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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]].
{{Technical data page}}
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]], the [[octaphore]] and the [[hemimage comma]].


== Unicorn ==
== 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.
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
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{{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}
{{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}


[[Badness]] (Dirichlet): 3.530
[[Badness]]:
* Smith: 0.150487
* Dirichlet: 3.530


[[Badness]]: 0.150487
== Septimal unicorn ==
 
{{See also| Octaphore }}
=== 2.3.5.7 subgroup (septimal unicorn) ===
{{ See also | The octaphore }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
{{Multival|legend=1| 8 13 23 2 14 17 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:
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{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}


Badness (Dirichlet): 1.035
Badness:  
 
* Smith: 0.040913
Badness: 0.040913
* Dirichlet: 1.035


==== 2.3.5.7.13 subgroup ====
=== 2.3.5.7.13 subgroup ===
[[Subgroup]]: 2.3.5.7.13
[[Subgroup]]: 2.3.5.7.13


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{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}
{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}


Badness (Dirichlet): 0.590
Badness (Sintel): 0.590


==== 2.3.5.7.13.29 subgroup ====
==== 2.3.5.7.13.29 subgroup ====
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{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 212cfn }}
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 212cfn }}


Badness (Dirichlet): 0.487
Badness (Sintel): 0.487


==== 2.3.5.7.13.29.43 subgroup ====
==== 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.
A notable tuning of unicorn not appearing in the [[optimal ET sequence]] here is [[96edo]] using the 96d val (with a 963[[cent|¢]] [[~]][[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
[[Subgroup]]: 2.3.5.7.13.29.43
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{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}


Badness (Dirichlet): 0.514
Badness (Sintel): 0.514


=== Alicorn ===
=== Alicorn ===
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{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
{{Multival|legend=1| 8 13 4 2 -16 -27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"