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{{Technical data page}}
{{Technical data page}}
The '''compton family''', otherwise known as the '''aristoxenean family''', of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[Pythagorean comma]] ([[ratio]]: 531441/524288, {{monzo|legend=1| -19 12 }}, and hence the fifths form a closed 12-note [[circle of fifths]], identical to [[12edo]]. While the tuning of the fifth will be that of 12edo, two [[cent]]s flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.
The '''compton family''', otherwise known as the '''aristoxenean family''', of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[Pythagorean comma]] ([[ratio]]: 531441/524288, {{monzo|legend=1| -19 12 }}, and hence the fifths form a closed 12-note [[circle of fifths]], identical to [[12edo]]. While the tuning of the fifth is fixed to 7 steps of 12edo, about 2{{cent}} flat of [[just]], these temperaments aim to add tunings for higher primes which are more in tune than in 12edo.


== Compton ==
== Compton ==
Line 59: Line 59:
Comma list: 225/224, 441/440, 4375/4356
Comma list: 225/224, 441/440, 4375/4356


Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}
{{Mapping|legend=0| 12 19 0 -22 -42 | 0 0 1 2 3 }}


Optimal tunings:  
Optimal tunings:  
Line 74: Line 74:
Comma list: 225/224, 351/350, 364/363, 441/440
Comma list: 225/224, 351/350, 364/363, 441/440


Mapping: {{mapping| 12 19 0 -22 -42 -67 | 0 0 1 2 3 4 }}
{{Mapping|legend=0| 12 19 0 -22 -42 -67 | 0 0 1 2 3 4 }}


Optimal tunings:  
Optimal tunings:  
Line 89: Line 89:
Comma list: 221/220, 225/224, 289/288, 351/350, 441/440
Comma list: 221/220, 225/224, 289/288, 351/350, 441/440


Mapping: {{mapping| 12 19 0 -22 -42 -67 49 | 0 0 1 2 3 4 0 }}
{{Mapping|legend=0| 12 19 0 -22 -42 -67 49 | 0 0 1 2 3 4 0 }}


Optimal tunings:  
Optimal tunings:  
Line 104: Line 104:
Comma list: 225/224, 325/324, 441/440, 1001/1000
Comma list: 225/224, 325/324, 441/440, 1001/1000


Mapping: {{mapping| 12 19 0 -22 -42 100 | 0 0 1 2 3 -2 }}
{{Mapping|legend=0| 12 19 0 -22 -42 100 | 0 0 1 2 3 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 119: Line 119:
Comma list: 225/224, 273/272, 289/288, 325/324, 441/440
Comma list: 225/224, 273/272, 289/288, 325/324, 441/440


Mapping: {{mapping| 12 19 0 -22 -42 100 49 | 0 0 1 2 3 -2 0 }}
{{Mapping|legend=0| 12 19 0 -22 -42 100 49 | 0 0 1 2 3 -2 0 }}


Optimal tunings:  
Optimal tunings:  
Line 155: Line 155:
Comma list: 81/80, 99/98, 128/125
Comma list: 81/80, 99/98, 128/125


Mapping: {{mapping| 12 19 28 0 -26 | 0 0 0 1 2 }}
{{Mapping|legend=0| 12 19 28 0 -26 | 0 0 0 1 2 }}


Optimal tunings:  
Optimal tunings:  
Line 170: Line 170:
Comma list: 81/80, 128/125, 540/539
Comma list: 81/80, 128/125, 540/539


Mapping: {{mapping| 12 19 28 0 109 | 0 0 0 1 -2 }}
{{Mapping|legend=0| 12 19 28 0 109 | 0 0 0 1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 185: Line 185:
Comma list: 56/55, 81/80, 128/125
Comma list: 56/55, 81/80, 128/125


Mapping: {{mapping| 12 19 28 0 8 | 0 0 0 1 1 }}
{{Mapping|legend=0| 12 19 28 0 8 | 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 200: Line 200:
Comma list: 56/55, 66/65, 81/80, 105/104
Comma list: 56/55, 66/65, 81/80, 105/104


Mapping: {{mapping| 12 19 28 0 8 11 | 0 0 0 1 1 1 }}
{{Mapping|legend=0| 12 19 28 0 8 11 | 0 0 0 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 215: Line 215:
Comma list: 51/50, 56/55, 66/65, 81/80, 105/104
Comma list: 51/50, 56/55, 66/65, 81/80, 105/104


Mapping: {{mapping| 12 19 28 0 8 11 49 | 0 0 0 1 1 1 0 }}
{{Mapping|legend=0| 12 19 28 0 8 11 49 | 0 0 0 1 1 1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 230: Line 230:
Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95
Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95


Mapping: {{mapping| 12 19 28 0 8 11 49 51 | 0 0 0 1 1 1 0 0 }}
{{Mapping|legend=0| 12 19 28 0 8 11 49 51 | 0 0 0 1 1 1 0 0 }}


Optimal tunings:  
Optimal tunings:  
Line 241: Line 241:


==== Duodecic ====
==== Duodecic ====
{{Distinguish|Duodecim}}
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 81/80, 91/90, 128/125
Comma list: 56/55, 81/80, 91/90, 128/125


Mapping: {{mapping| 12 19 28 0 8 78 | 0 0 0 1 1 -1 }}
{{Mapping|legend=0| 12 19 28 0 8 78 | 0 0 0 1 1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 275: Line 277:
Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95
Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95


Mapping: {{mapping| 12 19 28 0 8 78 49 51 | 0 0 0 1 1 -1 0 0 }}
{{Mapping|legend=0| 12 19 28 0 8 78 49 51 | 0 0 0 1 1 -1 0 0 }}


Optimal tunings:  
Optimal tunings:  
Line 284: Line 286:


Badness (Sintel): 1.27
Badness (Sintel): 1.27
=== Paly ===
{{Todo|inline=1|complete section|unify precision|comment= Sort the comma basis. Normalize the mapping. Unify the cent values in optimal tunings to 4 decimal places. Add the optimal ET sequence. <br>Add the data for the 11-, 13-, and 17-limit versions. }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 91/90, 51/50, 128/125, 76/75, 121/120
{{Mapping| 24 38 56 48 83 108 98 102 | 0 0 0 1 0 -1 0 0 }}
Optimal tunings:
* WE: ~33/32 = 49.994{{c}}, ~7/4 = 965.340{{c}}
: error map: {{val| -0.144 -2.183 +13.350 -3.774 -1.816 -6.516 -5.543 +1.875 }}
* CWE: ~33/32 = 50.000{{c}}, ~7/4 = 965.449{{c}}
: error map: {{val| -0.000 -1.955 +13.686 -3.377 -1.318 -5.977 -4.955 +2.487 }}
Badness (Sintel): 2.383


== Duodecim ==
== Duodecim ==
{{Distinguish|Duodecic}}
Duodecim uses exactly the same mapping as the 7-limit of 12edo, only correcting its poor approximation of prime 11.  
Duodecim uses exactly the same mapping as the 7-limit of 12edo, only correcting its poor approximation of prime 11.  


Line 332: Line 354:
Comma list: 243/242, 385/384, 9801/9800
Comma list: 243/242, 385/384, 9801/9800


Mapping: {{mapping| 24 38 0 123 83 | 0 0 1 -1 0 }}
{{Mapping|legend=0| 24 38 0 123 83 | 0 0 1 -1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 347: Line 369:
Comma list: 243/242, 351/350, 364/363, 385/384
Comma list: 243/242, 351/350, 364/363, 385/384


Mapping: {{mapping| 24 38 0 123 83 33 | 0 0 1 -1 0 1 }}
{{Mapping|legend=0| 24 38 0 123 83 33 | 0 0 1 -1 0 1 }}


Optimal tunings:  
Optimal tunings:  
Line 385: Line 407:
Comma list: 540/539, 1029/1024, 4000/3993
Comma list: 540/539, 1029/1024, 4000/3993


Mapping: {{mapping| 36 57 0 101 41 | 0 0 1 0 1 }}
{{Mapping|legend=0| 36 57 0 101 41 | 0 0 1 0 1 }}


Optimal tunings:  
Optimal tunings:  
Line 414: Line 436:
[[Badness]] (Sintel): 1.24
[[Badness]] (Sintel): 1.24


[[Category:Temperament families]]
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton| ]] <!-- key article -->
[[Category:Compton| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Compton_family"