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Gamelismic clan: Difference between revisions

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==== Subgroup extensions ====
==== Subgroup extensions ====
No-five subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-five rodan, considered below, euslendric, a 2.3.7.13-subgroup extension, baladic, a weak 2.3.7.13.17-subgroup extension, and gigapyth, a 2.3.7.85-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[Subgroup temperaments #Trisect|trisect]] in the 2.3.7.11/5 subgroup.
No-5 subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-5 rodan, euslendric, a 2.3.7.13-subgroup extension, baladic, a weak 2.3.7.13-subgroup extension, trisect, a weak 2.3.7.11/5-subgroup extension, and gigapyth, a 2.3.7.85-subgroup extension, considered in [[#Subgroup extensions_2|#Subgroup extensions]].  
 
=== Radon ===
{{See also|Chromatic pairs #Radon}}
 
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].
 
Subgroup: 2.3.7.11
 
Comma list: 896/891, 1029/1024
 
Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}
 
Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }}
 
Optimal tunings:
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}
 
{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}
 
Badness (Sintel): 0.619


== Mothra ==
== Mothra ==
Line 1,936: Line 1,915:
Badness (Sintel): 0.894
Badness (Sintel): 0.894


== Other subgroup extensions ==
== Subgroup extensions ==
=== Euslendric (2.3.7.13) ===
=== Radon (2.3.7.11) ===
Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.
{{See also|Chromatic pairs #Radon}}
 
Radon is the no-5's version of [[rodan]], equating the diatonic major third to [[14/11]].


Subgroup: 2.3.7.13
Subgroup: 2.3.7.11


Comma list: 729/728, 1029/1024
Comma list: 896/891, 1029/1024


Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}
{{Mapping|legend=2| 1 1 3 6 | 0 3 -1 -13 }}


Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}
{{Mapping|legend=4| 1 1 0 3 6 | 0 3 0 -1 -13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}
 
{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}
 
Badness (Sintel): 0.619
 
=== Euslendric (2.3.7.13) ===
Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.
 
Subgroup: 2.3.7.13
 
Comma list: 729/728, 1029/1024
 
{{Mapping|legend=2| 1 1 3 0 | 0 3 -1 19 }}
 
{{Mapping|legend=4| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}
 
Optimal tunings:
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}


{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}
{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}
Line 1,961: Line 1,961:
Comma list: 273/272, 729/728, 833/832
Comma list: 273/272, 729/728, 833/832


Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}
{{Mapping|legend=2| 1 1 3 0 0 | 0 3 -1 19 21 }}


Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}
{{Mapping|legend=4| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}


Optimal tunings:  
Optimal tunings:  
Line 1,978: Line 1,978:
Comma list: 273/272, 343/342, 513/512, 729/728
Comma list: 273/272, 343/342, 513/512, 729/728


Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}
{{Mapping|legend=2| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}


Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}
{{Mapping|legend=4| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 1,995: Line 1,995:
Comma list: 273/272, 343/342, 392/391, 513/512, 729/728
Comma list: 273/272, 343/342, 392/391, 513/512, 729/728


Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}
{{Mapping|legend=2| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}


Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}
{{Mapping|legend=4| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}


Optimal tunings:  
Optimal tunings:  
Line 2,012: Line 2,012:
Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608
Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608


Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}
{{Mapping|legend=2| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}


Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}
{{Mapping|legend=4| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}


Optimal tunings:  
Optimal tunings:  
Line 2,031: Line 2,031:
Comma list: 169/168, 1029/1024
Comma list: 169/168, 1029/1024


Subgroup-val mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }}
{{Mapping|legend=2| 2 2 6 7 | 0 3 -1 1 }}


Gencom mapping: {{mapping| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }}
{{Mapping|legend=4| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }}
: mapping generators: ~91/64, ~8/7
: mapping generators: ~91/64, ~8/7


Line 2,049: Line 2,049:
Comma list: 169/168, 273/272, 289/288
Comma list: 169/168, 273/272, 289/288


Subgroup-val mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }}
{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }}


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


Optimal tunings:  
Optimal tunings:  
Line 2,060: Line 2,060:


Badness (Sintel): 0.253
Badness (Sintel): 0.253
=== Trisect (2.3.7.11/5) ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup [[restriction]] of [[augmented family #Trisected|trisected]].
Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[gamelismic family #Portent|portent]] or [[landscape family #Landscape|landscape]] results in the [[weak extension]] known as [[kleismic family #Tritikleismic|tritikleismic]].
Subgroup: 2.3.7.11/5
Comma list: 1029/1024, 4000/3993
{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}
: mapping generators: ~44/35, ~175/121
Optimal tunings:
* Subgroup WE: ~44/35 = 400.1400{{c}}, ~175/121 = 633.9635{{c}}
* Subgroup CWE: ~44/35 = 400.0000{{c}}, ~175/121 = 633.7885{{c}}
{{Optimal ET sequence|legend=0| 15, 21, 36, 123, 159, 195, 231 }}
==== 2.3.7.11/5.13 subgroup ====
Subgroup: 2.3.7.11/5.13
Comma list: 1029/1024, 1575/1573, 2080/2079
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}
Optimal tunings:
* Subgroup WE: ~44/35 = 400.1532{{c}}, ~13/9 = 634.1597{{c}}
* Subgroup CWE: ~44/35 = 400.0000{{c}}, ~13/9 = 633.9765{{c}}
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159, 354df }}
==== 2.3.7.11/5.13.17 subgroup ====
Subgroup: 2.3.7.11/5.13.17
Comma list: 273/272, 595/594, 833/832, 1575/1573
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}
Optimal tunings:
* Subgroup WE: ~34/27 = 400.1534{{c}}, ~13/9 = 634.0635{{c}}
* Subgroup CWE: ~34/27 = 400.0000{{c}}, ~13/9 = 633.8797{{c}}
{{Optimal ET sequence|legend=0| 15g, 21fg, 36, 123, 159, 195f }}
===== Trisector =====
Subgroup: 2.3.7.11/5.13.17.19
Comma list: 210/209, 273/272, 286/285, 595/594, 833/832
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}
Optimal tunings:
* Subgroup WE: ~34/27 = 400.0514{{c}}, ~13/9 = 633.9749{{c}}
* Subgroup CWE: ~34/27 = 400.0000{{c}}, ~13/9 = 633.9126{{c}}
{{Optimal ET sequence|legend=0| 15g, 21fg, 36, 123h, 159h }}
===== 2.3.7.11/5.13.17.19.23 subgroup =====
Subgroup: 2.3.7.11/5.13.17.19.23
Comma list: 208/207, 210/209, 231/230, 273/272, 286/285, 595/594
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}
Optimal tunings:
* Subgroup WE: ~34/27 = 399.9410{{c}}, ~13/9 = 633.9438{{c}}
* Subgroup CWE: ~34/27 = 400.0000{{c}}, ~13/9 = 634.0183{{c}}
{{Optimal ET sequence|legend=0| 15g, 21fg, 36, 87, 123hi }}
===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
Subgroup: 2.3.7.11/5.13.17.19.23.29
Comma list: 208/207, 210/209, 231/230, 273/272, 286/285, 320/319, 595/594
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}
Optimal tunings:
* Subgroup WE: ~29/23 = 399.8924{{c}}, ~13/9 = 633.9310{{c}}
* Subgroup CWE: ~29/23 = 400.0000{{c}}, ~13/9 = 634.0711{{c}}
{{Optimal ET sequence|legend=0| 15g, 21fg, 36, 87, 123hi }}


=== Gigapyth (2.3.7.85) ===
=== Gigapyth (2.3.7.85) ===
Line 2,066: Line 2,149:
Comma list: 1029/1024, 7225/7203
Comma list: 1029/1024, 7225/7203


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


Optimal tunings:  
Optimal tunings:  
Line 2,079: Line 2,162:


[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Gamelismic| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Listen]]
[[Category:Listen]]
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