Unicorn family: Difference between revisions

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Line 34: Line 34:
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}


Badness (Sintel): 1.035
[[Badness]] (Sintel): 1.035


=== 2.3.5.7.13 subgroup ===
=== 2.3.5.7.13 subgroup ===
Line 49: Line 49:
{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}
{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}


Badness (Sintel): 0.590
[[Badness]] (Sintel): 0.590


==== 2.3.5.7.13.29 subgroup ====
==== 2.3.5.7.13.29 subgroup ====
Line 58: Line 58:
{{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, ~26/25 = 62.334
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7715¢, ~26/25 = 62.2860¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3141¢


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


Badness (Sintel): 0.487
[[Badness]] (Sintel): 0.487


==== 2.3.5.7.13.29.43 subgroup ====
==== 2.3.5.7.13.29.43 subgroup ====
Line 73: Line 75:
{{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, ~26/25 = 62.339
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6269¢, ~26/25 = 62.2584¢
* [[CWE]]: ~2 = 1200.0000¢, ~26/25 = 62.3008¢


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


Badness (Sintel): 0.514
[[Badness]] (Sintel): 0.514


=== Alicorn ===
=== Alicorn ===
Line 191: Line 195:
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 tunings:
* WE: ~55/39 = 599.8267¢, ~26/25 = 62.2833¢
* WE: ~55/39 = 599.8267¢, ~26/25 = 62.2833¢
* CWE: ~55/39 = 600.0000¢, ~26/25 = 62.3262¢
* CWE: ~55/39 = 600.0000¢, ~26/25 = 62.3262¢
Line 205: Line 210:
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}


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

Latest revision as of 04:18, 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 tunings:

  • WE: ~2 = 1199.7715¢, ~26/25 = 62.2860¢
  • CWE: ~2 = 1200.0000¢, ~26/25 = 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

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

Optimal tunings:

  • WE: ~2 = 1199.6269¢, ~26/25 = 62.2584¢
  • CWE: ~2 = 1200.0000¢, ~26/25 = 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): 1.294

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]]

Optimal tunings:

  • 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]]

Optimal tunings:

  • 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

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