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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]] and [[the octaphore]].
{{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.
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
Line 10: Line 11:
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
* [[WE]]: ~2 = 1200.0889¢, ~250/243 =  62.4623¢
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458
* [[CWE]]: ~2 = 1200.0000¢, ~250/243 = 62.4494¢


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


[[Badness]] (Dirichlet): 3.530
[[Badness]] (Sintel): 3.530


[[Badness]]: 0.150487
== Septimal unicorn ==
 
{{See also| Octaphore }}
=== 2.3.5.7 subgroup (septimal unicorn) ===
{{ See also | The octaphore }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 29: Line 28:
{{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¢, ~28/27 = 62.2621¢
[[Optimal tuning]]s:
* [[CWE]]: ~2 = 1200.0000¢, ~28/27 = 62.2996¢
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278


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


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


Badness: 0.040913
=== 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 }}
{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}


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


Badness (Dirichlet): 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 (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.
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]], [[729/728]], [[784/783]], [[841/840]], 216/215
[[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 }}
{{Mapping|legend=1| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}


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


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


=== Alicorn ===
=== Alicorn ===
Line 89: Line 90:
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¢, ~28/27 = 62.0316¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.1435¢


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


Badness: 0.039156
Badness (Sintel): 1.294


==== 13-limit ====
==== 13-limit ====
Line 102: Line 105:
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¢, ~26/25 = 62.0480¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.1636¢


{{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 120:
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¢, ~28/27 = 62.4576¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.4252¢


{{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): 2.180


==== 13-limit ====
==== 13-limit ====
Line 128: Line 135:
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¢, ~26/25 = 62.4603¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.4277¢


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


Badness: 0.036155
Badness (Sintel): 1.494


=== Qilin ===
=== Qilin ===
Line 141: Line 150:
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¢, ~28/27 = 62.1645¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.2199¢


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


Badness: 0.041426
Badness (Sintel): 1.370


==== 13-limit ====
==== 13-limit ====
Line 154: Line 165:
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¢, ~26/25 = 62.1601¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.2251¢


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


Badness: 0.022842
Badness (Sintel): 0.944


=== Monocerus ===
=== Monocerus ===
Line 167: Line 180:
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¢, ~28/27 = 62.2737¢
* CWE: ~99/70 = 600.0000¢, ~28/27 = 62.3180¢


{{Optimal ET sequence|legend=1| 58, 96d, 154, 212ce, 366cce }}
{{Optimal ET sequence|legend=0| 58, 96d, 154, 212ce, 366cce }}


Badness: 0.052757
Badness (Sintel): 1.744


==== 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¢, ~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


{{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¢, ~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 ==
== Rhinoceros ==
Line 193: Line 225:
{{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¢, ~21/20 = 63.0830¢
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
* [[CWE]]: ~2 = 1200.0000¢, ~21/20 = 62.6687¢


{{Optimal ET sequence|legend=1| 1c, 19 }}
{{Optimal ET sequence|legend=1| 1c, 19 }}


[[Badness]]: 0.081864
[[Badness]] (Sintel): 2.072


=== 11-limit ===
=== 11-limit ===
Line 208: Line 240:
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¢, ~21/20 = 62.9338¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.7783¢


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


Badness: 0.059319
Badness (Sintel): 1.961


=== 13-limit ===
=== 13-limit ===
Line 221: Line 255:
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¢, ~21/20 = 63.1591¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.8727¢


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


Badness: 0.039343
Badness (Sintel): 1.626


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"