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The '''unicorn family''' tempers out the [[unicorn comma]], 1594323/1562500 = {{monzo| -2 13 -8 }}.  
{{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]].


= Five-limit unicorn =
== Unicorn ==
Subgroup: 2.3.5
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


[[Comma list]]: 1594323/1562500
[[Comma list]]: 1594323/1562500


[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -8 -13 }}]
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}
 
[[Optimal tuning]]s:
* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458
 
{{Optimal ET sequence|legend=1| 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|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
 
[[Optimal tuning]]s:
* [[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 }}
 
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|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
 
{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}


[[POTE generator]]: ~250/243 = 62.458
Badness (Sintel): 0.590


{{Val list|legend=1| 19, 58, 77, 96, 173, 269 }}
==== 2.3.5.7.13.29 subgroup ====
[[Subgroup]]: 2.3.5.7.13.29


[[Badness]]: 0.150487
[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]]


= Septimal unicorn =
{{Mapping|legend=1| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}
Subgroup: 2.3.5.7


[[Comma list]]: 126/125, 10976/10935
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.334


[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -8 -13 -23 }}]
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 212cfn }}


{{Multival|legend=1| 8 13 23 2 14 17 }}
Badness (Sintel): 0.487


[[POTE generator]]: ~28/27 = 62.278
==== 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]].


{{Val list|legend=1| 19, 39d, 58, 77, 135c }}
[[Subgroup]]: 2.3.5.7.13.29.43


[[Badness]]: 0.040913
[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]], 216/215


== Alicorn ==
{{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 ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}
 
Badness (Sintel): 0.514
 
=== Alicorn ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 2 3 4 3 }}, {{val| 0 -8 -13 -23 9 }}]
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}


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


Vals: {{Val list| 19, 39d, 58 }}
{{Optimal ET sequence|legend=1| 19, 39d, 58 }}


Badness: 0.039156
Badness: 0.039156


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 2 3 4 3 5 }}, {{val| 0 -8 -13 -23 9 -25 }}]
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}


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


Vals: {{Val list| 19, 39df, 58 }}
{{Optimal ET sequence|legend=1| 19, 39df, 58 }}


Badness: 0.023667
Badness: 0.023667


== Camahueto ==
=== Camahueto ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 2 3 4 2 }}, {{val| 0 -8 -13 -23 28 }}]
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}


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


Vals: {{Val list| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}


Badness: 0.065940
Badness: 0.065940


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 2 3 4 2 5 }}, {{val| 0 -8 -13 -23 28 -25 }}]
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}


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


Vals: {{Val list| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}


Badness: 0.036155
Badness: 0.036155


== Qilin ==
=== Qilin ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 2 3 4 6 }}, {{val| 0 -8 -13 -23 -49 }}]
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}


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


Vals: {{Val list| 58, 77, 135c, 193c, 328cc }}
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193c, 328cc }}


Badness: 0.041426
Badness: 0.041426


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 2 3 4 6 5 }}, {{val| 0 -8 -13 -23 -49 -25 }}]
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}


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


Vals: {{Val list| 58, 77, 135c, 193cf, 328ccff }}
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193cf, 328ccff }}


Badness: 0.022842
Badness: 0.022842


== Monocerus ==
=== Monocerus ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 2 4 6 8 9 }}, {{val| 0 -8 -13 -23 -20 }}]
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}


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


Vals: {{Val list| 58, 96d, 154, 212ce, 366cce }}
{{Optimal ET sequence|legend=1| 58, 96d, 154, 212ce, 366cce }}


Badness: 0.052757
Badness: 0.052757


=== 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, 364/363, 676/675


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


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


Vals: {{Val list| 58, 96d, 154, 366ccef }}
{{Optimal ET sequence|legend=1| 58, 96d, 154, 366ccef }}


Badness: 0.028795
Badness: 0.028795


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


[[Comma list]]: 49/48, 4375/4374
[[Comma list]]: 49/48, 4375/4374


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 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]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920


[[POTE generator]]: ~21/20 = 62.920
{{Optimal ET sequence|legend=1| 1c, 19 }}
 
{{Val list|legend=1| 1c, 19 }}


[[Badness]]: 0.081864
[[Badness]]: 0.081864


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -8 -13 -4 -10 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -8 -13 -4 -10 }}


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


Vals: {{Val list| 1ce, 19 }}
{{Optimal ET sequence|legend=1| 1ce, 19 }}


Badness: 0.059319
Badness: 0.059319


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -8 -13 -4 -10 -6 }}]
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}


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


Vals: {{Val list| 1ce, 19 }}
{{Optimal ET sequence|legend=1| 1ce, 19 }}


Badness: 0.039343
Badness: 0.039343


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Unicorn]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Rank 2]]
[[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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