Unicorn family: Difference between revisions

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m Units; misc. cleanup; - redundant category
Review data for each temp. + intro to rhinoceros
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0889{{c}}, ~250/243 =  62.4623{{c}}
* [[WE]]: ~2 = 1200.0889{{c}}, ~250/243 =  62.4623{{c}}
: [[error map]]: {{val| +0.089 -1.476 +1.943 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 62.4494{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 62.4494{{c}}
: error map: {{val| 0.000 -1.550 +1.844 }}


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


[[Badness]] (Sintel): 3.530
[[Badness]] (Sintel): 3.53


== Septimal unicorn ==
== Septimal unicorn ==
Line 32: Line 34:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}
* [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}
: [[error map]]: {{val| -0.305 -0.662 +3.364 -2.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}
: error map: {{val| 0.000 -0.352 +3.792 -1.717 }}


{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
Line 46: Line 50:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7013{{c}}, ~26/25 = 62.2785{{c}}
* WE: ~2 = 1199.7013{{c}}, ~28/27 = 62.2785{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3151{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3151{{c}}


{{Optimal ET sequence|legend=0| 19, 39df, 58, 77, 212cf }}
{{Optimal ET sequence|legend=0| 19, 39df, 58, 77, 212cf }}
Line 61: Line 65:


Optimal tunings:  
Optimal tunings:  
* [[WE]]: ~2 = 1199.7715{{c}}, ~26/25 = 62.2860{{c}}
* WE: ~2 = 1199.7715{{c}}, ~28/27 = 62.2860{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/25 = 62.3141{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3141{{c}}


{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}
Line 78: Line 82:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.6269{{c}}, ~26/25 = 62.2584{{c}}
* WE: ~2 = 1199.6269{{c}}, ~28/27 = 62.2584{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.3008{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3008{{c}}


{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 135c, 212cfn }}
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 135c, 212cfn }}
Line 98: Line 102:
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}


Badness (Sintel): 1.294
Badness (Sintel): 1.29


==== 13-limit ====
==== 13-limit ====
Line 108: Line 112:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.6298{{c}}, ~26/25 = 62.0480{{c}}
* WE: ~2 = 1198.6298{{c}}, ~28/27 = 62.0480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1636{{c}}


{{Optimal ET sequence|legend=0| 19, 39df, 58 }}
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}
Line 128: Line 132:
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}


Badness (Sintel): 2.180
Badness (Sintel): 2.18


==== 13-limit ====
==== 13-limit ====
Line 138: Line 142:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5004{{c}}, ~26/25 = 62.4603{{c}}
* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.4277{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}


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


Badness (Sintel): 1.494
Badness (Sintel): 1.49


=== Qilin ===
=== Qilin ===
Line 156: Line 160:
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}


{{Optimal ET sequence|legend=0| 58, 77, 135c, 193c, 328cc }}
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193c }}


Badness (Sintel): 1.370
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 168: Line 172:


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.2874{{c}}, ~26/25 = 62.1601{{c}}
* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 62.2251{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}


{{Optimal ET sequence|legend=0| 58, 77, 135c, 193cf, 328ccff }}
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193cf }}


Badness (Sintel): 0.944
Badness (Sintel): 0.944
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* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}


{{Optimal ET sequence|legend=0| 58, 96d, 154, 212ce, 366cce }}
{{Optimal ET sequence|legend=0| 38d, 58, 96d, 154, 212ce }}


Badness (Sintel): 1.744
Badness (Sintel): 1.74


==== 13-limit ====
==== 13-limit ====
Line 198: Line 202:


Optimal tunings:  
Optimal tunings:  
* WE: ~55/39 = 599.8267{{c}}, ~26/25 = 62.2833{{c}}
* WE: ~55/39 = 599.8267{{c}}, ~28/27 = 62.2833{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~26/25 = 62.3262{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~28/27 = 62.3262{{c}}


{{Optimal ET sequence|legend=0| 58, 96d, 154, 366ccef }}
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}


Badness (Sintel): 1.190
Badness (Sintel): 1.19


==== 17-limit ====
==== 17-limit ====
Line 213: Line 217:


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.0715{{c}}, ~26/25 = 62.3767{{c}}
* WE: ~17/12 = 600.0715{{c}}, ~28/27 = 62.3767{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/25 = 62.3605{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/27 = 62.3605{{c}}


{{Optimal ET sequence|legend=0| 58, 96d, 154 }}
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}


Badness (Sintel): 1.237
Badness (Sintel): 1.24


== Rhinoceros ==
== Rhinoceros ==
Rhinoceros tempers out the [[semaphoresma]] and the [[ragisma]], identifying the generator as a [[~]][[21/20]], rather than the [[28/27]] of septimal unicorn. It may be described as the {{nowrap| 19 & 20c }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 229: Line 235:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.1031{{c}}, ~21/20 = 63.0830{{c}}
* [[WE]]: ~2 = 1203.1031{{c}}, ~21/20 = 63.0830{{c}}
: [[error map]]: {{val| +3.103 -0.412 +2.917 -11.848 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 62.6687{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 62.6687{{c}}
: error map: {{val| 0.000 -3.305 -1.007 -19.501 }}


{{Optimal ET sequence|legend=1| 1c, 19 }}
{{Optimal ET sequence|legend=1| 1c, …, 18bcd, 19 }}


[[Badness]] (Sintel): 2.072
[[Badness]] (Sintel): 2.07


=== 11-limit ===
=== 11-limit ===
Line 246: Line 254:
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}


{{Optimal ET sequence|legend=0| 1ce, 19 }}
{{Optimal ET sequence|legend=0| 1ce, 18bcd, 19 }}


Badness (Sintel): 1.961
Badness (Sintel): 1.96


=== 13-limit ===
=== 13-limit ===
Line 261: Line 269:
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.8727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.8727{{c}}


{{Optimal ET sequence|legend=0| 1ce, 19 }}
{{Optimal ET sequence|legend=0| 1ce, 18bcdf, 19 }}


Badness (Sintel): 1.626
Badness (Sintel): 1.63


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 19:06, 4 August 2026

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 (monzo: [-2 13 -8⟩, ratio: 1594323/1562500).

Unicorn

Subgroup: 2.3.5

Comma list: 1594323/1562500

Mapping: [⟨1 2 3], ⟨0 -8 -13]]

mapping generators: ~2, ~250/243

Optimal tunings:

  • WE: ~2 = 1200.0889 ¢, ~250/243 = 62.4623 ¢
error map: ⟨+0.089 -1.476 +1.943]
  • CWE: ~2 = 1200.0000 ¢, ~250/243 = 62.4494 ¢
error map: ⟨0.000 -1.550 +1.844]

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

Badness (Sintel): 3.53

Septimal unicorn

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.

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.

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 10976/10935

Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]

Optimal tunings:

  • WE: ~2 = 1199.6949 ¢, ~28/27 = 62.2621 ¢
error map: ⟨-0.305 -0.662 +3.364 -2.074]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2996 ¢
error map: ⟨0.000 -0.352 +3.792 -1.717]

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

Badness (Sintel): 1.035

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

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

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

Optimal tunings:

  • WE: ~2 = 1199.7013 ¢, ~28/27 = 62.2785 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.3151 ¢

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, 196/195, 261/260, 377/375

Subgroup-val mapping: [⟨1 2 3 4 5 6], ⟨0 -8 -13 -23 -25 -22]]

Optimal tunings:

  • WE: ~2 = 1199.7715 ¢, ~28/27 = 62.2860 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.3141 ¢

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

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, 196/195, 216/215, 261/260, 377/375

Subgroup-val mapping: [⟨1 2 3 4 5 6 6], ⟨0 -8 -13 -23 -25 -22 -11]]

Optimal tunings:

  • WE: ~2 = 1199.6269 ¢, ~28/27 = 62.2584 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.3008 ¢

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 tunings:

  • WE: ~2 = 1198.6510 ¢, ~28/27 = 62.0316 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.1435 ¢

Optimal ET sequence: 19, 39d, 58

Badness (Sintel): 1.29

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 tunings:

  • WE: ~2 = 1198.6298 ¢, ~28/27 = 62.0480 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.1636 ¢

Optimal ET sequence: 19, 39df, 58

Badness (Sintel): 0.978

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 tunings:

  • WE: ~2 = 1200.5186 ¢, ~28/27 = 62.4576 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.4252 ¢

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

Badness (Sintel): 2.18

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 tunings:

  • WE: ~2 = 1200.5004 ¢, ~28/27 = 62.4603 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.4277 ¢

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

Badness (Sintel): 1.49

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 tunings:

  • WE: ~2 = 1199.3865 ¢, ~28/27 = 62.1645 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2199 ¢

Optimal ET sequence: 19e, …, 58, 135c, 193c

Badness (Sintel): 1.37

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 tunings:

  • WE: ~2 = 1199.2874 ¢, ~28/27 = 62.1601 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2251 ¢

Optimal ET sequence: 19e, …, 58, 135c, 193cf

Badness (Sintel): 0.944

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 tunings:

  • WE: ~99/70 = 599.8223 ¢, ~28/27 = 62.2737 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~28/27 = 62.3180 ¢

Optimal ET sequence: 38d, 58, 96d, 154, 212ce

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242, 364/363

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

Optimal tunings:

  • WE: ~55/39 = 599.8267 ¢, ~28/27 = 62.2833 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~28/27 = 62.3262 ¢

Optimal ET sequence: 38df, 58, 96d, 154

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 196/195, 221/220, 243/242, 289/288

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

Optimal tunings:

  • WE: ~17/12 = 600.0715 ¢, ~28/27 = 62.3767 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~28/27 = 62.3605 ¢

Optimal ET sequence: 38df, 58, 96d, 154

Badness (Sintel): 1.24

Rhinoceros

Rhinoceros tempers out the semaphoresma and the ragisma, identifying the generator as a ~21/20, rather than the 28/27 of septimal unicorn. It may be described as the 19 & 20c temperament.

Subgroup: 2.3.5.7

Comma list: 49/48, 4375/4374

Mapping: [⟨1 2 3 3], ⟨0 -8 -13 -4]]

Optimal tunings:

  • WE: ~2 = 1203.1031 ¢, ~21/20 = 63.0830 ¢
error map: ⟨+3.103 -0.412 +2.917 -11.848]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.6687 ¢
error map: ⟨0.000 -3.305 -1.007 -19.501]

Optimal ET sequence: 1c, …, 18bcd, 19

Badness (Sintel): 2.07

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 tunings:

  • WE: ~2 = 1201.1458 ¢, ~21/20 = 62.9338 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.7783 ¢

Optimal ET sequence: 1ce, 18bcd, 19

Badness (Sintel): 1.96

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 tunings:

  • WE: ~2 = 1202.2047 ¢, ~21/20 = 63.1591 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.8727 ¢

Optimal ET sequence: 1ce, 18bcdf, 19

Badness (Sintel): 1.63

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