensions === | ||
Temperaments discussed elsewhere are: | |||
* ''[[Urania]]'' (+81/80 or 121/120) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]] | |||
* ''[[Triglav]]'' (+3025/3024 or 3136/3125) → [[Hemimean family #Triglav|Hemimean family]] | |||
* ''[[Semicathart]]'' (+9801/9800) → [[Cathartic family #Semicathart|Cathartic family]] | |||
Considered below are syndeute and hemireiwa, in addition to the add-17 extension of deuteromere. | |||
[[ | == Syndeute == | ||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 14641/14580, 19712/19683 | |||
{{Mapping|legend=1| 1 0 -2 -8 0 | 0 1 2 7 2 | 0 0 4 -1 1 }} | |||
{{ | [[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| 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: {{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{{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 }} | ||
Badness (Sintel): 2.82 | |||
=== | === 17-limit === | ||
Subgroup: 2.3.5.11.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 442/441, 561/560, 715/714, 14641/14580 | ||
Mapping: {{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{{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 | {{Optimal ET sequence|legend=0| 94, 111, 205, 212g, 323 }} | ||
Badness: | Badness (Sintel): 1.61 | ||
=== | === 19-limit === | ||
Subgroup: 2.3.5.11.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 442/441, 561/560, 715/714, 1216/1215, 1445/1444 | ||
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 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 | {{Optimal ET sequence|legend=0| 94, 111, 205, 212gh, 323 }} | ||
Badness: | Badness (Sintel): 1.31 | ||
== | == Hemireiwa == | ||
Hemireiwa is in a certain sense dual to [[semicathart]]. It splits the perfect twelfth into two, whereas semicathart splits the octave. | |||
Subgroup: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
[[Comma list]]: 14641/14580, 160083/160000 | [[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]]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 }} | |||
[[Badness]] (Sintel): 4.74 | |||
[[Badness]]: | |||
=== 13-limit === | === 13-limit === | ||
| Line 88: | Line 120: | ||
Comma list: 676/675, 1001/1000, 14641/14580 | Comma list: 676/675, 1001/1000, 14641/14580 | ||
Mapping: | Mapping: {{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{{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 | {{Optimal ET sequence|legend=0| 87, 111, 198, 323, 410 }} | ||
Badness: 1. | Badness (Sintel): 1.77 | ||
=== 17-limit === | === 17-limit === | ||
| Line 101: | Line 135: | ||
Comma list: 676/675, 715/714, 1001/1000, 14641/14580 | Comma list: 676/675, 715/714, 1001/1000, 14641/14580 | ||
Mapping: | 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: | |||
* 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 | {{Optimal ET sequence|legend=0| 87, 111, 198g, 212g, 299, 323, 410 }} | ||
Badness: 1. | Badness (Sintel): 1.81 | ||
=== 19-limit === | === 19-limit === | ||
| Line 114: | Line 150: | ||
Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444 | Comma list: 676/675, 715/714, 1001/1000, 1216/1215, 1445/1444 | ||
Mapping: | 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: | |||
* 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 }} | |||
Badness (Sintel): 1.78 | |||
== Subgroup extensions == | |||
=== Deuteromere (2.3.5.11.17) === | |||
Subgroup: 2.3.5.11.17 | |||
Comma list: 1089/1088, 14641/14580 | |||
Subgroup-val mapping: {{mapping| 1 0 -2 0 -6 | 0 1 2 2 6 | 0 0 4 1 2 }} | |||
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 }} | |||
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: {{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{{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 | {{Optimal ET sequence|legend=0| 24, 55gh, 63, 70c, 80gghh, 87, 94, 111, 118, 205, 323, 528e }} | ||
Badness: | Badness (Sintel): 0.630 | ||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category: | [[Category:Semicathartismic clan| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
Latest revision as of 13:08, 10 August 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 semicathartismic clan of rank-3 temperaments tempers out the semicathartisma (monzo: [-2 -6 -1 0 4⟩, ratio: 14641/14580).
For the rank-4 semicathartismic temperament, see Catalog of rank-4 temperaments #Semicathartismic (14641/14580).
Deuteromere


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
- 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
Overview to extensions
Temperaments discussed elsewhere are:
- Urania (+81/80 or 121/120) → Rastmic rank-3 clan
- Triglav (+3025/3024 or 3136/3125) → Hemimean family
- Semicathart (+9801/9800) → Cathartic family
Considered below are syndeute and hemireiwa, in addition to the add-17 extension of deuteromere.
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]]
- 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 semicathart. It splits the perfect twelfth into two, whereas semicathart 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
- 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
Subgroup extensions
Deuteromere (2.3.5.11.17)
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