Unicorn family: Difference between revisions

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== Unicorn ==

Unicorn divides the [[4/3|perfect fourth]] into eight equal parts, three for [[10/9]] and five for [[6/5]]. Its [[ploidacot]] is omega-octacot.

[[Subgroup]]: 2.3.5

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== Septimal unicorn ==

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]].

The canonical extension to the 7-limit is by interpreting the generator as a very slightly flattened [[~]][[28/27]], corresponding to tempering out 126/125, the [[starling comma]], and 10976/10935, the [[hemimage comma]]. As it equates the perfect fourth with a stack of eight 28/27's, it also tempers out the [[octaphore comma]] in the [[2.3.7 subgroup]].

The hemifourth, reached by four generator steps, can be interpreted as ~[[15/13]]. This gives rise to a natural [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] extension where [[196/195]] and [[676/675]] vanish.

From 10976/10935 = ([[784/783]])<sup>2</sup>⋅([[841/840]]) we obtain an add-29 extension for free, so the generator triples as 28/27~[[29/28]]~[[30/29]], and as 196/195 vanishes already, so does [[729/728]], leading to its [[S-expression]]-based comma list of {[[676/675|S26]], [[729/728|S27]], [[784/783|S28]], [[841/840|S29]]}.

Experimentation shows we can find a reasonable mapping for prime 43 at -11 generator steps, tempering out [[216/215]]. 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~~.

In all the subgroups above, the optimum is between [[58edo]] and [[77edo]], though a notable tuning not appearing in the [[optimal ET sequence]]s here is [[96edo]] using the 96d val (with a 963{{c}} [[~]][[7/4]] similar to that of [[meanpop]]), which sacrifices the accuracy of prime 7 in favour of a more accurate 5.

=== 7-limit ===

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[[Badness]] (Sintel): 1.~~035~~

[[Badness]] (Sintel): 1.04

==== 2.3.5.7.13 subgroup ====

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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[[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~~

~~Comma list: 126/125, 196/195, 216/215, 261/260, 377/375~~

~~Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1199.6269{{c}}, ~28/27 = 62.2584{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3008{{c}}

~~{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 135c, 212cfn }}~~

~~Badness (Sintel): 0.514~~

=== Alicorn ===

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== Rhinoceros ==

Rhinoceros tempers out the [[semaphoresma]] and the [[ragisma]], identifying the generator as a [[~]][[21/20]], rather than the [[28/27]] of septimal unicorn. It may be described as the {{nowrap| 19 & 20c }} temperament.

Rhinoceros tempers out the [[semaphoresma]] and the [[ragisma]], identifying the generator as a [[~]][[21/20]], rather than the [[28/27]] of septimal unicorn. It may be described as the {{nowrap| 19 & 20c }} temperament. [[19edo]] is a good tuning, though [[39edo]] is a possible alternative.

[[Subgroup]]: 2.3.5.7

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 == Unicorn ==
+Unicorn divides the [[4/3|perfect fourth]] into eight equal parts, three for [[10/9]] and five for [[6/5]]. Its [[ploidacot]] is omega-octacot.
 [[Subgroup]]: 2.3.5
@@ Line 21: / Line 23: @@
 == Septimal unicorn ==
-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]].
+The canonical extension to the 7-limit is by interpreting the generator as a very slightly flattened [[~]][[28/27]], corresponding to tempering out 126/125, the [[starling comma]], and 10976/10935, the [[hemimage comma]]. As it equates the perfect fourth with a stack of eight 28/27's, it also tempers out the [[octaphore comma]] in the [[2.3.7 subgroup]].
+The hemifourth, reached by four generator steps, can be interpreted as ~[[15/13]]. This gives rise to a natural [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] extension where [[196/195]] and [[676/675]] vanish.
+From 10976/10935 = ([[784/783]])<sup>2</sup>⋅([[841/840]]) we obtain an add-29 extension for free, so the generator triples as 28/27~[[29/28]]~[[30/29]], and as 196/195 vanishes already, so does [[729/728]], leading to its [[S-expression]]-based comma list of {[[676/675|S26]], [[729/728|S27]], [[784/783|S28]], [[841/840|S29]]}.
+Experimentation shows we can find a reasonable mapping for prime 43 at -11 generator steps, tempering out [[216/215]]. 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.
+In all the subgroups above, the optimum is between [[58edo]] and [[77edo]], though a notable tuning not appearing in the [[optimal ET sequence]]s here is [[96edo]] using the 96d val (with a 963{{c}} [[~]][[7/4]] similar to that of [[meanpop]]), which sacrifices the accuracy of prime 7 in favour of a more accurate 5.
 === 7-limit ===
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 {{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
-[[Badness]] (Sintel): 1.035
+[[Badness]] (Sintel): 1.04
 ==== 2.3.5.7.13 subgroup ====
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 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[[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
-Comma list: 126/125, 196/195, 216/215, 261/260, 377/375
-Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}
-Optimal tunings:
-* WE: ~2 = 1199.6269{{c}}, ~28/27 = 62.2584{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3008{{c}}
-{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 135c, 212cfn }}
-Badness (Sintel): 0.514
 === Alicorn ===
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 == Rhinoceros ==
-Rhinoceros tempers out the [[semaphoresma]] and the [[ragisma]], identifying the generator as a [[~]][[21/20]], rather than the [[28/27]] of septimal unicorn. It may be described as the {{nowrap| 19 & 20c }} temperament.
+Rhinoceros tempers out the [[semaphoresma]] and the [[ragisma]], identifying the generator as a [[~]][[21/20]], rather than the [[28/27]] of septimal unicorn. It may be described as the {{nowrap| 19 & 20c }} temperament. [[19edo]] is a good tuning, though [[39edo]] is a possible alternative.
 [[Subgroup]]: 2.3.5.7