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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]], 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.


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 }}


[[POTE generator]]: ~250/243 = 62.458
[[Optimal tuning]]s:
* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458


{{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]]:
* Smith: 0.150487
* Dirichlet: 3.530


= Alicorn =
== Septimal unicorn ==
{{see also| Starling temperaments #Alicorn }}
{{See also| Octaphore }}


Subgroup: 2.3.5.7
[[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:
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278


[[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:
* Smith: 0.040913
* Dirichlet: 1.035


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


== 11-limit ==
[[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 (Sintel): 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 (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]], [[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, 39dfj, 58, 77, 135c, 212cfn }}
Badness (Sintel): 0.514
=== 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
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101


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


Badness: 0.0392
Badness: 0.039156
 
=== 13-limit ===


==== 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


{{Val list|legend=1| 19, 39df, 58 }}
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.119


Badness: 0.0237
{{Optimal ET sequence|legend=1| 19, 39df, 58 }}


== Camahueto ==
Badness: 0.023667


=== 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


{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.431


Badness: 0.0659
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}


=== 13-limit ===
Badness: 0.065940


==== 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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.434


Badness: 0.0362
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}


== Qilin ==
Badness: 0.036155


=== 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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.196


Badness: 0.0414
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193c, 328cc }}


=== 13-limit ===
Badness: 0.041426


==== 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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.197


Badness: 0.0228
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193cf, 328ccff }}


== Monocerus ==
Badness: 0.022842


=== 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 }}
 
POTE generator: ~28/27 = 62.292


{{Val list|legend=1| 58, 96d, 154, 212ce, 366de }}
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.292


Badness: 0.0528
{{Optimal ET sequence|legend=1| 58, 96d, 154, 212ce, 366cce }}


=== 13-limit ===
Badness: 0.052757


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


Comma list: 126/125, 196/195, 364/363, 676/675
Comma list: 126/125, 196/195, 364/363, 676/675


Mapping: [{{val| 2 4 6 8 9 10 }}, {{val| 0 -8 -13 -23 -20 -25 }}]
Mapping: {{mapping| 2 4 6 8 9 10 | 0 -8 -13 -23 -20 -25 }}
 
POTE generator: ~28/27 = 62.301


{{Val list|legend=1| 58, 96d, 154, 366cef }}
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.301


Badness: 0.0288
{{Optimal ET sequence|legend=1| 58, 96d, 154, 366ccef }}


= Rhinoceros =
Badness: 0.028795


Subgroup: 2.3.5.7
== Rhinoceros ==
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -8 -13 -4 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
 
{{Multival|legend=1| 8 13 4 2 -16 -27 }}
 
[[POTE generator]]: ~21/20 = 62.920


{{Val list|legend=1| 19 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920


[[Badness]]: 0.0819
{{Optimal ET sequence|legend=1| 1c, 19 }}


== 11-limit ==
[[Badness]]: 0.081864


=== 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 tuning (POTE): ~2 = 1\1, ~21/20 = 62.874


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


Badness: 0.0593
Badness: 0.059319
 
== 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 tuning (POTE): ~2 = 1\1, ~21/20 = 63.043


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


Badness: 0.0393
Badness: 0.039343


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Unicorn]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"