Xenharmonic Wiki

Unicorn family: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m recat
m Recategorize
 
(34 intermediate revisions by 11 users not shown)
Line 1: Line 1:
__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]] ({{monzo|legend=1| -2 13 -8 }}, [[ratio]]: 1594323/1562500).  


=Unicorn=
== Unicorn ==
Comma: 1594323/1562500
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.


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 }}
: mapping generators: ~2, ~250/243


Badness: 0.1505
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0889{{c}}, ~250/243 =  62.4623{{c}}
: [[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 }}


=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): 3.53


Map: [<1 2 3 4|, <0 -8 -13 -23|]
== Septimal unicorn ==
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]].


EDOs: 19, 58, 77, 96d
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.


Badness: 0.0409
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]]}.  


==11-limit==
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.
Commas: 126/125, 540/539, 896/891


POTE generator: ~28/27 = 62.101
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.  


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


EDOs: 19, 58
[[Comma list]]: 126/125, 10976/10935


Badness: 0.0392
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}


==13-limit==
[[Optimal tuning]]s:
Commas: 126/125, 144/143, 196/195, 676/675
* [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}
: [[error map]]: {{val| -0.305 -0.662 +3.364 -2.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}
: error map: {{val| 0.000 -0.352 +3.792 -1.717 }}


POTE generator: ~28/27 = 62.119
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}


Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
[[Badness]] (Sintel): 1.04


EDOs: 19, 58
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13


Badness: 0.0237
Comma list: 126/125, 196/195, 676/675


==Camahueto==
Subgroup-val mapping: {{mapping| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
Commas: 126/125, 10976/10935, 385/384


POTE generator: ~28/27 = 62.431
Optimal tunings:  
* WE: ~2 = 1199.7013{{c}}, ~28/27 = 62.2785{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3151{{c}}


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
{{Optimal ET sequence|legend=0| 19, 39df, 58, 77, 212cf }}


EDOs: 19, 58, 77, 96d
Badness (Sintel): 0.590


Badness: 0.0659
==== 2.3.5.7.13.29 subgroup ====
Subgroup: 2.3.5.7.13.29


==13-limit==
Comma list: 126/125, 196/195, 261/260, 377/375
Commas: 126/125, 196/195, 385/384, 676/675


POTE generator: ~28/27 = 62.434
Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
Optimal tunings:  
* WE: ~2 = 1199.7715{{c}}, ~28/27 = 62.2860{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3141{{c}}


EDOs: 19, 58, 77, 96d
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}


Badness: 0.0362
Badness (Sintel): 0.487


=Monocerus=
=== Alicorn ===
Commas: 126/125, 243/242, 5488/5445
Subgroup: 2.3.5.7.11


POTE generator: ~28/27 = 62.292
Comma list: 126/125, 540/539, 896/891


Map: [&lt;2 4 6 8 9|, &lt;0 -8 -13 -23 -20|]
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}


EDOs: 58, 96d, 154, 212ce, 366de
Optimal tunings:  
* WE: ~2 = 1198.6510{{c}}, ~28/27 = 62.0316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1435{{c}}


Badness: 0.0528
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}


==13-limit==
Badness (Sintel): 1.29
Commas: 126/125, 196/195, 364/363, 676/675


POTE generator: ~28/27 = 62.301
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 4 6 8 9 10|, &lt;0 -8 -13 -23 -20 -25|]
Comma list: 126/125, 144/143, 196/195, 676/675


EDOs: 58, 96d, 154, 366cef
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}


Badness: 0.0288
Optimal tunings:  
* WE: ~2 = 1198.6298{{c}}, ~28/27 = 62.0480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1636{{c}}


=Qilin=
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}
Commas: 126/125, 176/175, 10976/10935


POTE generator: ~28/27 = 62.196
Badness (Sintel): 0.978


Map: [&lt;1 2 3 4 6|, &lt;0 -8 -13 -23 -49|]
=== Camahueto ===
Subgroup: 2.3.5.7.11


EDOs: 58, 77, 135c, 193c, 328c
Comma list: 126/125, 385/384, 10976/10935


Badness: 0.0414
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}


==13-limit==
Optimal tunings:
Commas: 126/125, 176/175, 196/195, 2200/2197
* WE: ~2 = 1200.5186{{c}}, ~28/27 = 62.4576{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4252{{c}}


POTE generator: ~28/27 = 62.197
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}


Map: [&lt;1 2 3 4 6 5|, &lt;0 -8 -13 -23 -49 -25|]
Badness (Sintel): 2.18


EDOs: 58, 77, 135c, 193cf, 328cf
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0228
Comma list: 126/125, 196/195, 385/384, 676/675


=Rhinoceros=
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}
Commas: 49/48, 4375/4374


POTE generator: ~21/20 = 62.920
Optimal tunings:  
* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}


Map: [&lt;1 2 3 3]|, &lt;0 -8 -13 -4|]
{{Optimal ET sequence|legend=0| 19, 77, 96d, 173d }}


Wedgie: &lt;&lt;8 13 4 2 -16 -27||
Badness (Sintel): 1.49


EDOs: 19
=== Qilin ===
Subgroup: 2.3.5.7.11


Badness: 0.0819
Comma list: 126/125, 176/175, 10976/10935


==11-limit==
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
Commas: 49/48, 100/99, 126/121


POTE generator: ~21/20 = 62.874
Optimal tunings:  
* WE: ~2 = 1199.3865{{c}}, ~28/27 = 62.1645{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}


Map: [&lt;1 2 3 3 4]|, &lt;0 -8 -13 -4 -10|]
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193c }}


EDOs: 19
Badness (Sintel): 1.37


Badness: 0.0593
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 126/125, 176/175, 196/195, 2200/2197
Commas: 49/48, 78/77, 100/99, 126/121


POTE generator: ~21/20 = 63.043
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}


Map: [&lt;1 2 3 3 4 4]|, &lt;0 -8 -13 -4 -10 -6|]
Optimal tunings:  
* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}


EDOs: 19
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193cf }}


Badness: 0.0393
Badness (Sintel): 0.944


[[Category:Theory]]
=== Monocerus ===
[[Category:Temperament family]]
Subgroup: 2.3.5.7.11
[[Category:Unicorn]]
 
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{{c}}, ~28/27 = 62.2737{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}
 
{{Optimal ET sequence|legend=0| 38d, 58, 96d, 154, 212ce }}
 
Badness (Sintel): 1.74
 
==== 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{{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
 
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{{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 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
 
[[Comma list]]: 49/48, 4375/4374
 
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
 
[[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 ET sequence|legend=1| 1c, …, 18bcd, 19 }}
 
[[Badness]] (Sintel): 2.07
 
=== 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{{c}}, ~21/20 = 62.9338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 18bcd, 19 }}
 
Badness (Sintel): 1.96
 
=== 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{{c}}, ~21/20 = 63.1591{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.8727{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 18bcdf, 19 }}
 
Badness (Sintel): 1.63
 
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"