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Unicorn family: Difference between revisions

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The '''unicorn family''' tempers out the [[unicorn comma]], 1594323/1562500 = {{monzo| -2 13 -8 }}.  
{{Technical data page}}
The '''unicorn family''' tempers out the [[unicorn comma]] ({{monzo|legend=1| -2 13 -8 }}, [[ratio]]: 1594323/1562500).  


= Unicorn =
== 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
[[Subgroup]]: 2.3.5


[[Comma list]]: 1594323/1562500
[[Comma list]]: 1594323/1562500


[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -8 -13 }}]
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}
: mapping generators: ~2, ~250/243


[[POTE generator]]: ~250/243 = 62.458
[[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 }}


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


[[Badness]]: 0.1505
[[Badness]] (Sintel): 3.53


= Alicorn =
== Septimal unicorn ==
{{see also| Starling temperaments #Alicorn }}
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]].


Subgroup: 2.3.5.7
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.
 
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 ===
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 10976/10935
[[Comma list]]: 126/125, 10976/10935


[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 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:
* [[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 }}


[[POTE generator]]: ~28/27 = 62.278
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}


{{Val list|legend=1| 19, 39d, 58, 77, 135c }}
[[Badness]] (Sintel): 1.04


[[Badness]]: 0.0409
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13


== 11-limit ==
Comma list: 126/125, 196/195, 676/675


Subgroup-val mapping: {{mapping| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
Optimal tunings:
* WE: ~2 = 1199.7013{{c}}, ~28/27 = 62.2785{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3151{{c}}
{{Optimal ET sequence|legend=0| 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, 196/195, 261/260, 377/375
Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}
Optimal tunings:
* WE: ~2 = 1199.7715{{c}}, ~28/27 = 62.2860{{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
=== Alicorn ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 540/539, 896/891
Comma list: 126/125, 540/539, 896/891


Mapping: [{{val| 1 2 3 4 3 }}, {{val| 0 -8 -13 -23 9 }}]
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}
 
POTE generator: ~28/27 = 62.101


{{Val list|legend=1| 19, 39d, 58 }}
Optimal tunings:
* WE: ~2 = 1198.6510{{c}}, ~28/27 = 62.0316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1435{{c}}


Badness: 0.0392
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}


=== 13-limit ===
Badness (Sintel): 1.29


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 144/143, 196/195, 676/675
Comma list: 126/125, 144/143, 196/195, 676/675


Mapping: [{{val| 1 2 3 4 3 5 }}, {{val| 0 -8 -13 -23 9 -25 }}]
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}


POTE generator: ~28/27 = 62.119
Optimal tunings:  
* WE: ~2 = 1198.6298{{c}}, ~28/27 = 62.0480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1636{{c}}


{{Val list|legend=1| 19, 39df, 58 }}
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}


Badness: 0.0237
Badness (Sintel): 0.978
 
== Camahueto ==


=== Camahueto ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 385/384, 10976/10935
Comma list: 126/125, 385/384, 10976/10935


Mapping: [{{val| 1 2 3 4 2 }}, {{val| 0 -8 -13 -23 28 }}]
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}


POTE generator: ~28/27 = 62.431
Optimal tunings:  
* WE: ~2 = 1200.5186{{c}}, ~28/27 = 62.4576{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4252{{c}}


{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}


Badness: 0.0659
Badness (Sintel): 2.18
 
=== 13-limit ===


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 196/195, 385/384, 676/675
Comma list: 126/125, 196/195, 385/384, 676/675


Mapping: [{{val| 1 2 3 4 2 5 }}, {{val| 0 -8 -13 -23 28 -25 }}]
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}
 
POTE generator: ~28/27 = 62.434


{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
Optimal tunings:
* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}


Badness: 0.0362
{{Optimal ET sequence|legend=0| 19, 77, 96d, 173d }}


== Qilin ==
Badness (Sintel): 1.49


=== Qilin ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 10976/10935
Comma list: 126/125, 176/175, 10976/10935


Mapping: [{{val| 1 2 3 4 6 }}, {{val| 0 -8 -13 -23 -49 }}]
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
 
POTE generator: ~28/27 = 62.196


{{Val list|legend=1| 58, 77, 135c, 193c, 328c }}
Optimal tunings:
* WE: ~2 = 1199.3865{{c}}, ~28/27 = 62.1645{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}


Badness: 0.0414
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193c }}


=== 13-limit ===
Badness (Sintel): 1.37


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 176/175, 196/195, 2200/2197
Comma list: 126/125, 176/175, 196/195, 2200/2197


Mapping: [{{val| 1 2 3 4 6 5 }}, {{val| 0 -8 -13 -23 -49 -25 }}]
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}
 
POTE generator: ~28/27 = 62.197


{{Val list|legend=1| 58, 77, 135c, 193cf, 328cf }}
Optimal tunings:
* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}


Badness: 0.0228
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193cf }}


== Monocerus ==
Badness (Sintel): 0.944


=== Monocerus ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 243/242, 5488/5445
Comma list: 126/125, 243/242, 5488/5445


Mapping: [{{val| 2 4 6 8 9 }}, {{val| 0 -8 -13 -23 -20 }}]
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8223{{c}}, ~28/27 = 62.2737{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}
 
{{Optimal ET sequence|legend=0| 38d, 58, 96d, 154, 212ce }}
 
Badness (Sintel): 1.74


POTE generator: ~28/27 = 62.292
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 58, 96d, 154, 212ce, 366de }}
Comma list: 126/125, 196/195, 243/242, 364/363


Badness: 0.0528
Mapping: {{mapping| 2 4 6 8 9 10 | 0 -8 -13 -23 -20 -25 }}


=== 13-limit ===
Optimal tunings:
* WE: ~55/39 = 599.8267{{c}}, ~28/27 = 62.2833{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~28/27 = 62.3262{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}


Comma list: 126/125, 196/195, 364/363, 676/675
Badness (Sintel): 1.19


Mapping: [{{val| 2 4 6 8 9 10 }}, {{val| 0 -8 -13 -23 -20 -25 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~28/27 = 62.301
Comma list: 126/125, 196/195, 221/220, 243/242, 289/288


{{Val list|legend=1| 58, 96d, 154, 366cef }}
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}


Badness: 0.0288
Optimal tunings:  
* WE: ~17/12 = 600.0715{{c}}, ~28/27 = 62.3767{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/27 = 62.3605{{c}}


= Rhinoceros =
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.24


[[Comma list]]: 49/48, 4375/4374
== 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.


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -8 -13 -4 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 8 13 4 2 -16 -27 }}
[[Comma list]]: 49/48, 4375/4374


[[POTE generator]]: ~21/20 = 62.920
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}


{{Val list|legend=1| 19 }}
[[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]]: 0.0819
{{Optimal ET sequence|legend=1| 1c, …, 18bcd, 19 }}


== 11-limit ==
[[Badness]] (Sintel): 2.07


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 100/99, 126/121
Comma list: 49/48, 100/99, 126/121


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -8 -13 -4 -10 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -8 -13 -4 -10 }}


POTE generator: ~21/20 = 62.874
Optimal tunings:  
* WE: ~2 = 1201.1458{{c}}, ~21/20 = 62.9338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}


{{Val list|legend=1| 19 }}
{{Optimal ET sequence|legend=0| 1ce, 18bcd, 19 }}


Badness: 0.0593
Badness (Sintel): 1.96
 
== 13-limit ==


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 78/77, 100/99, 126/121
Comma list: 49/48, 78/77, 100/99, 126/121


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -8 -13 -4 -10 -6 }}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}


POTE generator: ~21/20 = 63.043
Optimal tunings:  
* WE: ~2 = 1202.2047{{c}}, ~21/20 = 63.1591{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.8727{{c}}


{{Val list|legend=1| 19 }}
{{Optimal ET sequence|legend=0| 1ce, 18bcdf, 19 }}


Badness: 0.0393
Badness (Sintel): 1.63


[[Category:Regular temperament theory]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Unicorn]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"