Llywelynsmic clan

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.

• Gorgo (+36/35) → Gamelismic clan
• Gorgik (+2401/2400) → Breedsmic temperaments
• Semaja (+3125/3087) → Gariboh clan

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

• Ultrapyth (+64/63) → Archytas clan
• Misneb (+225/224) → Marvel temperaments
• Buzzard (+1728/1715) → Buzzardsmic clan

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