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__FORCETOC__
{{Technical data page}}
The unicorn family tempers out the unicorn comma, 1594323/1562500 = |-2 13 -8>.
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 ==
Comma: 1594323/1562500
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.


POTE generator: ~250/243 = 62.458
[[Subgroup]]: 2.3.5


Map: [<1 2 3|, <0 -8 -13|]
[[Comma list]]: 1594323/1562500


EDOs: 19, 58, 77, 96, 173, 269
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}


Badness: 0.1505
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0889¢, ~250/243 =  62.4623¢
* [[CWE]]: ~2 = 1200.0000¢, ~250/243 = 62.4494¢


=Alicorn=
{{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}
Commas: 126/125, 10976/10935


POTE generator: ~28/27 = 62.279
[[Badness]] (Sintel): 3.530


Map: [<1 2 3 4|, <0 -8 -13 -23|]
== Septimal unicorn ==
{{See also| Octaphore }}


EDOs: 19, 58, 77, 96d
[[Subgroup]]: 2.3.5.7


Badness: 0.0409
[[Comma list]]: [[126/125]], [[10976/10935]]


==11-limit==
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
Commas: 126/125, 540/539, 896/891


POTE generator: ~28/27 = 62.101
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6949¢, ~28/27 = 62.2621¢
* [[CWE]]: ~2 = 1200.0000¢, ~28/27 = 62.2996¢


Map: [<1 2 3 4 3|, <0 -8 -13 -23 9|]
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}


EDOs: 19, 58
[[Badness]] (Sintel): 1.035


Badness: 0.0392
=== 2.3.5.7.13 subgroup ===
[[Subgroup]]: 2.3.5.7.13


==13-limit==
[[Comma list]]: [[126/125]], [[196/195]], [[676/675]]
Commas: 126/125, 144/143, 196/195, 676/675


POTE generator: ~28/27 = 62.119
{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}


Map: [<1 2 3 4 3 5|, <0 -8 -13 -23 9 -25|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7013¢, ~26/25 = 62.2785¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3151¢


EDOs: 19, 58
{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}


Badness: 0.0237
[[Badness]] (Sintel): 0.590


==Camahueto==
==== 2.3.5.7.13.29 subgroup ====
Commas: 126/125, 10976/10935, 385/384
[[Subgroup]]: 2.3.5.7.13.29


POTE generator: ~28/27 = 62.431
[[Comma list]]: [[126/125]], [[196/195]], [[261/260]], [[377/375]]


Map: [<1 2 3 4 2|, <0 -8 -13 -23 28|]
{{Mapping|legend=1| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}


EDOs: 19, 58, 77, 96d
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7715¢, ~26/25 = 62.2860¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3141¢


Badness: 0.0659
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 212cf }}


==13-limit==
[[Badness]] (Sintel): 0.487
Commas: 126/125, 196/195, 385/384, 676/675


POTE generator: ~28/27 = 62.434
==== 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]].


Map: [<1 2 3 4 2 5|, <0 -8 -13 -23 28 -25|]
[[Subgroup]]: 2.3.5.7.13.29.43


EDOs: 19, 58, 77, 96d
[[Comma list]]: [[126/125]], [[196/195]], [[216/215]], [[261/260]], [[377/375]]


Badness: 0.0362
{{Mapping|legend=1| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}


=Monocerus=
[[Optimal tuning]]s:
Commas: 126/125, 243/242, 5488/5445
* [[WE]]: ~2 = 1199.6269¢, ~26/25 = 62.2584¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3008¢


POTE generator: ~28/27 = 62.292
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}


Map: [<2 4 6 8 9|, <0 -8 -13 -23 -20|]
[[Badness]] (Sintel): 0.514


EDOs: 58, 96d, 154, 212ce, 366de
=== Alicorn ===
Subgroup: 2.3.5.7.11


Badness: 0.0528
Comma list: 126/125, 540/539, 896/891


==13-limit==
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}
Commas: 126/125, 196/195, 364/363, 676/675


POTE generator: ~28/27 = 62.301
Optimal tunings:  
* WE: ~2 = 1198.6510¢, ~28/27 = 62.0316¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.1435¢


Map: [<2 4 6 8 9 10|, <0 -8 -13 -23 -20 -25|]
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}


EDOs: 58, 96d, 154, 366cef
Badness (Sintel): 1.294


Badness: 0.0288
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Qilin=
Comma list: 126/125, 144/143, 196/195, 676/675
Commas: 126/125, 176/175, 10976/10935


POTE generator: ~28/27 = 62.196
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}


Map: [<1 2 3 4 6|, <0 -8 -13 -23 -49|]
Optimal tunings:  
* WE: ~2 = 1198.6298¢, ~26/25 = 62.0480¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.1636¢


EDOs: 58, 77, 135c, 193c, 328c
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}


Badness: 0.0414
Badness (Sintel): 0.978


==13-limit==
=== Camahueto ===
Commas: 126/125, 176/175, 196/195, 2200/2197
Subgroup: 2.3.5.7.11


POTE generator: ~28/27 = 62.197
Comma list: 126/125, 385/384, 10976/10935


Map: [<1 2 3 4 6 5|, <0 -8 -13 -23 -49 -25|]
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}


EDOs: 58, 77, 135c, 193cf, 328cf
Optimal tunings:  
* WE: ~2 = 1200.5186¢, ~28/27 = 62.4576¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.4252¢


Badness: 0.0228
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}


=Rhinoceros=
Badness (Sintel): 2.180
Commas: 49/48, 4375/4374


POTE generator: ~21/20 = 62.920
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<1 2 3 3]|, <0 -8 -13 -4|]
Comma list: 126/125, 196/195, 385/384, 676/675


Wedgie: <<8 13 4 2 -16 -27||
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}


EDOs: 19
Optimal tunings:  
* WE: ~2 = 1200.5004¢, ~26/25 = 62.4603¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.4277¢


Badness: 0.0819
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}


==11-limit==
Badness (Sintel): 1.494
Commas: 49/48, 100/99, 126/121


POTE generator: ~21/20 = 62.874
=== Qilin ===
Subgroup: 2.3.5.7.11


Map: [<1 2 3 3 4]|, <0 -8 -13 -4 -10|]
Comma list: 126/125, 176/175, 10976/10935


EDOs: 19
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}


Badness: 0.0593
Optimal tunings:  
* WE: ~2 = 1199.3865¢, ~28/27 = 62.1645¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.2199¢


==13-limit==
{{Optimal ET sequence|legend=0| 58, 77, 135c, 193c, 328cc }}
Commas: 49/48, 78/77, 100/99, 126/121


POTE generator: ~21/20 = 63.043
Badness (Sintel): 1.370


Map: [<1 2 3 3 4 4]|, <0 -8 -13 -4 -10 -6|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 19
Comma list: 126/125, 176/175, 196/195, 2200/2197


Badness: 0.0393
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}
[[Category:family]]
 
[[Category:temperament_family]]
Optimal tunings:
[[Category:unicorn]]
* WE: ~2 = 1199.2874¢, ~26/25 = 62.1601¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.2251¢
 
{{Optimal ET sequence|legend=0| 58, 77, 135c, 193cf, 328ccff }}
 
Badness (Sintel): 0.944
 
=== Monocerus ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 5488/5445
 
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¢
 
{{Optimal ET sequence|legend=0| 58, 96d, 154, 212ce, 366cce }}
 
Badness (Sintel): 1.744
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 2 4 6 8 9 10 | 0 -8 -13 -23 -20 -25 }}
 
Optimal tunings:
* WE: ~55/39 = 599.8267¢, ~26/25 = 62.2833¢
* CWE: ~55/39 = 600.0000¢, ~26/25 = 62.3262¢
 
{{Optimal ET sequence|legend=0| 58, 96d, 154, 366ccef }}
 
Badness (Sintel): 1.190
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 126/125, 196/195, 221/220, 243/242, 289/288
 
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}
 
Optimal tunings:
* WE: ~17/12 = 600.0715¢, ~26/25 = 62.3767¢
* CWE: ~17/12 = 600.0000¢, ~26/25 = 62.3605¢
 
{{Optimal ET sequence|legend=0| 58, 96d, 154 }}
 
Badness (Sintel): 1.237
 
== Rhinoceros ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 4375/4374
 
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.1031¢, ~21/20 = 63.0830¢
* [[CWE]]: ~2 = 1200.0000¢, ~21/20 = 62.6687¢
 
{{Optimal ET sequence|legend=1| 1c, 19 }}
 
[[Badness]] (Sintel): 2.072
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 100/99, 126/121
 
Mapping: {{mapping|  1 2 3 3 4 | 0 -8 -13 -4 -10 }}
 
Optimal tunings:
* WE: ~2 = 1201.1458¢, ~21/20 = 62.9338¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.7783¢
 
{{Optimal ET sequence|legend=0| 1ce, 19 }}
 
Badness (Sintel): 1.961
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 78/77, 100/99, 126/121
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}
 
Optimal tunings:
* WE: ~2 = 1202.2047¢, ~21/20 = 63.1591¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.8727¢
 
{{Optimal ET sequence|legend=0| 1ce, 19 }}
 
Badness (Sintel): 1.626
 
[[Category:Temperament families]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"