Unicorn family: Difference between revisions
extend unicorn to its logical subgroup based on interpreting the generator as a streak of equated superparticular intervals |
m →Unicorn: name change |
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== Unicorn == | == Unicorn == | ||
By noticing that the generator is very close to [[28/27]] we find the extension to the 7-limit by tempering [[ | By noticing that the generator is very close to [[28/27]] we find the extension to the 7-limit by tempering the [[octaphore]] (which finds [[~]][[9/7]] at 7 gens and [[~]][[4/3]] at 8 gens, hence its name) and [[126/125]] (finding [[~]][[6/5]] at 5 gens). From this we can observe that the most natural extension is by equating adjacent [[superparticular interval]]s, by tempering the [[square-particular]]s between them, leading to its S-expression-based comma list of {[[676/675|S26]], [[729/728|S27]], [[784/783|S28]], [[841/840|S29]]}, to which experimentation shows we can find a reasonable mapping for prime 43 at -11 gens while all other primes require either quite complex mappings (being significantly positive rather than negative) or require high error or both. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Revision as of 19:00, 3 October 2024
The unicorn family tempers out the unicorn comma, 1594323/1562500 = [-2 13 -8⟩. The canonical extension to the 7-limit is by interpreting the generator as a slightly flattened ~28/27 so that a flat ~6/5 is found at 5 generators, corresponding to tempering 126/125 and the octaphore.
Unicorn
By noticing that the generator is very close to 28/27 we find the extension to the 7-limit by tempering the octaphore (which finds ~9/7 at 7 gens and ~4/3 at 8 gens, hence its name) and 126/125 (finding ~6/5 at 5 gens). From this we can observe that the most natural extension is by equating adjacent superparticular intervals, by tempering the square-particulars between them, leading to its S-expression-based comma list of {S26, S27, S28, S29}, to which experimentation shows we can find a reasonable mapping for prime 43 at -11 gens while all other primes require either quite complex mappings (being significantly positive rather than negative) or require high error or both.
Subgroup: 2.3.5
Comma list: 1594323/1562500
Mapping: [⟨1 2 3], ⟨0 -8 -13]]
Optimal ET sequence: 19, 58, 77, 96, 173, 269
Badness (Dirichlet): 3.530
Badness: 0.150487
2.3.5.7 subgroup (septimal unicorn)
Subgroup: 2.3.5.7
Comma list: 126/125, 10976/10935
Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]
Wedgie: ⟨⟨ 8 13 23 2 14 17 ]]
Optimal ET sequence: 19, 39d, 58, 77, 135c, 212c
Badness (Dirichlet): 1.035
Badness: 0.040913
2.3.5.7.13 subgroup
Subgroup: 2.3.5.7.13
Comma list: 126/125, 351/350, 676/675
Mapping: [⟨1 2 3 4 5], ⟨0 -8 -13 -23 -25]]
Optimal tuning (CTE): 2 = 1\1, ~28/27 = 62.339
Optimal ET sequence: 19, 39df, 58, 77, 212cf
Badness (Dirichlet): 0.590
2.3.5.7.13.29 subgroup
Subgroup: 2.3.5.7.13.29
Comma list: 126/125, 729/728, 784/783, 841/840
Mapping: [⟨1 2 3 4 5 6], ⟨0 -8 -13 -23 -25 -22]]
Optimal tuning (CTE): 2 = 1\1, ~28/27 = 62.334
Optimal ET sequence: 19, 39dfj, 58, 77, 212cfn
Badness (Dirichlet): 0.487
2.3.5.7.13.29.43 subgroup
A notable tuning of unicorn not appearing in the optimal ET sequence here is 96edo using the 96d val (meaning accepting 16edo's ~7/4 of 975 ¢), which serves as a nice alternative to 77edo that sacrifices the accuracy of prime 7 in favour of more accurate other primes usually not found accurately in good rank 1 unicorn tunings.
Subgroup: 2.3.5.7.13.29.43
Comma list: 126/125, 729/728, 784/783, 841/840, 216/215
Mapping: [⟨1 2 3 4 5 6 6], ⟨0 -8 -13 -23 -25 -22 -11]]
Optimal tuning (CTE): 2 = 1\1, ~28/27 = 62.339
Optimal ET sequence: 19, 39dfj, 58, 77, 135c, 212cfn
Badness (Dirichlet): 0.514
Alicorn
Subgroup: 2.3.5.7.11
Comma list: 126/125, 540/539, 896/891
Mapping: [⟨1 2 3 4 3], ⟨0 -8 -13 -23 9]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101
Optimal ET sequence: 19, 39d, 58
Badness: 0.039156
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 196/195, 676/675
Mapping: [⟨1 2 3 4 3 5], ⟨0 -8 -13 -23 9 -25]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.119
Optimal ET sequence: 19, 39df, 58
Badness: 0.023667
Camahueto
Subgroup: 2.3.5.7.11
Comma list: 126/125, 385/384, 10976/10935
Mapping: [⟨1 2 3 4 2], ⟨0 -8 -13 -23 28]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.431
Optimal ET sequence: 19, 58e, 77, 96d, 173d
Badness: 0.065940
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 385/384, 676/675
Mapping: [⟨1 2 3 4 2 5], ⟨0 -8 -13 -23 28 -25]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.434
Optimal ET sequence: 19, 58e, 77, 96d, 173d
Badness: 0.036155
Qilin
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 10976/10935
Mapping: [⟨1 2 3 4 6], ⟨0 -8 -13 -23 -49]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.196
Optimal ET sequence: 58, 77, 135c, 193c, 328cc
Badness: 0.041426
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 196/195, 2200/2197
Mapping: [⟨1 2 3 4 6 5], ⟨0 -8 -13 -23 -49 -25]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.197
Optimal ET sequence: 58, 77, 135c, 193cf, 328ccff
Badness: 0.022842
Monocerus
Subgroup: 2.3.5.7.11
Comma list: 126/125, 243/242, 5488/5445
Mapping: [⟨2 4 6 8 9], ⟨0 -8 -13 -23 -20]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.292
Optimal ET sequence: 58, 96d, 154, 212ce, 366cce
Badness: 0.052757
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 364/363, 676/675
Mapping: [⟨2 4 6 8 9 10], ⟨0 -8 -13 -23 -20 -25]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.301
Optimal ET sequence: 58, 96d, 154, 366ccef
Badness: 0.028795
Rhinoceros
Subgroup: 2.3.5.7
Comma list: 49/48, 4375/4374
Mapping: [⟨1 2 3 3], ⟨0 -8 -13 -4]]
Wedgie: ⟨⟨ 8 13 4 2 -16 -27 ]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 62.920
Badness: 0.081864
11-limit
Subgroup: 2.3.5.7.11
Comma list: 49/48, 100/99, 126/121
Mapping: [⟨1 2 3 3 4], ⟨0 -8 -13 -4 -10]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 62.874
Badness: 0.059319
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 78/77, 100/99, 126/121
Mapping: [⟨1 2 3 3 4 4], ⟨0 -8 -13 -4 -10 -6]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 63.043
Badness: 0.039343