Unicorn family: Difference between revisions

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Listing septimal unicorn on the same level as rhinoceros. Consolidate badness measures
 
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{{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]].
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
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{{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:
[[Optimal tuning]]s:
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* Dirichlet: 1.035
* Dirichlet: 1.035


==== 2.3.5.7.13 subgroup ====
=== 2.3.5.7.13 subgroup ===
[[Subgroup]]: 2.3.5.7.13
[[Subgroup]]: 2.3.5.7.13


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


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