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Alphatricot family: Difference between revisions

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{{Technical data page}}
{{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).  
Strong 7-limit extensions of this temperament include alphatrimot ({{nowrap| 53 & 70 }}), alphatrident ({{nowrap| 53 & 229 }}) and alphatrillium ({{nowrap| 53 & 441 }}). Tempering out [[5120/5103|hemifamity comma]] (5120/5103) leads to alphatrimot, [[6144/6125|porwell comma]] (6144/6125) leads to alphatrident, and [[4375/4374|ragisma]] (4375/4374) leads to alphatrillium.


== Alphatricot ==
== Alphatricot ==
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* [[Alphatricot19]] – improper [[17L 2s]]
* [[Alphatricot19]] – improper [[17L 2s]]


=== Alphatrimot (2.3.5.13 subgroup) ===
=== Overview to extensions ===
{{See also| No-fives subgroup temperaments #Threedic }}
==== 7-limit extensions ====
 
Strong 7-limit extensions of this temperament include alphatrillium ({{nowrap| 53 & 441 }}), alphatrident ({{nowrap| 53 & 229 }}), and alphatrimot ({{nowrap| 53 & 70 }}), which temper out the [[4375/4374|ragisma]] (4375/4374), [[6144/6125|porwell comma]] (6144/6125), and [[5120/5103|aberschisma]] (5120/5103), respectively.
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
 
Comma list: 2197/2187, 41067/40960
 
{{Mapping|legend=0| 1 0 -13 0 | 0 3 29 7 }}
 
Optimal tunings:
* WE: ~2 = 1200.2092{{c}}, ~13/9 = 634.1076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0032{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 36c, 53, 335f, 388f, …, 653ff }}
 
Badness (Sintel): 1.26
 
=== Alphatrillium (2.3.5.13 subgroup) ===
However, alphatricot in the 5-limit is far more accurate than threedic. Alphatrillium 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
 
Comma list: 256000/255879, 140625/140608
 
{{Mapping|legend=0| 1 0 -13 -28 | 0 3 29 60 }}
 
Optimal tunings:
* WE: ~2 = 1199.9796{{c}}, ~75/52 = 634.0000{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0103{{c}}
 
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935, 6051f, 6986f, …, 10726bff }}


Badness (Sintel): 0.181
==== Subgroup extensions ====
This family has two notable [[2.3.5.13 subgroup|2.3.5.13-subgroup]] extensions, discussed in [[#Subgroup extenions_2|#Subgroup extensions]].  


== Alphatrillium ==
== Alphatrillium ==
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== Alphatrident ==
== Alphatrident ==
Alphatrident, also named by [[Xenllium]] in 2021 as ''trident'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 229 }} temperament. It tempers out the [[garischisma]], 33554432/33480783 ({{monzo| 25 -14 0 1 }}), and finds the harmonic 7 at -14 fifths or {{nowrap| (-14) × 3 {{=}} -42 }} generator steps, so that the smallest mos scale that includes it is the 53-note one.  
Alphatrident, also named by [[Xenllium]] in 2021 as ''trident'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 229 }} temperament. It tempers out the [[septischisma]], 33554432/33480783 ({{monzo| 25 -14 0 1 }}), and finds the harmonic 7 at -14 fifths or {{nowrap| (-14) × 3 {{=}} -42 }} generator steps, so that the smallest mos scale that includes it is the 53-note one.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness (Sintel): 1.40
Badness (Sintel): 1.40
== Subgroup extensions ==
=== Alphatrillium (2.3.5.13) ===
Alphatrillium interprets the generator as [[~]][[75/52]], making the vanishing of [[140625/140608]], the catasma, instead of the threedie. It also tempers out [[256000/255879]], the phaotisma.
Subgroup: 2.3.5.13
Comma list: 140625/140608, 256000/255879
{{Mapping|legend=0| 1 0 -13 -28 | 0 3 29 60 }}
Optimal tunings:
* WE: ~2 = 1199.9796{{c}}, ~75/52 = 634.0000{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0103{{c}}
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935, 6051f, 6986f, …, 10726bff }}
Badness (Sintel): 0.181
=== Alphatrimot (2.3.5.13) ===
{{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
Comma list: 2197/2187, 41067/40960
{{Mapping|legend=0| 1 0 -13 0 | 0 3 29 7 }}
Optimal tunings:
* WE: ~2 = 1200.2092{{c}}, ~13/9 = 634.1076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0032{{c}}
{{Optimal ET sequence|legend=0| 17c, 36c, 53, 335f, 388f, …, 653ff }}
Badness (Sintel): 1.26


== References ==
== References ==


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