Unicorn family: Difference between revisions

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 ==

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

[[Optimal tuning]]s:

* [[POTE]]: 2 = 1\1, ~250/243 = 62.458

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

[[Badness]]:  

* Dirichlet: 3.530

{{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 }}

* [[POTE]]: 2 = 1\1, ~28/27 = 62.278

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

Badness:  

[[Subgroup]]: 2.3.5.7.13

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

{{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

Badness (Sintel): 0.487

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

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 896/891

Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101

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

Badness: 0.039156

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 676/675

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

Badness: 0.023667

=== Camahueto ===

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 10976/10935

Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.431

Badness: 0.065940

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 676/675

Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.434

Badness: 0.036155

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 10976/10935

Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.196

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

Badness: 0.041426

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 2200/2197

Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Unicorn family| ]] <!-- main article -->

[[Category:Unicorn| ]] <!-- key article -->

[[Category:Rank 2]]