Unicorn family
- 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 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, the octaphore and the hemimage comma.
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
- Smith: 0.150487
- Dirichlet: 3.530
Septimal unicorn
Subgroup: 2.3.5.7
Comma list: 126/125, 10976/10935
Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]
Optimal ET sequence: 19, 39d, 58, 77, 135c, 212c
Badness:
- Smith: 0.040913
- Dirichlet: 1.035
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 (Sintel): 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 (Sintel): 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 (with a 963¢ ~7/4 similar to that of meanpop), an alternative to 77edo that sacrifices the accuracy of prime 7 in favour of a more accurate 5/4 and 43/32.
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 (Sintel): 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]]
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