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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 }}. 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]].
{{Technical data page}}
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.
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
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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:  
* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
* [[WE]]: ~2 = 1200.0889{{c}}, ~250/243 =  62.4623{{c}}
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458
: [[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 }}


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


[[Badness]]:
[[Badness]] (Sintel): 3.53
* Smith: 0.150487
* Dirichlet: 3.530


== Septimal unicorn ==
== Septimal unicorn ==
{{See also| Octaphore }}
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]].


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
[[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 }}


{{Multival|legend=1| 8 13 23 2 14 17 }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}
[[Optimal tuning]]s:
: [[error map]]: {{val| -0.305 -0.662 +3.364 -2.074 }}
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278
: error map: {{val| 0.000 -0.352 +3.792 -1.717 }}


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


Badness:
[[Badness]] (Sintel): 1.04
* Smith: 0.040913
* Dirichlet: 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]], [[351/350]], [[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]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
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=1| 19, 39df, 58, 77, 212cf }}
{{Optimal ET sequence|legend=0| 19, 39df, 58, 77, 212cf }}


Badness (Dirichlet): 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]], [[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 (Dirichlet): 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 (meaning accepting [[16edo]]'s [[~]][[7/4]] of 975{{cent}}), which serves as a nice alternative to [[77edo]] that sacrifices the accuracy of prime 7 in favour of more accurate other primes usually not found accurately in good rank 1 unicorn tunings.
 
[[Subgroup]]: 2.3.5.7.13.29.43


[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]], 216/215
Comma list: 126/125, 196/195, 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 | 0 -8 -13 -23 -25 -22 }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
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=1| 19, 39dfj, 58, 77, 135c, 212cfn }}
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}


Badness (Dirichlet): 0.514
Badness (Sintel): 0.487


=== Alicorn ===
=== Alicorn ===
Line 89: Line 87:
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101
Optimal tunings:
* WE: ~2 = 1198.6510{{c}}, ~28/27 = 62.0316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1435{{c}}


{{Optimal ET sequence|legend=1| 19, 39d, 58 }}
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}


Badness: 0.039156
Badness (Sintel): 1.29


==== 13-limit ====
==== 13-limit ====
Line 102: Line 102:
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.023667
Badness (Sintel): 0.978


=== Camahueto ===
=== Camahueto ===
Line 115: Line 117:
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.065940
Badness (Sintel): 2.18


==== 13-limit ====
==== 13-limit ====
Line 128: Line 132:
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.434
Optimal tunings:
* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}


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


Badness: 0.036155
Badness (Sintel): 1.49


=== Qilin ===
=== Qilin ===
Line 141: Line 147:
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.196
Optimal tunings:
* WE: ~2 = 1199.3865{{c}}, ~28/27 = 62.1645{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}


{{Optimal ET sequence|legend=1| 58, 77, 135c, 193c, 328cc }}
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193c }}


Badness: 0.041426
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 154: Line 162:
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.197
Optimal tunings:
* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}


{{Optimal ET sequence|legend=1| 58, 77, 135c, 193cf, 328ccff }}
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193cf }}


Badness: 0.022842
Badness (Sintel): 0.944


=== Monocerus ===
=== Monocerus ===
Line 167: Line 177:
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.292
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=1| 58, 96d, 154, 212ce, 366cce }}
{{Optimal ET sequence|legend=0| 38d, 58, 96d, 154, 212ce }}


Badness: 0.052757
Badness (Sintel): 1.74


==== 13-limit ====
==== 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, 243/242, 364/363


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


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.301
Optimal tunings:
* WE: ~55/39 = 599.8267{{c}}, ~28/27 = 62.2833{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~28/27 = 62.3262{{c}}
 
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}
 
Badness (Sintel): 1.19
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


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


Badness: 0.028795
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}
 
Optimal tunings:
* WE: ~17/12 = 600.0715{{c}}, ~28/27 = 62.3767{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/27 = 62.3605{{c}}
 
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}
 
Badness (Sintel): 1.24


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


Line 193: Line 224:
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}


{{Multival|legend=1| 8 13 4 2 -16 -27 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
{{Optimal ET sequence|legend=1| 1c, …, 18bcd, 19 }}


{{Optimal ET sequence|legend=1| 1c, 19 }}
[[Badness]] (Sintel): 2.07
 
[[Badness]]: 0.081864


=== 11-limit ===
=== 11-limit ===
Line 208: Line 241:
Mapping: {{mapping|  1 2 3 3 4 | 0 -8 -13 -4 -10 }}
Mapping: {{mapping|  1 2 3 3 4 | 0 -8 -13 -4 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.059319
Badness (Sintel): 1.96


=== 13-limit ===
=== 13-limit ===
Line 221: Line 256:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~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}}


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


Badness: 0.039343
Badness (Sintel): 1.63


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