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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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0889{{c}}, ~250/243 = 62.4623{{c}}

: [[error map]]: {{val| +0.089 -1.476 +1.943 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 62.4494{{c}}

: error map: {{val| 0.000 -1.550 +1.844 }}

[[Badness]] (Sintel): 3.~~530~~

[[Badness]] (Sintel): 3.53

== 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 tuning]]s:

* [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}

: [[error map]]: {{val| -0.305 -0.662 +3.364 -2.074 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}

: error map: {{val| 0.000 -0.352 +3.792 -1.717 }}

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

[[Badness]] (Sintel): 1.04

==== 2.3.5.7.13 subgroup ====

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Optimal tunings:

* WE: ~2 = 1199.7013{{c}}, ~26/25 = 62.2785{{c}}

* WE: ~2 = 1199.7013{{c}}, ~28/27 = 62.2785{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3151{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3151{{c}}

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Optimal tunings:

* [[WE]]: ~2 = 1199.7715{{c}}, ~26/25 = 62.2860{{c}}

* WE: ~2 = 1199.7715{{c}}, ~28/27 = 62.2860{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/25 = 62.3141{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3141{{c}}

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}}, ~26/25 = 62.2584{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3008{{c}}

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

~~Badness (Sintel): 0.514~~

=== Alicorn ===

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Badness (Sintel): 1.~~294~~

Badness (Sintel): 1.29

==== 13-limit ====

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Optimal tunings:

* WE: ~2 = 1198.6298{{c}}, ~26/25 = 62.0480{{c}}

* WE: ~2 = 1198.6298{{c}}, ~28/27 = 62.0480{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.1636{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1636{{c}}

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Badness (Sintel): 2.~~180~~

Badness (Sintel): 2.18

==== 13-limit ====

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Optimal tunings:

* WE: ~2 = 1200.5004{{c}}, ~26/25 = 62.4603{{c}}

* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.4277{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}

Badness (Sintel): 1.~~494~~

Badness (Sintel): 1.49

=== Qilin ===

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* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}

{{Optimal ET sequence|legend=0| 58~~, 77~~, 135c, 193c~~, 328cc~~ }}

Badness (Sintel): 1.~~370~~

Badness (Sintel): 1.37

==== 13-limit ====

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Optimal tunings:

* WE: ~2 = 1199.2874{{c}}, ~26/25 = 62.1601{{c}}

* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.2251{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}

{{Optimal ET sequence|legend=0| 58~~, 77~~, 135c, 193cf~~, 328ccff~~ }}

Badness (Sintel): 0.944

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* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}

{{Optimal ET sequence|legend=0| 58, 96d, 154, 212ce~~, 366cce~~ }}

Badness (Sintel): 1.~~744~~

Badness (Sintel): 1.74

==== 13-limit ====

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Optimal tunings:

* WE: ~55/39 = 599.8267{{c}}, ~26/25 = 62.2833{{c}}

* WE: ~55/39 = 599.8267{{c}}, ~28/27 = 62.2833{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~26/25 = 62.3262{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~28/27 = 62.3262{{c}}

Badness (Sintel): 1.~~190~~

Badness (Sintel): 1.19

==== 17-limit ====

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Optimal tunings:

* WE: ~17/12 = 600.0715{{c}}, ~26/25 = 62.3767{{c}}

* WE: ~17/12 = 600.0715{{c}}, ~28/27 = 62.3767{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~26/25 = 62.3605{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~28/27 = 62.3605{{c}}

Badness (Sintel): 1.~~237~~

Badness (Sintel): 1.24

== 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. [[19edo]] is a good tuning, though [[39edo]] is a possible alternative.

[[Subgroup]]: 2.3.5.7

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1203.1031{{c}}, ~21/20 = 63.0830{{c}}

: [[error map]]: {{val| +3.103 -0.412 +2.917 -11.848 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 62.6687{{c}}

: error map: {{val| 0.000 -3.305 -1.007 -19.501 }}

[[Badness]] (Sintel): 2.~~072~~

[[Badness]] (Sintel): 2.07

=== 11-limit ===

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* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}

Badness (Sintel): 1.~~961~~

Badness (Sintel): 1.96

=== 13-limit ===

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* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.8727{{c}}

Badness (Sintel): 1.~~626~~

Badness (Sintel): 1.63

[[Category:Unicorn family| ]]

[[Category:Temperament families]]

[[Category:~~Unicorn family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 3: / Line 3: @@
 == 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 12: / Line 14: @@
 [[Optimal tuning]]s:
 * [[WE]]: ~2 = 1200.0889{{c}}, ~250/243 =  62.4623{{c}}
+: [[error map]]: {{val| +0.089 -1.476 +1.943 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 62.4494{{c}}
+: error map: {{val| 0.000 -1.550 +1.844 }}
 {{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}
-[[Badness]] (Sintel): 3.530
+[[Badness]] (Sintel): 3.53
 == 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 ===
@@ Line 32: / Line 42: @@
 [[Optimal tuning]]s:
 * [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}
+: [[error map]]: {{val| -0.305 -0.662 +3.364 -2.074 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}
+: error map: {{val| 0.000 -0.352 +3.792 -1.717 }}
 {{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 ====
@@ Line 46: / Line 58: @@
 Optimal tunings:
-* WE: ~2 = 1199.7013{{c}}, ~26/25 = 62.2785{{c}}
+* WE: ~2 = 1199.7013{{c}}, ~28/27 = 62.2785{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3151{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3151{{c}}
 {{Optimal ET sequence|legend=0| 19, 39df, 58, 77, 212cf }}
@@ Line 61: / Line 73: @@
 Optimal tunings:
-* [[WE]]: ~2 = 1199.7715{{c}}, ~26/25 = 62.2860{{c}}
+* WE: ~2 = 1199.7715{{c}}, ~28/27 = 62.2860{{c}}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/25 = 62.3141{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3141{{c}}
 {{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}
 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}}, ~26/25 = 62.2584{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3008{{c}}
-{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 135c, 212cfn }}
-Badness (Sintel): 0.514
 === Alicorn ===
@@ Line 98: / Line 93: @@
 {{Optimal ET sequence|legend=0| 19, 39d, 58 }}
-Badness (Sintel): 1.294
+Badness (Sintel): 1.29
 ==== 13-limit ====
@@ Line 108: / Line 103: @@
 Optimal tunings:
-* WE: ~2 = 1198.6298{{c}}, ~26/25 = 62.0480{{c}}
+* WE: ~2 = 1198.6298{{c}}, ~28/27 = 62.0480{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.1636{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1636{{c}}
 {{Optimal ET sequence|legend=0| 19, 39df, 58 }}
@@ Line 128: / Line 123: @@
 {{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
-Badness (Sintel): 2.180
+Badness (Sintel): 2.18
 ==== 13-limit ====
@@ Line 138: / Line 133: @@
 Optimal tunings:
-* WE: ~2 = 1200.5004{{c}}, ~26/25 = 62.4603{{c}}
+* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.4277{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}
-{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
+{{Optimal ET sequence|legend=0| 19, 77, 96d, 173d }}
-Badness (Sintel): 1.494
+Badness (Sintel): 1.49
 === Qilin ===
@@ Line 156: / Line 151: @@
 * CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}
-{{Optimal ET sequence|legend=0| 58, 77, 135c, 193c, 328cc }}
+{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193c }}
-Badness (Sintel): 1.370
+Badness (Sintel): 1.37
 ==== 13-limit ====
@@ Line 168: / Line 163: @@
 Optimal tunings:
-* WE: ~2 = 1199.2874{{c}}, ~26/25 = 62.1601{{c}}
+* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.2251{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}
-{{Optimal ET sequence|legend=0| 58, 77, 135c, 193cf, 328ccff }}
+{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193cf }}
 Badness (Sintel): 0.944
@@ Line 186: / Line 181: @@
 * CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}
-{{Optimal ET sequence|legend=0| 58, 96d, 154, 212ce, 366cce }}
+{{Optimal ET sequence|legend=0| 38d, 58, 96d, 154, 212ce }}
-Badness (Sintel): 1.744
+Badness (Sintel): 1.74
 ==== 13-limit ====
@@ Line 198: / Line 193: @@
 Optimal tunings:
-* WE: ~55/39 = 599.8267{{c}}, ~26/25 = 62.2833{{c}}
+* WE: ~55/39 = 599.8267{{c}}, ~28/27 = 62.2833{{c}}
-* CWE: ~55/39 = 600.0000{{c}}, ~26/25 = 62.3262{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~28/27 = 62.3262{{c}}
-{{Optimal ET sequence|legend=0| 58, 96d, 154, 366ccef }}
+{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}
-Badness (Sintel): 1.190
+Badness (Sintel): 1.19
 ==== 17-limit ====
@@ Line 213: / Line 208: @@
 Optimal tunings:
-* WE: ~17/12 = 600.0715{{c}}, ~26/25 = 62.3767{{c}}
+* WE: ~17/12 = 600.0715{{c}}, ~28/27 = 62.3767{{c}}
-* CWE: ~17/12 = 600.0000{{c}}, ~26/25 = 62.3605{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~28/27 = 62.3605{{c}}
-{{Optimal ET sequence|legend=0| 58, 96d, 154 }}
+{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}
-Badness (Sintel): 1.237
+Badness (Sintel): 1.24
 == 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. [[19edo]] is a good tuning, though [[39edo]] is a possible alternative.
 [[Subgroup]]: 2.3.5.7
@@ Line 229: / Line 226: @@
 [[Optimal tuning]]s:
 * [[WE]]: ~2 = 1203.1031{{c}}, ~21/20 = 63.0830{{c}}
+: [[error map]]: {{val| +3.103 -0.412 +2.917 -11.848 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 62.6687{{c}}
+: error map: {{val| 0.000 -3.305 -1.007 -19.501 }}
-{{Optimal ET sequence|legend=1| 1c, 19 }}
+{{Optimal ET sequence|legend=1| 1c, …, 18bcd, 19 }}
-[[Badness]] (Sintel): 2.072
+[[Badness]] (Sintel): 2.07
 === 11-limit ===
@@ Line 246: / Line 245: @@
 * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}
-{{Optimal ET sequence|legend=0| 1ce, 19 }}
+{{Optimal ET sequence|legend=0| 1ce, 18bcd, 19 }}
-Badness (Sintel): 1.961
+Badness (Sintel): 1.96
 === 13-limit ===
@@ Line 261: / Line 260: @@
 * CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.8727{{c}}
-{{Optimal ET sequence|legend=0| 1ce, 19 }}
+{{Optimal ET sequence|legend=0| 1ce, 18bcdf, 19 }}
-Badness (Sintel): 1.626
+Badness (Sintel): 1.63
+[[Category:Unicorn family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Unicorn family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]