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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:  
* [[WE]]: ~2 = 1200.0889¢, ~250/243 =  62.4623¢
* [[CWE]]: ~2 = 1200.0000¢, ~250/243 = 62.4494¢


{{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): 3.530


= 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 }}


[[POTE generator]]: ~28/27 = 62.279
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6949¢, ~28/27 = 62.2621¢
* [[CWE]]: ~2 = 1200.0000¢, ~28/27 = 62.2996¢


{{Val list|legend=1| 19, 39d, 58, 77, 135c }}
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}


[[Badness]]: 0.0409
[[Badness]] (Sintel): 1.035


== 11-limit ==
=== 2.3.5.7.13 subgroup ===
[[Subgroup]]: 2.3.5.7.13


[[Comma list]]: [[126/125]], [[196/195]], [[676/675]]
{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7013¢, ~26/25 = 62.2785¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3151¢
{{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]], [[196/195]], [[261/260]], [[377/375]]
{{Mapping|legend=1| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7715¢, ~26/25 = 62.2860¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3141¢
{{Optimal ET sequence|legend=1| 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]]
{{Mapping|legend=1| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6269¢, ~26/25 = 62.2584¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3008¢
{{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


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


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


=== 13-limit ===
Badness (Sintel): 1.294


==== 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¢, ~26/25 = 62.0480¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.1636¢


{{Val list|legend=1| 19, 39d, 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, 10976/10935, 385/384
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¢, ~28/27 = 62.4576¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.4252¢


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


Badness: 0.0659
Badness (Sintel): 2.180
 
=== 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
Optimal tunings:  
* WE: ~2 = 1200.5004¢, ~26/25 = 62.4603¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.4277¢


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


Badness: 0.0362
Badness (Sintel): 1.494
 
== Qilin ==


=== 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¢, ~28/27 = 62.1645¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.2199¢


Badness: 0.0414
{{Optimal ET sequence|legend=0| 58, 77, 135c, 193c, 328cc }}


=== 13-limit ===
Badness (Sintel): 1.370


==== 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¢, ~26/25 = 62.1601¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.2251¢


Badness: 0.0228
{{Optimal ET sequence|legend=0| 58, 77, 135c, 193cf, 328ccff }}


== 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¢, ~28/27 = 62.2737¢
* CWE: ~99/70 = 600.0000¢, ~28/27 = 62.3180¢


POTE generator: ~28/27 = 62.292
{{Optimal ET sequence|legend=0| 58, 96d, 154, 212ce, 366cce }}


{{Val list|legend=1| 58, 96d, 154, 212ce, 366de }}
Badness (Sintel): 1.744


Badness: 0.0528
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 126/125, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 2 4 6 8 9 10 | 0 -8 -13 -23 -20 -25 }}


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~55/39 = 599.8267¢, ~26/25 = 62.2833¢
* CWE: ~55/39 = 600.0000¢, ~26/25 = 62.3262¢


Comma list: 126/125, 196/195, 364/363, 676/675
{{Optimal ET sequence|legend=0| 58, 96d, 154, 366ccef }}


Mapping: [{{val| 2 4 6 8 9 10 }}, {{val| 0 -8 -13 -23 -20 -25 }}]
Badness (Sintel): 1.190


POTE generator: ~28/27 = 62.301
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 58, 96d, 154, 366cef }}
Comma list: 126/125, 196/195, 221/220, 243/242, 289/288


Badness: 0.0288
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}


= Rhinoceros =
Optimal tunings:
* WE: ~17/12 = 600.0715¢, ~26/25 = 62.3767¢
* CWE: ~17/12 = 600.0000¢, ~26/25 = 62.3605¢


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 58, 96d, 154 }}


[[Comma list]]: 49/48, 4375/4374
Badness (Sintel): 1.237


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -8 -13 -4 }}]
== Rhinoceros ==
[[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¢, ~21/20 = 63.0830¢
* [[CWE]]: ~2 = 1200.0000¢, ~21/20 = 62.6687¢


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


== 11-limit ==
[[Badness]] (Sintel): 2.072


=== 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¢, ~21/20 = 62.9338¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.7783¢


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


Badness: 0.0593
Badness (Sintel): 1.961
 
== 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¢, ~21/20 = 63.1591¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.8727¢


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


Badness: 0.0393
Badness (Sintel): 1.626


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