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Alphatricot family: Difference between revisions

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{{Technical data page}}
The '''alphatricot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[alphatricot comma]] ({{monzo|legend=1| 39 -29 3 }}, [[ratio]]: 68 719 476 736 000 / 68 630 377 364 883).  
The '''alphatricot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[alphatricot comma]] ({{monzo|legend=1| 39 -29 3 }}, [[ratio]]: 68 719 476 736 000 / 68 630 377 364 883).  


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: mapping generators: ~2, ~59049/40960
: mapping generators: ~2, ~59049/40960
{{Multival|legend=1| 3 29 39 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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[[Badness]] (Smith): 0.046093
[[Badness]] (Smith): 0.046093
; Scales
* [[Alphatricot17]] – proper [[2L 15s]]
* [[Alphatricot19]] – improper [[17L 2s]]


=== 2.3.5.13 subgroup ===
=== 2.3.5.13 subgroup ===
{{See also| No-fives subgroup temperaments #Threedic }}
{{See also| No-fives subgroup temperaments #Threedic }}
This extension identifies the generator with [[13/9]] by tempering out the threedie, [[2197/2187]], providing a relatively low-complexity mapping for 13.


Subgroup: 2.3.5.13
Subgroup: 2.3.5.13
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Comma list: 2197/2187, 41067/40960
Comma list: 2197/2187, 41067/40960


Sval mapping: {{mapping| 1 0 -13 0 | 0 3 29 7 }}
Mapping: {{mapping| 1 0 -13 0 | 0 3 29 7 }}
 
Gencom mapping]: {{mapping| 1 0 -13 0 0 0 | 0 3 29 0 0 7 }}
 
: gencom: [2 13/9; 2197/2187, 41067/40960]


Optimal tunings:  
Optimal tunings:  
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{{Optimal ET sequence|legend=0| 17c, 36c, 53 }}
{{Optimal ET sequence|legend=0| 17c, 36c, 53 }}


RMS error: 0.2342 cents
Badness (Sintel): 1.262
 
=== Catatricot ===
However, alphatricot in the 5-limit is far more accurate than threedic. Catatricot interprets the generator as ~[[75/52]] instead of 13/9, making the tempering of [[140625/140608]], the catasma, instead of the threedie. It also tempers out [[256000/255879]], the phaotisma.
 
Subgroup: 2.3.5.13


; Scales
Comma list: 140628/140625, 256000/255879
* [[Alphatricot17]] – proper [[2L 15s]]
 
* [[Alphatricot19]] – improper [[17L 2s]]
Mapping: {{mapping| 1 0 -13 -28 | 0 3 29 60 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~75/52 = 634.009
* POTE: ~2 = 1200.000, ~75/52 = 634.0108
 
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }}
 
Badness (Sintel): 0.181


== Alphatrillium ==
== Alphatrillium ==
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{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}
{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}
{{Multival|legend=1| 3 29 -95 39 -159 -302 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}
{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}
{{Multival|legend=1| 3 29 -42 39 -75 -179 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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== Alphatrimot ==
== Alphatrimot ==
Alphatrimot, named by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref> but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 70 }} temperament. It finds prime 7 at only 11 generators up so that the generator is interpreted as a sharp ~[[81/56]], but is more of a full 13-limit system in its own right. [[123edo]] in the 123de val is a great tuning for it. Mos scales of 5, 7, 9, 11, 13, 15, 17, 19, 36 or 53 notes are available.  
Alphatrimot, named by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref> but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 70 }} temperament. It finds prime 7 at only 11 generators up so that the generator is interpreted as a flat ~[[81/56]], but is more of a full 13-limit system in its own right. [[123edo]] in the 123de val is a great tuning for it. Mos scales of 5, 7, 9, 11, 13, 15, 17, 19, 36 or 53 notes are available.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}
{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}
{{Multival|legend=1| 3 29 11 39 9 -56 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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{{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }}
{{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }}
{{Multival|legend=1| 9 87 156 117 222 118 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000, ~100352/91125 = 165.9837
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000, ~100352/91125 = 165.9837
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Alphatricot family| ]] <!-- main article -->
[[Category:Alphatricot family| ]] <!-- main article -->
[[Category:Alphatricot| ]] <!-- key article -->
[[Category:Alphatricot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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