Shibboleth family

The shibboleth family tempers out the shibboleth comma, 1953125/1889568.

Temperaments discussed elsewhere include:

• superthird, {245/243, 78125/76832} → Sensamagic clan #Superthird

Shibboleth

Subgroup: 2.3.5

Comma list: 1953125/1889568

Mapping: [⟨1 4 5], ⟨0 -9 -10]]

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 321.773
• POTE: ~2 = 1\1, ~6/5 = 321.852

Optimal ET sequence: 15, 26, 41, 179c, 220c, 261cc, 302cc

Badness: 0.227553

Superkleismic

Main article: Superkleismic

The S-expression-based comma list of superkleismic is {S5/S6, S7/S8, S10, S12(, S21)}. Through careful observation of the equivalences therein one can derive that a sharpened ~6/5 is the generator as well as the mapping of the full 13-limit.

Subgroup: 2.3.5.7

Comma list: 875/864, 1029/1024

Mapping: [⟨1 4 5 2], ⟨0 -9 -10 3]]

Wedgie: ⟨⟨ 9 10 -3 -5 -30 -35 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 321.798
• POTE: ~2 = 1\1, ~6/5 = 321.930

Optimal ET sequence: 11c, 15, 26, 41

Badness: 0.047932

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 385/384

Mapping: [⟨1 4 5 2 4], ⟨0 -9 -10 3 -2]]

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 321.815
• POTE: ~2 = 1\1, ~6/5 = 321.847

Optimal ET sequence: 11c, 15, 26, 41, 179cde, 220cde, 261ccdee

Badness: 0.025659

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 245/242

Mapping: [⟨1 4 5 2 4 8], ⟨0 -9 -10 3 -2 -16]]

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 321.986
• POTE: ~2 = 1\1, ~6/5 = 321.994

Optimal ET sequence: 11cf, 15, 26, 41

Badness: 0.021478

Gilead

Subgroup: 2.3.5.7

Comma list: 126/125, 343/324

Mapping: [⟨1 4 5 6], ⟨0 -9 -10 -12]]

Wedgie: ⟨⟨ 9 10 12 -5 -6 0 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 321.109
• POTE: ~2 = 1\1, ~6/5 = 321.423

Optimal ET sequence: 11cd, 15, 41dd, 56dd

Badness: 0.115292