Llywelynsmic clan: Difference between revisions

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

Latest revision as of 13:22, 9 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 llywelynsmic clan of temperaments tempers out the no-threes llywelyn comma (monzo: [22 0 -1 -7⟩, ratio: 4194304/4117715 }}.

Llywelyn a.k.a. shoe

Llywelyn a.k.a. shoe can be described as the 16 & 21 temperament, generated by a flattened ~8/7 whole tone, seven of which give the interval class of 5. Its ploidacot is alpha-heptaseph.

Subgroup: 2.5.7

Comma list: 4194304/4117715

Subgroup-val mapping: [⟨1 1 3], ⟨0 7 -1]]

Gencom mapping: [⟨1 0 1 3], ⟨0 0 7 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1199.1974 ¢, ~8/7 = 226.7586 ¢
error map: ⟨-0.803 +0.194 +2.008]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.8388 ¢
error map: ⟨0.000 +1.558 +4.335]

Optimal ET sequence: 5, 11c, 16, 21, 37, 164d

Badness (Sintel): 2.06

Scales: Shoe5, Shoe6, Shoe11, Shoe16

Overview to extensions

Strong 7-limit extensions of llywelyn include gorgo (16 & 21), gorgik (21 & 37), and semaja (16 & 37). These temperaments are distributed into different family pages.

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.

Discussed elsewhere are

Considered immediately below is the canonical 2.5.7.11-subgroup to 2.5.7.11.13.17-subgroup extension.

2.5.7.11 subgroup

Subgroup: 2.5.7.11

Comma list: 176/175, 1310720/1294139

Subgroup-val mapping: [⟨1 1 3 1], ⟨0 7 -1 13]]

Gencom mapping: [⟨1 0 1 3 1], ⟨0 0 7 -1 13]]

Optimal tunings:

  • WE: ~2 = 1199.1784 ¢, ~8/7 = 226.9582 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.0438 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 3 1 2], ⟨0 7 -1 13 9]]

Gencom mapping: [⟨1 0 1 3 1 2], ⟨0 0 7 -1 13 9]]

Optimal tunings:

  • WE: ~2 = 1199.1643 ¢, ~8/7 = 226.9495 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.0315 ¢

Optimal ET sequence: 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

Subgroup-val mapping: [⟨1 1 3 1 2 2], ⟨0 7 -1 13 9 11]]

Gencom mapping: [⟨1 0 1 3 1 2 2], ⟨0 0 7 -1 13 9 11]]

Optimal tunings:

  • WE: ~2 = 1199.3162 ¢, ~8/7 = 227.1125 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.1774 ¢

Optimal ET sequence: 16, 21, 37, 95

Badness (Sintel): 0.458

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