Alphatricot family: Difference between revisions

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: mapping generators: ~2, ~59049/40960

~~{{Multival|legend=1| 3 29 39 }}~~

[[Optimal tuning]]s:

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Badness (~~Dirichlet~~): 1.262

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 ~~far more accurate~~ tempering of [[140625/140608]], the catasma, instead of the threedie. It also tempers out [[256000/255879]], the phaotisma.

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

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{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }}

Badness (~~Dirichlet~~): 0.181

Badness (Sintel): 0.181

== Alphatrillium ==

Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator [[233\441]]. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] is the 123-tone one. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].

Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] is the 123-tone one. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].

It can be extended to the 11-limit by tempering out [[131072/130977]], and to the 13-limit by tempering out [[2080/2079]], [[4096/4095]] and [[4225/4224]]. The optimal tunings in the 11- and 13-limit lean towards [[494edo]]; [[935edo]] and especially [[1429edo]] are recommendable tunings.

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~~{{Multival|legend=1| 3 29 -95 39 -159 -302 }}~~

[[Optimal tuning]]s:

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~~{{Multival|legend=1| 3 29 -42 39 -75 -179 }}~~

[[Optimal tuning]]s:

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~~{{Multival|legend=1| 3 29 11 39 9 -56 }}~~

[[Optimal tuning]]s:

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~~{{Multival|legend=1| 9 87 156 117 222 118 }}~~

[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000, ~100352/91125 = 165.9837

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Alphatricot family| ]]

[[Category:Alphatricot| ]]

[[Category:Rank 2]]

@@ Line 16: / Line 16: @@
 : mapping generators: ~2, ~59049/40960
-{{Multival|legend=1| 3 29 39 }}
 [[Optimal tuning]]s:
@@ Line 50: / Line 48: @@
 {{Optimal ET sequence|legend=0| 17c, 36c, 53 }}
-Badness (Dirichlet): 1.262
+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 far more accurate tempering of [[140625/140608]], the catasma, instead of the threedie. It also tempers out [[256000/255879]], the phaotisma.
+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
@@ Line 67: / Line 65: @@
 {{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }}
-Badness (Dirichlet): 0.181
+Badness (Sintel): 0.181
 == Alphatrillium ==
-Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator [[233\441]]. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] is the 123-tone one. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].
+Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] is the 123-tone one. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].
 It can be extended to the 11-limit by tempering out [[131072/130977]], and to the 13-limit by tempering out [[2080/2079]], [[4096/4095]] and [[4225/4224]]. The optimal tunings in the 11- and 13-limit lean towards [[494edo]]; [[935edo]] and especially [[1429edo]] are recommendable tunings.
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 {{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}
-{{Multival|legend=1| 3 29 -95 39 -159 -302 }}
 [[Optimal tuning]]s:
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 {{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}
-{{Multival|legend=1| 3 29 -42 39 -75 -179 }}
 [[Optimal tuning]]s:
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 {{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}
-{{Multival|legend=1| 3 29 11 39 9 -56 }}
 [[Optimal tuning]]s:
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 {{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
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 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Alphatricot family| ]] <!-- main article -->
 [[Category:Alphatricot| ]] <!-- key article -->
 [[Category:Rank 2]]