Llywelynsmic clan: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''llywelyn clan''' tempers out the no-threes [[llywelyn comma]], {{monzo| 22 0 -1 -7 }} = 4194304/4117715.
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:
* [[Gorgo]] (+36/35) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* [[Ultrapyth]] (+64/63) → [[Archytas clan #Ultrapyth|Archytas clan]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan]]
* [[Semaja]] (+3125/3087) → [[Porwell temperaments #Semaja|Porwell temperaments]]
* [[Misneb]] (+225/224) → [[Marvel temperaments #Misneb|Marvel temperaments]]
* [[Gorgik]] (+2401/2400) → [[Breedsmic temperaments #Gorgik|Breedsmic temperaments]]
 
Discussed below is the 2.5.7 subgroup temperament llywelyn a.k.a. shoe.  


== Llywelyn a.k.a. shoe ==
== Llywelyn a.k.a. shoe ==
Llywelyn a.k.a. shoe can be described as the 16 & 21 temperament, starting with the 2.5.7 subgroup up to the no-3 17-limit.  
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
[[Subgroup]]: 2.5.7
Line 21: Line 11:
{{Mapping|legend=2| 1 1 3 | 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
: mapping generators: ~2, ~8/7


{{Mapping|legend=3| 1 0 1 3 | 0 0 7 -1 }}
[[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]]


: [[Gencom]]: [2 8/7; 4194304/4117715]
=== Overview to extensions ===
Strong 7-limit [[extension]]s of llywelyn include gorgo (16 & 21), gorgik (21 & 37), and semaja (16 & 37). 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]]): ~2 = 1\1, ~8/7 = 226.910
Temperaments discussed elsewhere are
* ''[[Gorgo]]'' (+36/35) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[Gorgik]]'' (+2401/2400) → [[Breedsmic temperaments #Gorgik|Breedsmic temperaments]]
* ''[[Semaja]]'' (+3125/3087) → [[Porwell temperaments #Semaja|Porwell temperaments]]
* [[Ultrapyth]] (+64/63) → [[Archytas clan #Ultrapyth|Archytas clan]]
* ''[[Misneb]]'' (+225/224) → [[Marvel temperaments #Misneb|Marvel temperaments]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan]]


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37, 164d }}
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 36: Line 44:
Comma list: 176/175, 1310720/1294139
Comma list: 176/175, 1310720/1294139


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


Gencom mapping: {{mapping| 1 0 1 3 1 | 0 0 7 -1 13 }}
Gencom mapping: {{mapping| 1 0 1 3 1 | 0 0 7 -1 13 }}


: gencom: [2 8/7; 176/175 1310720/1294139]
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): ~2 = 1\1, ~8/7 = 227.114
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}


{{Optimal ET sequence|legend=1| 16, 21, 37, 206cd }}
Badness (Sintel): 0.651


=== 2.5.7.11.13 subgroup ===
=== 2.5.7.11.13 subgroup ===
Line 51: Line 61:
Comma list: 176/175, 640/637, 847/845
Comma list: 176/175, 640/637, 847/845


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


Gencom mapping: {{mapping| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}
Gencom mapping: {{mapping| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}


: gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
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): ~2 = 1\1, ~8/7 = 227.108
{{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }}


{{Optimal ET sequence|legend=1| 16, 21, 37, 206cdf }}
Badness (Sintel): 0.319
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5433 cents


=== 2.5.7.11.13.17 subgroup ===
=== 2.5.7.11.13.17 subgroup ===
Line 68: Line 78:
Comma list: 176/175, 221/220, 640/637, 847/845
Comma list: 176/175, 221/220, 640/637, 847/845


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


Gencom mapping: {{mapping| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}
Gencom mapping: {{mapping| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.242
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 }}


{{Optimal ET sequence|legend=1| 16, 21, 37, 95 }}
Badness (Sintel): 0.458


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

Latest revision as of 10:14, 28 May 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). 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.

Temperaments 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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