Octagar temperaments
Octagar temperaments temper out the octagar comma, [5 -4 3 -2⟩ = 4000/3969.
Temperaments discussed elsewhere include garibaldi, augene, octacot, quartonic, superkleismic, injera, bidia, hemikleismic, pluto, nautilus, roman, ripple, and hamity.
Slendi
Subgroup: 2.3.5.7
Comma list: 4000/3969, 84035/82944
Mapping: [⟨1 2 3 3], ⟨0 -17 -28 -8]]
Wedgie: ⟨⟨ 17 28 8 5 -35 -60 ]]
POTE generator: ~49/48 = 29.181
Badness: 0.142903
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 245/242, 46656/46585
Mapping: [⟨1 2 3 3 4], ⟨0 -17 -28 -8 -22]]
POTE generator: ~49/48 = 29.180
Vals: Template:Val list
Badness: 0.063217
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 144/143, 245/242, 343/338
Mapping: [⟨1 2 3 3 4 4], ⟨0 -17 -28 -8 -22 -1]]
POTE generator: ~49/48 = 29.125
Vals: Template:Val list
Badness: 0.043644
Slithy
Subgroup: 2.3.5.7
Comma list: 4000/3969, 420175/419904
Mapping: [⟨1 3 5 4], ⟨0 -19 -36 -16]]
Wedgie: ⟨⟨ 19 36 16 13 -28 -64 ]]
POTE generator: ~21/20 = 89.282
Badness: 0.107482
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 2200/2187, 4375/4356
Mapping: [⟨1 3 5 4 8], ⟨0 -19 -36 -16 -61]]
POTE generator: ~21/20 = 89.291
Vals: Template:Val list
Badness: 0.045895
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351, 540/539, 1575/1573
Mapping: [⟨1 3 5 4 8 4], ⟨0 -19 -36 -16 -61 -4]]
POTE generator: ~21/20 = 89.286
Vals: Template:Val list
Badness: 0.028492
Quintupole
- See also: 16ed5/2 #Regular temperaments or 34ed7 #Regular temperaments
The quintupole temperament tempers out the octagar comma and the mistisma (458752/455625) in the 7-limit; 896/891 (pentacircle), 1375/1372 (moctdel), and 4375/4356 (fantares, luluzoquadyo) in the 11-limit. The word "quintupole" means five poles, but also a play on the words "quintuple" and "polypyth". It is so named because the generator is 1/5 of the polypyth fourth (~4/3, about 495.8 cents). Proper pronunciation of the word "quintupole" is /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". Not to be confused with quintapole temperament (12&85).
Subgroup: 2.3.5.7
Comma list: 4000/3969, 458752/455625
Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]
Wedgie: ⟨⟨ 5 -16 -34 -37 -68 -34 ]]
POTE generator: ~135/128 = 99.175
Optimal ET sequence: 12, 97, 109, 121
Badness: 0.111620
11-limit
Subgroup: 2.3.5.7.11
Comma list: 896/891, 1375/1372, 4375/4356
Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]
POTE generator: ~35/33 = 99.156
Vals: Template:Val list
Badness: 0.056501
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 625/624, 2704/2695
Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]
POTE generator: ~35/33 = 99.165
Vals: Template:Val list
Badness: 0.038431
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]
POTE generator: ~18/17 = 99.172
Vals: Template:Val list
Badness: 0.028721
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374
Mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]
POTE generator: ~18/17 = 99.164
Vals: Template:Val list
Badness: 0.023818