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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]].


== 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 }}


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


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


[[Badness]]: 0.150487
[[Badness]]:
* Smith: 0.150487
* Dirichlet: 3.530


== Septimal unicorn ==
== Septimal unicorn ==
Subgroup: 2.3.5.7
{{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 }}


[[Comma list]]: 126/125, 10976/10935
[[Optimal tuning]]s:
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278


[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -8 -13 -23 }}]
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}


{{Multival|legend=1| 8 13 23 2 14 17 }}
Badness:
* Smith: 0.040913
* Dirichlet: 1.035


[[POTE generator]]: ~28/27 = 62.278
=== 2.3.5.7.13 subgroup ===
[[Subgroup]]: 2.3.5.7.13


{{Val list|legend=1| 19, 39d, 58, 77, 135c }}
[[Comma list]]: [[126/125]], [[351/350]], [[676/675]]


[[Badness]]: 0.040913
{{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 }}
 
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|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 (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[[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
 
[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]], 216/215
 
{{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 ===
=== Alicorn ===
Line 34: Line 86:
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
Line 47: Line 99:
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
Line 60: Line 112:
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
Line 73: Line 125:
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
Line 86: Line 138:
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
Line 99: Line 151:
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
Line 112: Line 164:
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
Line 125: Line 177:
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 }}


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


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


[[Badness]]: 0.081864
[[Badness]]: 0.081864
Line 153: Line 203:
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
Line 166: Line 216:
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:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Unicorn family| ]] <!-- main article -->
[[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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