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Alphatricot family: Difference between revisions

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== Alphatricot ==
== Alphatricot ==
Alphatricot is a [[microtemperament]] whose generator is the real cube root of [[3/1|3rd]] [[harmonic]], 3<sup>1/3</sup>, tuned between 63/44 and 13/9. Its [[ploidacot]] is alpha-tricot. It is a member of the [[schismic–Mercator equivalence continuum]] with {{nowrap|''n'' {{=}} 3 }}, so unless 53edo is used as a tuning, the schisma is always observed.  
Alphatricot is a [[microtemperament]] whose generator is the real cube root of the [[3/1|3rd]] [[harmonic]], 3<sup>1/3</sup>, tuned between [[63/44]] and [[13/9]] and representing the acute augmented fourth of 59049/40960, that is, a [[729/512|Pythagorean augmented fourth]] plus a [[81/80|syntonic comma]]. Its [[ploidacot]] is alpha-tricot. It is a member of the [[schismic–Mercator equivalence continuum]] with {{nowrap|''n'' {{=}} 3 }}, so unless 53edo is used as a tuning, the [[schisma]] is always observed.  


The temperament was named by [[Paul Erlich]] in 2002 as ''tricot''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5041.html Yahoo! Tuning Group | ''Paul's new names'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5080.html#5113 Yahoo! Tuning Group | ''Ultimate 5-limit comma list'']</ref>, but renamed in 2025 following the specifications of ploidacot.  
The temperament was named by [[Paul Erlich]] in 2002 as ''tricot''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5041.html Yahoo! Tuning Group | ''Paul's new names'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5080.html#5113 Yahoo! Tuning Group | ''Ultimate 5-limit comma list'']</ref>, but renamed in 2025 following the specifications of ploidacot.  
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* [[Alphatricot19]] – improper [[17L 2s]]
* [[Alphatricot19]] – improper [[17L 2s]]


== Alphatrimot ==
== Alphatrillium ==
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>, can be described as the {{nowrap| 53 & 70 }} temperament.  
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]].  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2430/2401, 5120/5103
[[Comma list]]: 4375/4374, 1099511627776/1098337086315


{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}
{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}


{{Multival|legend=1| 3 29 11 39 9 -56 }}
{{Multival|legend=1| 3 29 -95 39 -159 -302 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000, ~81/56 = 634.0259
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000, ~23625/16384 = 634.0118


{{Optimal ET sequence|legend=1| 17c, 36c, 53, 70, 229dd, 282dd }}
{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1376, 3193, 4569 }}


[[Badness]] (Smith): 0.100127
[[Badness]] (Smith): 0.030852


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 121/120, 5120/5103
Comma list: 4375/4374, 131072/130977, 759375/758912
 
Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}
 
Optimal tuning (POTE): ~2 = 1200.0000, ~3888/2695 = 634.0094
 
{{Optimal ET sequence|legend=0| 53, 441, 494, 935, 1429 }}
 
Badness (Smith): 0.046758
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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 tuning (POTE): ~2 = 1200.0000, ~75/52 = 634.0095
 
{{Optimal ET sequence|legend=0| 53, 441, 494, 935, 1429 }}
 
Badness (Smith): 0.019393
 
=== Pseudotrillium ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 5632/5625, 4108797/4096000


Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}
Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~63/44 = 634.0273
Optimal tuning (POTE): ~2 = 1200.0000, ~231/160 = 634.0190


{{Optimal ET sequence|legend=0| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}


Badness (Smith): 0.056134
Badness (Smith): 0.111931


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 121/120, 169/168, 352/351
Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374


Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}
Mapping: {{mapping| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~13/9 = 634.0115
Optimal tuning (POTE): ~2 = 1200.0000, ~75/52 = 634.0181


{{Optimal ET sequence|legend=0| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}


Badness (Smith): 0.032102
Badness (Smith): 0.054837


== Alphatrident ==
== Alphatrident ==
Alphatrident, named by [[Xenllium]] in 2021 as ''trident'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 229 }} temperament.  
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness (Smith): 0.046593
Badness (Smith): 0.046593


== Alphatrillium ==
== Alphatrimot ==
Alphatrillium, also named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament.  
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 1099511627776/1098337086315
[[Comma list]]: 2430/2401, 5120/5103


{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}
{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}


{{Multival|legend=1| 3 29 -95 39 -159 -302 }}
{{Multival|legend=1| 3 29 11 39 9 -56 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000, ~23625/16384 = 634.0118
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000, ~81/56 = 634.0259


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1376, 3193, 4569 }}
{{Optimal ET sequence|legend=1| 17c, 36c, 53, 70, 229dd, 282dd }}


[[Badness]] (Smith): 0.030852
[[Badness]] (Smith): 0.100127


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 131072/130977, 759375/758912
Comma list: 99/98, 121/120, 5120/5103


Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}
Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~3888/2695 = 634.0094
Optimal tuning (POTE): ~2 = 1200.0000, ~63/44 = 634.0273


{{Optimal ET sequence|legend=0| 53, 441, 494, 935, 1429 }}
{{Optimal ET sequence|legend=0| 17c, 36ce, 53, 70, 123de }}


Badness (Smith): 0.046758
Badness (Smith): 0.056134


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078
Comma list: 99/98, 121/120, 169/168, 352/351


Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}
Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~75/52 = 634.0095
Optimal tuning (POTE): ~2 = 1200.0000, ~13/9 = 634.0115


{{Optimal ET sequence|legend=0| 53, 441, 494, 935, 1429 }}
{{Optimal ET sequence|legend=0| 17c, 36ce, 53, 70, 123de }}


Badness (Smith): 0.019393
Badness (Smith): 0.032102
 
=== Pseudotrillium ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 5632/5625, 4108797/4096000
 
Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}
 
Optimal tuning (POTE): ~2 = 1200.0000, ~231/160 = 634.0190
 
{{Optimal ET sequence|legend=0| 53, 335, 388 }}
 
Badness (Smith): 0.111931
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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 tuning (POTE): ~2 = 1200.0000, ~75/52 = 634.0181
 
{{Optimal ET sequence|legend=0| 53, 335, 388 }}
 
Badness (Smith): 0.054837


== Tritricot ==
== Tritricot ==
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=== Noletaland ===
=== Noletaland ===
Noletaland is described as 282 & 1323, and it combines the smallest consistent edo in the 29-odd-limit with the smallest uniquely consistent. It reaches 4/3 in nine generators ([[noleta]]-…) and tempers out the landscape comma (…-land). Noletaland reaches [[13/11]] in 2 generators, and [[29/19]] in 5. Then there is [[44/25]] in 4, and [[152/115]] in also 4.
Noletaland is described as {{nowrap| 282 & 1323 }}, and it combines the smallest consistent edo in the 29-odd-limit with the smallest uniquely consistent. It reaches 4/3 in nine generators ([[noleta]]-…) and tempers out the landscape comma (…-land). Noletaland reaches [[13/11]] in 2 generators, and [[29/19]] in 5. Then there is [[44/25]] in 4, and [[152/115]] in also 4.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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