Unicorn family: Difference between revisions

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-The '''unicorn family''' tempers out the [[unicorn comma]], 1594323/1562500 = {{monzo| -2 13 -8 }}.
+The '''unicorn family''' tempers out the [[unicorn comma]], 1594323/1562500 = {{monzo| -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 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
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 {{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/243 = 62.458
+[[Optimal tuning]]s:
+* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
+* [[POTE]]: 2 = 1\1, ~250/243 = 62.458
 {{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}
+[[Badness]] (Dirichlet): 3.530
 [[Badness]]: 0.150487
-== Septimal unicorn ==
+=== 2.3.5.7 subgroup (septimal unicorn) ===
+{{ See also | The octaphore }}
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 10976/10935
+[[Comma list]]: [[126/125]], [[10976/10935]]
 {{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
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 {{Multival|legend=1| 8 13 23 2 14 17 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.278
+[[Optimal tuning]]s:
+* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
+* [[POTE]]: 2 = 1\1, ~28/27 = 62.278
+{{Optimal ET sequence|legend=1| 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|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
+[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
+{{Optimal ET sequence|legend=1| 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|legend=1| 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|legend=1| 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{{cent}}), 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|legend=1| 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|legend=1| 19, 39d, 58, 77, 135c }}
+{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}
-[[Badness]]: 0.040913
+Badness (Dirichlet): 0.514
 === Alicorn ===