Unicorn family: Difference between revisions

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Line 11: Line 11:
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
* [[WE]]: ~2 = 1200.0889¢, ~250/243 =  62.4623¢
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458
* [[CWE]]: ~2 = 1200.0000¢, ~250/243 = 62.4494¢


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


[[Badness]]:
[[Badness]] (Sintel): 3.530
* Smith: 0.150487
* Dirichlet: 3.530


== Septimal unicorn ==
== Septimal unicorn ==
Line 30: Line 28:
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
* [[WE]]: ~2 = 1199.6949¢, ~28/27 = 62.2621¢
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278
* [[CWE]]: ~2 = 1200.0000¢, ~28/27 = 62.2996¢


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


Badness:
Badness (Sintel): 1.035
* Smith: 0.040913
* 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


[[Comma list]]: [[126/125]], [[351/350]], [[676/675]]
[[Comma list]]: [[126/125]], [[196/195]], [[676/675]]


{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7013¢, ~26/25 = 62.2785¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3151¢


{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}
{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}
Line 56: Line 54:
[[Subgroup]]: 2.3.5.7.13.29
[[Subgroup]]: 2.3.5.7.13.29


[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]]
[[Comma list]]: [[126/125]], [[196/195]], [[261/260]], [[377/375]]


{{Mapping|legend=1| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}
{{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 tuning]] ([[CTE]]): 2 = 1\1, ~26/25 = 62.334


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


Badness (Sintel): 0.487
Badness (Sintel): 0.487
Line 71: Line 69:
[[Subgroup]]: 2.3.5.7.13.29.43
[[Subgroup]]: 2.3.5.7.13.29.43


[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]], 216/215
[[Comma list]]: [[126/125]], [[196/195]], [[216/215]], [[261/260]], [[377/375]]


{{Mapping|legend=1| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}
{{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 tuning]] ([[CTE]]): 2 = 1\1, ~26/25 = 62.339


{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}
Line 88: Line 86:
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101
Optimal tunings:
* WE: ~2 = 1198.6510¢, ~28/27 = 62.0316¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.1435¢


{{Optimal ET sequence|legend=1| 19, 39d, 58 }}
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}


Badness: 0.039156
Badness (Sintel): 1.294


==== 13-limit ====
==== 13-limit ====
Line 101: Line 101:
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.119
Optimal tunings:
* WE: ~2 = 1198.6298¢, ~26/25 = 62.0480¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.1636¢


{{Optimal ET sequence|legend=1| 19, 39df, 58 }}
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}


Badness: 0.023667
Badness (Sintel): 0.978


=== Camahueto ===
=== Camahueto ===
Line 114: Line 116:
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.431
Optimal tunings:
* WE: ~2 = 1200.5186¢, ~28/27 = 62.4576¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.4252¢


{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}


Badness: 0.065940
Badness (Sintel): 2.180


==== 13-limit ====
==== 13-limit ====
Line 127: Line 131:
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.434
Optimal tunings:
* WE: ~2 = 1200.5004¢, ~26/25 = 62.4603¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.4277¢


{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}


Badness: 0.036155
Badness (Sintel): 1.494


=== Qilin ===
=== Qilin ===
Line 140: Line 146:
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.196
Optimal tunings:
* WE: ~2 = 1199.3865¢, ~28/27 = 62.1645¢
* CWE: ~2 = 1200.0000¢, ~28/27 = 62.2199¢


{{Optimal ET sequence|legend=1| 58, 77, 135c, 193c, 328cc }}
{{Optimal ET sequence|legend=0| 58, 77, 135c, 193c, 328cc }}


Badness: 0.041426
Badness (Sintel): 1.370


==== 13-limit ====
==== 13-limit ====
Line 153: Line 161:
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.197
Optimal tunings:
* WE: ~2 = 1199.2874¢, ~26/25 = 62.1601¢
* CWE: ~2 = 1200.0000¢, ~26/25 = 62.2251¢


{{Optimal ET sequence|legend=1| 58, 77, 135c, 193cf, 328ccff }}
{{Optimal ET sequence|legend=0| 58, 77, 135c, 193cf, 328ccff }}


Badness: 0.022842
Badness (Sintel): 0.944


=== Monocerus ===
=== Monocerus ===
Line 166: Line 176:
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.292
Optimal tunings:
* WE: ~99/70 = 599.8223¢, ~28/27 = 62.2737¢
* CWE: ~99/70 = 600.0000¢, ~28/27 = 62.3180¢


{{Optimal ET sequence|legend=1| 58, 96d, 154, 212ce, 366cce }}
{{Optimal ET sequence|legend=0| 58, 96d, 154, 212ce, 366cce }}


Badness: 0.052757
Badness (Sintel): 1.744


==== 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, 243/242, 364/363


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


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.301
* WE: ~55/39 = 599.8267¢, ~26/25 = 62.2833¢
* CWE: ~55/39 = 600.0000¢, ~26/25 = 62.3262¢


{{Optimal ET sequence|legend=1| 58, 96d, 154, 366ccef }}
{{Optimal ET sequence|legend=0| 58, 96d, 154, 366ccef }}


Badness: 0.028795
Badness (Sintel): 1.190
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 126/125, 196/195, 221/220, 243/242, 289/288
 
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}
 
* WE: ~17/12 = 600.0715¢, ~26/25 = 62.3767¢
* CWE: ~17/12 = 600.0000¢, ~26/25 = 62.3605¢
 
{{Optimal ET sequence|legend=0| 58, 96d, 154 }}
 
Badness (Sintel): 1.237


== Rhinoceros ==
== Rhinoceros ==
Line 192: Line 219:
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.1031¢, ~21/20 = 63.0830¢
* [[CWE]]: ~2 = 1200.0000¢, ~21/20 = 62.6687¢


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


[[Badness]]: 0.081864
[[Badness]] (Sintel): 2.072


=== 11-limit ===
=== 11-limit ===
Line 205: Line 234:
Mapping: {{mapping|  1 2 3 3 4 | 0 -8 -13 -4 -10 }}
Mapping: {{mapping|  1 2 3 3 4 | 0 -8 -13 -4 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 62.874
Optimal tunings:
* WE: ~2 = 1201.1458¢, ~21/20 = 62.9338¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.7783¢


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


Badness: 0.059319
Badness (Sintel): 1.961


=== 13-limit ===
=== 13-limit ===
Line 218: Line 249:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}
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 tunings:
* WE: ~2 = 1202.2047¢, ~21/20 = 63.1591¢
* CWE: ~2 = 1200.0000¢, ~21/20 = 62.8727¢


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


Badness: 0.039343
Badness (Sintel): 1.626


[[Category:Temperament families]]
[[Category:Temperament families]]

Revision as of 04:09, 25 April 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, 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:

  • WE: ~2 = 1200.0889¢, ~250/243 = 62.4623¢
  • CWE: ~2 = 1200.0000¢, ~250/243 = 62.4494¢

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

Badness (Sintel): 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:

  • WE: ~2 = 1199.6949¢, ~28/27 = 62.2621¢
  • CWE: ~2 = 1200.0000¢, ~28/27 = 62.2996¢

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

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

Optimal tunings:

  • WE: ~2 = 1199.7013¢, ~26/25 = 62.2785¢
  • CWE: ~2 = 1200.0000¢, ~26/25 = 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

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

Optimal tuning (CTE): 2 = 1\1, ~26/25 = 62.334

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

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

Optimal tuning (CTE): 2 = 1\1, ~26/25 = 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 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): 1.294

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¢, ~26/25 = 62.0480¢
  • CWE: ~2 = 1200.0000¢, ~26/25 = 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): 2.180

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¢, ~26/25 = 62.4603¢
  • CWE: ~2 = 1200.0000¢, ~26/25 = 62.4277¢

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

Badness (Sintel): 1.494

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: 58, 77, 135c, 193c, 328cc

Badness (Sintel): 1.370

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¢, ~26/25 = 62.1601¢
  • CWE: ~2 = 1200.0000¢, ~26/25 = 62.2251¢

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

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: 58, 96d, 154, 212ce, 366cce

Badness (Sintel): 1.744

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]]

  • WE: ~55/39 = 599.8267¢, ~26/25 = 62.2833¢
  • CWE: ~55/39 = 600.0000¢, ~26/25 = 62.3262¢

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

Badness (Sintel): 1.190

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]]

  • WE: ~17/12 = 600.0715¢, ~26/25 = 62.3767¢
  • CWE: ~17/12 = 600.0000¢, ~26/25 = 62.3605¢

Optimal ET sequence: 58, 96d, 154

Badness (Sintel): 1.237

Rhinoceros

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¢
  • CWE: ~2 = 1200.0000¢, ~21/20 = 62.6687¢

Optimal ET sequence: 1c, 19

Badness (Sintel): 2.072

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, 19

Badness (Sintel): 1.961

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, 19

Badness (Sintel): 1.626

Retrieved from "https://en.xen.wiki/index.php?title=Unicorn_family&oldid=228442"