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No-threes subgroup temperaments: Difference between revisions

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Sort by complexity of the subgroup formed by the first two primes, which seems more useful
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* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]


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[[Badness]] (Sintel): 0.156
[[Badness]] (Sintel): 0.156
=== Llywelyn a.k.a. shoe ===
Llywelyn a.k.a. shoe is the common [[restriction]] of [[gamelismic clan #Gorgo|gorgo]] (16 & 21), [[breedsmic temperaments #Gorgik|gorgik]] (21 & 37), and [[gariboh clan #Semaja|semaja]] (16 & 37). It can be described as 16 & 21, and tempers out the [[llywelynsma]]. Its generator is a flattened [[~]][[8/7]] whole tone, seven of which give the interval class of [[5/1|5]]. Its [[ploidacot]] is alpha-heptaseph. [[Weak extension]]s to the full 7-limit include [[marvel temperaments #Misneb|misneb]] (21 & 32), which splits 8/7 into two for [[15/14]]~[[16/15]] each, [[archytas clan #Ultrapyth|ultrapyth]] (32 & 37), which splits [[16/7]] into two for a [[3/2|perfect fifth]], and [[buzzardsmic clan #Septimal buzzard|buzzard]] (53 & 58), which splits 16/7 into three, each for [[21/16]].
; Scales
* [[Shoe5]], [[Shoe6]], [[Shoe11]], [[Shoe16]]
[[Subgroup]]: 2.5.7
[[Comma list]]: 4194304/4117715
{{Mapping|legend=3| 1 1 3 | 0 7 -1 }}
{{Mapping|legend=5| 1 0 1 3 | 0 0 7 -1 }}
: mapping generators: ~2, ~8/7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1974{{c}}, ~8/7 = 226.7586{{c}}
: [[error map]]: {{val| -0.803 +0.194 +2.008 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.8388{{c}}
: error map: {{val| 0.000 +1.558 +4.335 }}
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37, 164d }}
[[Badness]] (Sintel): 2.06
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Comma list: 176/175, 1310720/1294139
{{Mapping|legend=2| 1 1 3 1 | 0 7 -1 13 }}
{{Mapping|legend=4| 1 0 1 3 1 | 0 0 7 -1 13 }}
Optimal tunings:
* WE: ~2 = 1199.1784{{c}}, ~8/7 = 226.9582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0438{{c}}
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}
Badness (Sintel): 0.651
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637, 847/845
{{Mapping|legend=2| 1 1 3 1 2 | 0 7 -1 13 9 }}
{{Mapping|legend=4| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}
Optimal tunings:
* WE: ~2 = 1199.1643{{c}}, ~8/7 = 226.9495{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0315{{c}}
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }}
Badness (Sintel): 0.319
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
Comma list: 176/175, 221/220, 640/637, 847/845
{{Mapping|legend=2| 1 1 3 1 2 2 | 0 7 -1 13 9 11 }}
{{Mapping|legend=4| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}
Optimal tunings:
* WE: ~2 = 1199.3162{{c}}, ~8/7 = 227.1125{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.1774{{c}}
{{Optimal ET sequence|legend=0| 16, 21, 37, 95 }}
Badness (Sintel): 0.458
=== Mercy ===
Named by {{u|KingHyperio}} in 2021, mercy is the common restriction of [[gamelismic clan #Miracle|miracle]] and [[porwell temperaments #Countermiracle|countermiracle]]. It tempers out the [[quince comma]]. In this temperament, two generators make [[~]][[8/7]], five make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph. [[Weak extension]]s of this temperament include [[starling temperaments #Cassablanca|cassablanca]], [[octagar temperaments #Slendi|slendi]], [[ragismic microtemperaments #Quincy|quincy]], [[septischismic clan #Cotoneum|cotoneum]], and [[varunismic temperaments #Octowerck|octowerck]].
[[Subgroup]]: 2.5.7
[[Comma list]]: 823543/819200
{{Mapping|legend=3| 1 3 3 | 0 -7 -2 }}
{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}
: mapping generators: ~2, ~343/320
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2169{{c}}, ~343/320 = 116.3116{{c}}
: [[error map]]: {{val| +0.217 +0.156 -0.798 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}
: error map: {{val| 0.000 -0.117 -1.341 }}
{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
[[Badness]] (Sintel): 0.428
==== 2.5.7.13 subgroup ====
Presented here is a low-accuracy extension, which may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].
Subgroup: 2.5.7.13
Comma list: 343/338, 640/637
{{Mapping|legend=2| 1 3 3 4 | 0 -7 -2 -3 }}
{{Mapping|legend=4| 1 0 3 3 0 4 | 0 0 -7 -2 0 -3 }}
Optimal tunings:
* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}
{{Optimal ET sequence|legend=0| 10, 21, 31 }}
Badness (Sintel): 0.546
==== 2.5.7.13.17 subgroup ====
Subgroup: 2.5.7.13.17
Comma list: 170/169, 224/221, 343/338
{{Mapping|legend=2| 1 3 3 4 4 | 0 -7 -2 -3 1 }}
{{Mapping|legend=4| 1 0 3 3 0 4 4 | 0 0 -7 -2 0 -3 1 }}
Optimal tunings:
* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}
{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}
Badness (Sintel): 0.451
==== 2.5.7.13.17.19 subgroup ====
Subgroup: 2.5.7.13.17.19
Comma list: 170/169, 224/221, 343/338, 476/475
{{Mapping|legend=2| 1 3 3 4 4 3 | 0 -7 -2 -3 1 13 }}
{{Mapping|legend=4| 1 0 3 3 0 4 4 3 | 0 0 -7 -2 0 -3 1 13 }}
Optimal tunings:
* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}
{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}
Badness (Sintel): 0.759


=== Huntington ===
=== Huntington ===
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[[Comma list]]: 26/25
[[Comma list]]: 26/25


{{Mapping|legend=3| 1 2 3 | 0 1 2 }}
{{Mapping|legend=3| 1 0 -1 | 0 1 2 }}
: mapping generators, ~2, ~5/4
: mapping generators, ~2, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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=== Argument ===
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].  
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup, equating [[14/11]] to a step of [[3edo]]. It is the no-3 [[restriction]] of [[#Augment|augment]].  


[[Subgroup]]: 2.7.11
[[Subgroup]]: 2.7.11
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{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}


Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
{{Mapping|legend=4| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}


Optimal tunings:  
Optimal tunings:  
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[[Comma list]]: 1352/1331
[[Comma list]]: 1352/1331


{{Mapping|legend=3| 1 3 3 | 0 2 3 }}
{{Mapping|legend=3| 1 1 0 | 0 2 3 }}


{{Mapping|legend=5| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
{{Mapping|legend=5| 1 0 0 0 1 0 | 0 0 0 0 2 3 }}
: mapping generators, ~2, ~13/11
: mapping generators: ~2, ~26/11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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