Llywelynsmic clan: Difference between revisions
Units, misc. cleanup |
+ badnesses. Adopt WE & CWE tunings |
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== Llywelyn a.k.a. shoe == | == Llywelyn a.k.a. shoe == | ||
Llywelyn a.k.a. shoe can be described as the {{nowrap| 16 & 21 }} temperament. | 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 10: | Line 10: | ||
{{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|legend=3| 1 0 1 3 | 0 0 7 -1 }} | ||
: mapping generators: ~2, ~8/7 | |||
[[Optimal tuning]] | [[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 }} | {{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37, 164d }} | ||
[[Badness]] (Sintel): 2.06 | |||
=== Overview to extensions === | === Overview to extensions === | ||
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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 }} | ||
Optimal | 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 }} | {{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }} | ||
Badness (Sintel): 0.651 | |||
=== 2.5.7.11.13 subgroup === | === 2.5.7.11.13 subgroup === | ||
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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 }} | ||
Optimal | 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 }} | {{Optimal ET sequence|legend=0| 16, 21, 37, 206cdf, 243cdf, …, 354ccddff }} | ||
Badness (Sintel): 0.319 | |||
=== 2.5.7.11.13.17 subgroup === | === 2.5.7.11.13.17 subgroup === | ||
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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 | 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=0| 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]] | ||
Revision as of 09:56, 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
- 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
Overview to extensions
Full 7-limit extensions of llywelyn temperament include:
- Gorgo (+36/35) → Gamelismic clan
- Ultrapyth (+64/63) → Archytas clan
- Buzzard (+1728/1715) → Buzzardsmic clan
- Semaja (+3125/3087) → Porwell temperaments
- Misneb (+225/224) → Marvel temperaments
- Gorgik (+2401/2400) → Breedsmic temperaments
Discussed 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