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The '''llywelyn clan''' tempers out the no-threes llywelyn comma, {{Monzo|22 0 -1 -7}} = [[4194304/4117715]].  
{{Technical data page}}
The '''llywelynsmic clan''' of [[regular temperament|temperaments]] tempers out the no-threes [[llywelyn comma]] ({{monzo|legend=1| 22 0 -1 -7 }}, [[ratio]]: 4194304/4117715 }}.  


== Llywelyn ==
== Llywelyn a.k.a. shoe ==
Subgroup: 2.5.7
Llywelyn a.k.a. shoe can be described as the {{nowrap| 16 & 21 }} temperament, generated by a flattened [[~]][[8/7]] whole tone, seven of which give the interval class of [[5/1|5]]. Its [[ploidacot]] is alpha-heptaseph.
 
[[Subgroup]]: 2.5.7


[[Comma list]]: 4194304/4117715
[[Comma list]]: 4194304/4117715


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
{{Mapping|legend=2| 1 1 3 | 0 7 -1 }}
 
{{Mapping|legend=3| 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
 
Scales: [[Shoe5]], [[Shoe6]], [[Shoe11]], [[Shoe16]]


Mapping generators: 2, ~8/7
=== Overview to extensions ===
Strong 7-limit [[extension]]s of llywelyn include gorgo (16 & 21), gorgik (21 & 37), and semaja (16 & 37). These temperaments are distributed into different family pages.
* ''[[Gorgo]]'' (+36/35) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[Gorgik]]'' (+2401/2400) → [[Breedsmic temperaments #Gorgik|Breedsmic temperaments]]
* ''[[Semaja]]'' (+3125/3087) → [[Gariboh clan #Semaja|Gariboh clan]]


Gencom mapping: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
The rest are weak extensions. Misneb (21 & 32) splits 8/7 into two for 15/14~16/15 each. Ultrapyth (32 & 37) splits 16/7 into two for a perfect fifth. Buzzard (53 & 58) splits 16/7 into three, each for 21/16.


[[Gencom]]: [2 8/7; 4194304/4117715]
Discussed elsewhere are
* [[Ultrapyth]] (+64/63) → [[Archytas clan #Ultrapyth|Archytas clan]]
* ''[[Misneb]]'' (+225/224) → [[Marvel temperaments #Misneb|Marvel temperaments]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan]]


[[POTE generator]]: ~8/7 = 226.910
Considered immediately below is the canonical 2.5.7.11-subgroup to 2.5.7.11.13.17-subgroup extension.


{{Val list|legend=1| 5, 11c, 16, 21, 37 }}
=== 2.5.7.11 subgroup ===
====2.5.7.11====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma]]: 176/175, 1310720/1294139
Comma list: 176/175, 1310720/1294139


[[Gencom]]: [2 8/7; 176/175 1310720/1294139]
Subgroup-val mapping: {{mapping| 1 1 3 1 | 0 7 -1 13 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1}}, {{val|0 0 7 -1 13}}]
Gencom mapping: {{mapping| 1 0 1 3 1 | 0 0 7 -1 13 }}


[[Mapping|Sval mapping]]: [{{val|1 1 3 1}}, {{val|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}}


[[Tp tuning|POL2 generator]]: ~8/7 = 227.114
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}


{{Val list|legend=1| 16, 21, 37 }}
Badness (Sintel): 0.651
====2.5.7.11.13====
 
Subgroup: 2.5.7.11
=== 2.5.7.11.13 subgroup ===
Subgroup: 2.5.7.11.13
 
Comma list: 176/175, 640/637, 847/845


[[Comma]]: 176/175, 640/637, 1304576/1294139
Subgroup-val mapping: {{mapping| 1 1 3 1 2 | 0 7 -1 13 9 }}


[[Gencom]]: [2 8/7; 176/175 640/637, 1304576/1294139]
Gencom mapping: {{mapping| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2}}, {{val|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}}


[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2}}, {{val|0 7 -1 13 9}}]
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }}


[[Tp tuning|POL2 generator]]: ~8/7 = 227.108
Badness (Sintel): 0.319


{{Val list|legend=1| 16, 21, 37 }}
=== 2.5.7.11.13.17 subgroup ===
====2.5.7.11.13.17====
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11


[[Comma]]: 176/175, 221/200, 640/637, 833/832
Comma list: 176/175, 221/220, 640/637, 847/845


[[Gencom]]: [2 8/7; 176/175 221/200, 640/637, 833/832]
Subgroup-val mapping: {{mapping| 1 1 3 1 2 2 | 0 7 -1 13 9 11 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2 2}}, {{val|0 0 7 -1 13 9 11}}]
Gencom mapping: {{mapping| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}


[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2 2}}, {{val|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}}


[[Tp tuning|POL2 generator]]: ~8/7 = 227.242
{{Optimal ET sequence|legend=0| 16, 21, 37, 95 }}


{{Val list|legend=1| 16, 21, 37 }}
Badness (Sintel): 0.458
=== Extensions ===
Full 7-limit [[extension]]s of llywelyn temperament include:
* [[Gorgo]], {36/35, 1029/1024} ([[Gamelismic clan #Gorgo|Gamelismic clan]])
* [[Ultrapyth]], {64/63, 6860/6561} ([[Archytas clan #Ultrapyth|Archytas clan]])
* [[Buzzard]], {1728/1715, 5120/5103} ([[Vulture family|Vulture]]/[[Hemifamity temperaments #Buzzard|Hemifamity]])
* [[Semaja]], {3125/3087, 6144/6125} ([[Porwell temperaments #Semaja|Porwell temperaments]])
* [[Misneb]], {225/224, 4194304/4117715} ([[Marvel temperaments #Misneb|Marvel temperaments]])
* [[Gorgik]], {2401/2400, 28672/28125} ([[Breedsmic temperaments #Gorgik|Breedsmic temperaments]])


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Llywelynsmic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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