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


== 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


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


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


{{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}
{{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}
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== Septimal unicorn ==
== Septimal unicorn ==
{{See also| Octaphore }}
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]].


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.
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6949¢, ~28/27 = 62.2621¢
* [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}
* [[CWE]]: ~2 = 1200.0000¢, ~28/27 = 62.2996¢
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}


{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
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[[Badness]] (Sintel): 1.035
[[Badness]] (Sintel): 1.035


=== 2.3.5.7.13 subgroup ===
==== 2.3.5.7.13 subgroup ====
[[Subgroup]]: 2.3.5.7.13
Subgroup: 2.3.5.7.13


[[Comma list]]: [[126/125]], [[196/195]], [[676/675]]
Comma list: 126/125, 196/195, 676/675


{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
Subgroup-val mapping: {{mapping| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.7013¢, ~26/25 = 62.2785¢
* WE: ~2 = 1199.7013{{c}}, ~26/25 = 62.2785{{c}}
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3151¢
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3151{{c}}


{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}
{{Optimal ET sequence|legend=0| 19, 39df, 58, 77, 212cf }}


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


==== 2.3.5.7.13.29 subgroup ====
==== 2.3.5.7.13.29 subgroup ====
[[Subgroup]]: 2.3.5.7.13.29
Subgroup: 2.3.5.7.13.29


[[Comma list]]: [[126/125]], [[196/195]], [[261/260]], [[377/375]]
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 }}
Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.7715¢, ~26/25 = 62.2860¢
* [[WE]]: ~2 = 1199.7715{{c}}, ~26/25 = 62.2860{{c}}
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3141¢
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/25 = 62.3141{{c}}


{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 212cf }}
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}


[[Badness]] (Sintel): 0.487
Badness (Sintel): 0.487


==== 2.3.5.7.13.29.43 subgroup ====
==== 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]].
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
Subgroup: 2.3.5.7.13.29.43


[[Comma list]]: [[126/125]], [[196/195]], [[216/215]], [[261/260]], [[377/375]]
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 }}
Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.6269¢, ~26/25 = 62.2584¢
* WE: ~2 = 1199.6269{{c}}, ~26/25 = 62.2584{{c}}
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3008¢
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3008{{c}}


{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 135c, 212cfn }}


[[Badness]] (Sintel): 0.514
Badness (Sintel): 0.514


=== Alicorn ===
=== Alicorn ===
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.6510¢, ~28/27 = 62.0316¢
* WE: ~2 = 1198.6510{{c}}, ~28/27 = 62.0316{{c}}
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.1435¢
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1435{{c}}


{{Optimal ET sequence|legend=0| 19, 39d, 58 }}
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.6298¢, ~26/25 = 62.0480¢
* WE: ~2 = 1198.6298{{c}}, ~26/25 = 62.0480{{c}}
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.1636¢
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.1636{{c}}


{{Optimal ET sequence|legend=0| 19, 39df, 58 }}
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5186¢, ~28/27 = 62.4576¢
* WE: ~2 = 1200.5186{{c}}, ~28/27 = 62.4576{{c}}
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.4252¢
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4252{{c}}


{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5004¢, ~26/25 = 62.4603¢
* WE: ~2 = 1200.5004{{c}}, ~26/25 = 62.4603{{c}}
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.4277¢
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.4277{{c}}


{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.3865¢, ~28/27 = 62.1645¢
* WE: ~2 = 1199.3865{{c}}, ~28/27 = 62.1645{{c}}
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.2199¢
* 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| 58, 77, 135c, 193c, 328cc }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.2874¢, ~26/25 = 62.1601¢
* WE: ~2 = 1199.2874{{c}}, ~26/25 = 62.1601{{c}}
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.2251¢
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.2251{{c}}


{{Optimal ET sequence|legend=0| 58, 77, 135c, 193cf, 328ccff }}
{{Optimal ET sequence|legend=0| 58, 77, 135c, 193cf, 328ccff }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 599.8223¢, ~28/27 = 62.2737¢
* WE: ~99/70 = 599.8223{{c}}, ~28/27 = 62.2737{{c}}
* CWE: ~99/70 = 600.0000¢, ~28/27 = 62.3180¢
* 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| 58, 96d, 154, 212ce, 366cce }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~55/39 = 599.8267¢, ~26/25 = 62.2833¢
* WE: ~55/39 = 599.8267{{c}}, ~26/25 = 62.2833{{c}}
* CWE: ~55/39 = 600.0000¢, ~26/25 = 62.3262¢
* CWE: ~55/39 = 600.0000{{c}}, ~26/25 = 62.3262{{c}}


{{Optimal ET sequence|legend=0| 58, 96d, 154, 366ccef }}
{{Optimal ET sequence|legend=0| 58, 96d, 154, 366ccef }}
Line 211: Line 213:


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


{{Optimal ET sequence|legend=0| 58, 96d, 154 }}
{{Optimal ET sequence|legend=0| 58, 96d, 154 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.1031¢, ~21/20 = 63.0830¢
* [[WE]]: ~2 = 1203.1031{{c}}, ~21/20 = 63.0830{{c}}
* [[CWE]]: ~2 = 1200.0000¢, ~21/20 = 62.6687¢
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 62.6687{{c}}


{{Optimal ET sequence|legend=1| 1c, 19 }}
{{Optimal ET sequence|legend=1| 1c, 19 }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.1458¢, ~21/20 = 62.9338¢
* WE: ~2 = 1201.1458{{c}}, ~21/20 = 62.9338{{c}}
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.7783¢
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}


{{Optimal ET sequence|legend=0| 1ce, 19 }}
{{Optimal ET sequence|legend=0| 1ce, 19 }}
Line 256: Line 258:


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


{{Optimal ET sequence|legend=0| 1ce, 19 }}
{{Optimal ET sequence|legend=0| 1ce, 19 }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"