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 }}.  


Full 7-limit [[extension]]s of llywelyn temperament include:
== Llywelyn a.k.a. shoe ==
* [[Gorgo]], {36/35, 1029/1024} → [[Gamelismic clan #Gorgo|Gamelismic clan]]
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.
* [[Ultrapyth]], {64/63, 6860/6561} → [[Archytas clan #Ultrapyth|Archytas clan]]
* [[Buzzard]], {1728/1715, 5120/5103} → [[Vulture family]] / [[Hemifamity temperaments #Buzzard|Hemifamity temperaments]]
* [[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]]


Discussed below is the 2.5.7 subgroup temperament llywelyn aka shoe.
== Llywelyn aka shoe ==
[[Subgroup]]: 2.5.7
[[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


Mapping generators: 2, ~8/7
Scales: [[Shoe5]], [[Shoe6]], [[Shoe11]], [[Shoe16]]


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
=== 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]]: [2 8/7; 4194304/4117715]
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.


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
Discussed elsewhere are
* [[Ultrapyth]] (+64/63) → [[Archytas clan #Ultrapyth|Archytas clan]]
* ''[[Misneb]]'' (+225/224) → [[Marvel temperaments #Misneb|Marvel temperaments]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan]]


{{Val list|legend=1| 5, 11c, 16, 21, 37 }}
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 ===
=== 2.5.7.11 subgroup ===
Line 33: Line 46:
Comma list: 176/175, 1310720/1294139
Comma list: 176/175, 1310720/1294139


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


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


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 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 tuning (POTE): ~8/7 = 227.114
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}


Optimal GPV sequence: {{Val list| 16, 21, 37 }}
Badness (Sintel): 0.651


=== 2.5.7.11.13 subgroup ===
=== 2.5.7.11.13 subgroup ===
Line 48: Line 63:
Comma list: 176/175, 640/637, 847/845
Comma list: 176/175, 640/637, 847/845


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


Optimal tuning (POTE): ~8/7 = 227.108
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }}


Optimal GPV sequence: {{Val list| 16, 21, 37 }}
Badness (Sintel): 0.319


=== 2.5.7.11.13.17 subgroup ===
=== 2.5.7.11.13.17 subgroup ===
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11.13.17


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


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


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


Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 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 tuning (POTE): ~8/7 = 227.242
{{Optimal ET sequence|legend=0| 16, 21, 37, 95 }}


Optimal GPV sequence: {{Val list| 16, 21, 37 }}
Badness (Sintel): 0.458


[[Category:Llywelynsmic clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Llywelyn clan| ]] <!-- main article -->

Latest revision as of 07:36, 15 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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