Alphatricot family: Difference between revisions

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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 ==

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

=== ~~Alphatrimot (2.3.5.13 subgroup)~~ ===

=== Overview to extensions ===

==== 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: {{mapping~~| ~~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~~

==== 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 (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: {{mapping~~| ~~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~~

== Alphatrillium ==

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Comma list: 4375/4374, 131072/130977, 759375/758912

~~Mapping:~~ {{~~mapping~~| 1 0 -13 53 -89 | 0 3 29 -95 175 }}

Optimal tunings:

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Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078

~~Mapping:~~ {{~~mapping~~| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}

Optimal tunings:

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Comma list: 4375/4374, 5632/5625, 4108797/4096000

~~Mapping:~~ {{~~mapping~~| 1 0 -13 53 -61 | 0 3 29 -95 122 }}

Optimal tunings:

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Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374

~~Mapping:~~ {{~~mapping~~| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}

Optimal tunings:

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== 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

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Comma list: 3388/3375, 6144/6125, 8019/8000

~~Mapping:~~ {{~~mapping~~| 1 0 -13 25 -33 | 0 3 29 -42 69 }}

Optimal tunings:

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Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125

~~Mapping:~~ {{~~mapping~~| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}

Optimal tunings:

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Comma list: 99/98, 121/120, 5120/5103

~~Mapping:~~ {{~~mapping~~| 1 0 -13 -3 -5 | 0 3 29 11 16 }}

Optimal tunings:

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Comma list: 99/98, 121/120, 169/168, 352/351

~~Mapping:~~ {{~~mapping~~| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}

Optimal tunings:

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Comma list: 4000/3993, 166698/166375, 200704/200475

~~Mapping:~~ {{~~mapping~~| 3 0 -39 -74 -34 | 0 3 29 52 28 }}

Optimal tunings:

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Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875

~~Mapping:~~ {{~~mapping~~| 3 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}

Optimal tunings:

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Comma list: 936/935, 1575/1573, 1701/1700, 2025/2023, 8624/8619

~~Mapping:~~ {{~~mapping~~| 3 0 -39 -74 -34 -84 -2 | 0 3 29 52 28 60 9 }}

{{Mapping|legend=0| 3 0 -39 -74 -34 -84 -2 | 0 3 29 52 28 60 9 }}

Optimal tunings:

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Comma list: 250047/250000, 56723625/56689952, 78675968/78594219

~~Mapping:~~ {{~~mapping~~| 3 6 19 30 35 | 0 -9 -87 -156 -178 }}

: mappin generators: ~63/50, ~1936/1875

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Comma list: 10648/10647, 43904/43875, 85750/85683, 250047/250000

~~Mapping:~~ {{~~mapping~~| 3 6 19 30 35 36 | 0 -9 -87 -156 -178 -180 }}

Optimal tunings:

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Comma list: 2058/2057, 4914/4913, 8624/8619, 12376/12375, 250047/250000

~~Mapping:~~ {{~~mapping~~| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }}

{{Mapping|legend=0| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }}

Optimal tunings:

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Comma list: 2058/2057, 2926/2925, 3136/3135, 4200/4199, 4914/4913, 250047/250000

~~Mapping:~~ {{~~mapping~~| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }}

{{Mapping|legend=0| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }}

Optimal tunings:

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Comma list: 2058/2057, 2926/2925, 3136/3135, 3381/3380, 3520/3519, 4914/4913, 18515/18513

~~Mapping:~~ {{~~mapping~~| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }}

{{Mapping|legend=0| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }}

Optimal tunings:

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Comma list: 2058/2057, 2755/2754, 2926/2925, 3136/3135, 3381/3380, 3451/3450, 3520/3519, 4914/4913

~~Mapping:~~ {{~~mapping~~| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }}

{{Mapping|legend=0| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }}

Optimal tunings:

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Badness (Sintel): 1.40

== ~~Notes~~ ==

== 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

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) ===

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

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 ==

[[Category:Alphatricot family| ]]

[[Category:Alphatricot| ]]

[[Category:Temperament families]]

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

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 1: / Line 1: @@
 {{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).
-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 ==
@@ Line 30: / Line 28: @@
 * [[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: {{mapping| 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
+==== 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 (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: {{mapping| 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
 == Alphatrillium ==
@@ Line 92: / Line 61: @@
 Comma list: 4375/4374, 131072/130977, 759375/758912
-Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}
+{{Mapping|legend=0| 1 0 -13 53 -89 | 0 3 29 -95 175 }}
 Optimal tunings:
@@ Line 107: / Line 76: @@
 Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078
-Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}
+{{Mapping|legend=0| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}
 Optimal tunings:
@@ Line 122: / Line 91: @@
 Comma list: 4375/4374, 5632/5625, 4108797/4096000
-Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}
+{{Mapping|legend=0| 1 0 -13 53 -61 | 0 3 29 -95 122 }}
 Optimal tunings:
@@ Line 137: / Line 106: @@
 Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374
-Mapping: {{mapping| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}
+{{Mapping|legend=0| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}
 Optimal tunings:
@@ Line 148: / Line 117: @@
 == 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
@@ Line 171: / Line 140: @@
 Comma list: 3388/3375, 6144/6125, 8019/8000
-Mapping: {{mapping| 1 0 -13 25 -33 | 0 3 29 -42 69 }}
+{{Mapping|legend=0| 1 0 -13 25 -33 | 0 3 29 -42 69 }}
 Optimal tunings:
@@ Line 186: / Line 155: @@
 Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125
-Mapping: {{mapping| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}
+{{Mapping|legend=0| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}
 Optimal tunings:
@@ Line 220: / Line 189: @@
 Comma list: 99/98, 121/120, 5120/5103
-Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}
+{{Mapping|legend=0| 1 0 -13 -3 -5 | 0 3 29 11 16 }}
 Optimal tunings:
@@ Line 235: / Line 204: @@
 Comma list: 99/98, 121/120, 169/168, 352/351
-Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}
+{{Mapping|legend=0| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}
 Optimal tunings:
@@ Line 268: / Line 237: @@
 Comma list: 4000/3993, 166698/166375, 200704/200475
-Mapping: {{mapping| 3 0 -39 -74 -34 | 0 3 29 52 28 }}
+{{Mapping|legend=0| 3 0 -39 -74 -34 | 0 3 29 52 28 }}
 Optimal tunings:
@@ Line 283: / Line 252: @@
 Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875
-Mapping: {{mapping| 3 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}
+{{Mapping|legend=0| 3 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}
 Optimal tunings:
@@ Line 298: / Line 267: @@
 Comma list: 936/935, 1575/1573, 1701/1700, 2025/2023, 8624/8619
-Mapping: {{mapping| 3 0 -39 -74 -34 -84 -2 | 0 3 29 52 28 60 9 }}
+{{Mapping|legend=0| 3 0 -39 -74 -34 -84 -2 | 0 3 29 52 28 60 9 }}
 Optimal tunings:
@@ Line 315: / Line 284: @@
 Comma list: 250047/250000, 56723625/56689952, 78675968/78594219
-Mapping: {{mapping| 3 6 19 30 35 | 0 -9 -87 -156 -178 }}
+{{Mapping|legend=0| 3 6 19 30 35 | 0 -9 -87 -156 -178 }}
 : mappin generators: ~63/50, ~1936/1875
@@ Line 331: / Line 300: @@
 Comma list: 10648/10647, 43904/43875, 85750/85683, 250047/250000
-Mapping: {{mapping| 3 6 19 30 35 36 | 0 -9 -87 -156 -178 -180 }}
+{{Mapping|legend=0| 3 6 19 30 35 36 | 0 -9 -87 -156 -178 -180 }}
 Optimal tunings:
@@ Line 346: / Line 315: @@
 Comma list: 2058/2057, 4914/4913, 8624/8619, 12376/12375, 250047/250000
-Mapping: {{mapping| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }}
+{{Mapping|legend=0| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }}
 Optimal tunings:
@@ Line 361: / Line 330: @@
 Comma list: 2058/2057, 2926/2925, 3136/3135, 4200/4199, 4914/4913, 250047/250000
-Mapping: {{mapping| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }}
+{{Mapping|legend=0| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }}
 Optimal tunings:
@@ Line 376: / Line 345: @@
 Comma list: 2058/2057, 2926/2925, 3136/3135, 3381/3380, 3520/3519, 4914/4913, 18515/18513
-Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }}
+{{Mapping|legend=0| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }}
 Optimal tunings:
@@ Line 391: / Line 360: @@
 Comma list: 2058/2057, 2755/2754, 2926/2925, 3136/3135, 3381/3380, 3451/3450, 3520/3519, 4914/4913
-Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }}
+{{Mapping|legend=0| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }}
 Optimal tunings:
@@ Line 401: / Line 370: @@
 Badness (Sintel): 1.40
-== Notes ==
+== 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 ==
+[[Category:Alphatricot family| ]] <!-- main article -->
+[[Category:Alphatricot| ]] <!-- key article -->
 [[Category:Temperament families]]
-[[Category:Alphatricot family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]