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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:
* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458


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


POTE generator: ~28/27 = 62.279
[[Badness]]:  
* Smith: 0.150487
* Dirichlet: 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:
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278


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:  
* Smith: 0.040913
* Dirichlet: 1.035


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


==13-limit==
[[Comma list]]: [[126/125]], [[351/350]], [[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]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339


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]], [[729/728]], [[784/783]], [[841/840]]


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]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.334


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


==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]], [[729/728]], [[784/783]], [[841/840]], 216/215


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


=Monocerus=
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
Commas: 126/125, 243/242, 5488/5445


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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.101


Map: [<2 4 6 8 9 10|, <0 -8 -13 -23 -20 -25|]
{{Optimal ET sequence|legend=1| 19, 39d, 58 }}


EDOs: 58, 96d, 154, 366cef
Badness: 0.039156


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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.119


EDOs: 58, 77, 135c, 193c, 328c
{{Optimal ET sequence|legend=1| 19, 39df, 58 }}


Badness: 0.0414
Badness: 0.023667


==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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.431


Badness: 0.0228
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}


=Rhinoceros=
Badness: 0.065940
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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.434


Badness: 0.0819
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}


==11-limit==
Badness: 0.036155
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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.196


==13-limit==
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193c, 328cc }}
Commas: 49/48, 78/77, 100/99, 126/121


POTE generator: ~21/20 = 63.043
Badness: 0.041426


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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.197
[[Category:unicorn]]
 
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193cf, 328ccff }}
 
Badness: 0.022842
 
=== 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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.292
 
{{Optimal ET sequence|legend=1| 58, 96d, 154, 212ce, 366cce }}
 
Badness: 0.052757
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 364/363, 676/675
 
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 ET sequence|legend=1| 58, 96d, 154, 366ccef }}
 
Badness: 0.028795
 
== 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]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
 
{{Optimal ET sequence|legend=1| 1c, 19 }}
 
[[Badness]]: 0.081864
 
=== 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 tuning (POTE): ~2 = 1\1, ~21/20 = 62.874
 
{{Optimal ET sequence|legend=1| 1ce, 19 }}
 
Badness: 0.059319
 
=== 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 tuning (POTE): ~2 = 1\1, ~21/20 = 63.043
 
{{Optimal ET sequence|legend=1| 1ce, 19 }}
 
Badness: 0.039343
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 00:34, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The unicorn family tempers out the unicorn comma, 1594323/1562500 = [-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

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 intervals, by tempering the square-particulars between them, leading to its S-expression-based comma list of {S26, S27, S28, 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

Comma list: 1594323/1562500

Mapping: [⟨1 2 3], ⟨0 -8 -13]]

Optimal tunings:

  • CTE: 2 = 1\1, ~250/243 = 62.441
  • POTE: 2 = 1\1, ~250/243 = 62.458

Optimal ET sequence: 19, 58, 77, 96, 173, 269

Badness:

  • Smith: 0.150487
  • Dirichlet: 3.530

Septimal unicorn

See also: Octaphore

Subgroup: 2.3.5.7

Comma list: 126/125, 10976/10935

Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]

Optimal tunings:

  • CTE: 2 = 1\1, ~28/27 = 62.324
  • POTE: 2 = 1\1, ~28/27 = 62.278

Optimal ET sequence: 19, 39d, 58, 77, 135c, 212c

Badness:

  • Smith: 0.040913
  • Dirichlet: 1.035

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 126/125, 351/350, 676/675

Mapping: [⟨1 2 3 4 5], ⟨0 -8 -13 -23 -25]]

Optimal tuning (CTE): 2 = 1\1, ~28/27 = 62.339

Optimal ET sequence: 19, 39df, 58, 77, 212cf

Badness (Sintel): 0.590

2.3.5.7.13.29 subgroup

Subgroup: 2.3.5.7.13.29

Comma list: 126/125, 729/728, 784/783, 841/840

Mapping: [⟨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: 19, 39dfj, 58, 77, 212cfn

Badness (Sintel): 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 (with a 963¢ ~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

Comma list: 126/125, 729/728, 784/783, 841/840, 216/215

Mapping: [⟨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 ET sequence: 19, 39dfj, 58, 77, 135c, 212cfn

Badness (Sintel): 0.514

Alicorn

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 4 3], ⟨0 -8 -13 -23 9]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101

Optimal ET sequence: 19, 39d, 58

Badness: 0.039156

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 676/675

Mapping: [⟨1 2 3 4 3 5], ⟨0 -8 -13 -23 9 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.119

Optimal ET sequence: 19, 39df, 58

Badness: 0.023667

Camahueto

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 10976/10935

Mapping: [⟨1 2 3 4 2], ⟨0 -8 -13 -23 28]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.431

Optimal ET sequence: 19, 58e, 77, 96d, 173d

Badness: 0.065940

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 4 2 5], ⟨0 -8 -13 -23 28 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.434

Optimal ET sequence: 19, 58e, 77, 96d, 173d

Badness: 0.036155

Qilin

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 4 6], ⟨0 -8 -13 -23 -49]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.196

Optimal ET sequence: 58, 77, 135c, 193c, 328cc

Badness: 0.041426

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 4 6 5], ⟨0 -8 -13 -23 -49 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.197

Optimal ET sequence: 58, 77, 135c, 193cf, 328ccff

Badness: 0.022842

Monocerus

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 5488/5445

Mapping: [⟨2 4 6 8 9], ⟨0 -8 -13 -23 -20]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.292

Optimal ET sequence: 58, 96d, 154, 212ce, 366cce

Badness: 0.052757

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 676/675

Mapping: [⟨2 4 6 8 9 10], ⟨0 -8 -13 -23 -20 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.301

Optimal ET sequence: 58, 96d, 154, 366ccef

Badness: 0.028795

Rhinoceros

Subgroup: 2.3.5.7

Comma list: 49/48, 4375/4374

Mapping: [⟨1 2 3 3], ⟨0 -8 -13 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 62.920

Optimal ET sequence: 1c, 19

Badness: 0.081864

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 100/99, 126/121

Mapping: [⟨1 2 3 3 4], ⟨0 -8 -13 -4 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 62.874

Optimal ET sequence: 1ce, 19

Badness: 0.059319

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 78/77, 100/99, 126/121

Mapping: [⟨1 2 3 3 4 4], ⟨0 -8 -13 -4 -10 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 63.043

Optimal ET sequence: 1ce, 19

Badness: 0.039343

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