Alphatricot family: Difference between revisions

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The generator for '''tricot temperament''' is the real cube root of third harmonic, 3<sup>1/3</sup>, tuned between 63/44 and 13/9. Tricot temperament can be described as 53&amp;70 temperament, tempering out the [[tricot comma]], {{monzo| 39 -29 3 }} in the 5-limit. There are some mappings for 7-limit extension of this temperament: septimal tricot (53 &amp; 70, also called as "trimot"), trident (53 &amp; 229) and trillium (53 &amp; 441). Tempering out [[5120/5103|hemifamity comma]] (5120/5103) leads to septimal tricot, [[6144/6125|porwell comma]] (6144/6125) leads to trident, and [[4375/4374|ragisma]] (4375/4374) leads to trillium.
{{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 (53 & 70), alphatrident (53 & 229) and alphatrillium (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 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.  


== Tricot ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 39 -29 3 }} = 68719476736000/68630377364883
[[Comma list]]: {{monzo| 39 -29 3 }}


{{Mapping|legend=1| 1 0 -13 | 0 3 29 }}
{{Mapping|legend=1| 1 0 -13 | 0 3 29 }}
Line 10: Line 17:
: mapping generators: ~2, ~59049/40960
: mapping generators: ~2, ~59049/40960


{{Multival|legend=1| 3 29 39 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0102
: [[error map]]: {{val| 0.0000 +0.0757 -0.0168 }}
* [[POTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0124
: error map: {{val| 0.0000 +0.0821 +0.0454 }}
 
{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc }}
 
[[Badness]] (Smith): 0.046093
 
; Scales
* [[Alphatricot17]] – proper [[2L 15s]]
* [[Alphatricot19]] – improper [[17L 2s]]
 
=== 2.3.5.13 subgroup ===
{{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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~59049/40960 = 634.012
Mapping: {{mapping| 1 0 -13 0 | 0 3 29 7 }}


{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/9 = 634.0179
* POTE: ~2 = 1200.000, ~13/9 = 633.9970


[[Badness]]: 0.046093
{{Optimal ET sequence|legend=0| 17c, 36c, 53 }}


== Septimal tricot aka trimot ==
Badness (Sintel): 1.262
[[Subgroup]]: 2.3.5.7
 
=== 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


[[Comma list]]: 2430/2401, 5120/5103
Comma list: 140628/140625, 256000/255879


{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}
Mapping: {{mapping| 1 0 -13 -28 | 0 3 29 60 }}


{{Multival|legend=1| 3 29 11 39 9 -56 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~75/52 = 634.009
* POTE: ~2 = 1200.000, ~75/52 = 634.0108


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 634.0259
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }}


{{Optimal ET sequence|legend=1| 17c, 36c, 53, 70, 229dd, 282dd }}
Badness (Sintel): 0.181


[[Badness]]: 0.100127
== 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]].  


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


Comma list: 99/98, 121/120, 5120/5103
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}
[[Comma list]]: 4375/4374, 1099511627776/1098337086315


Optimal tuning (POTE): ~2 = 1\1, ~63/44 = 634.0273
{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}


{{Optimal ET sequence|legend=1| 17c, 36ce, 53, 70, 123de }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0121
: [[error map]]: {{val| 0.0000 +0.0813 +0.0372 +0.0247 }}
* [[POTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0118
: error map: {{val| 0.0000 +0.0804 +0.0283 +0.0537 }}


Badness: 0.056134
{{Optimal ET sequence|legend=1| 53, …, 335, 388, 441, 935, 1376, 3193, 4569, 5945, 10514b }}


=== 13-limit ===
[[Badness]] (Smith): 0.030852
Subgroup: 2.3.5.7.11.13


Comma list: 99/98, 121/120, 169/168, 352/351
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}
Comma list: 4375/4374, 131072/130977, 759375/758912


Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0115
Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}


{{Optimal ET sequence|legend=1| 17c, 36ce, 53, 70, 123de }}
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3888/2695 = 634.0091
* POTE: ~2 = 1200.0000, ~3888/2695 = 634.0094


Badness: 0.032102
{{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e }}


== Trident ==
Badness (Smith): 0.046758
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 14348907/14336000
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}
Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078


{{Multival|legend=1| 3 29 -42 39 -75 -179 }}
Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4096/2835 = 634.0480
Optimal tunings:
* CTE: ~2 = 1200.0000, ~75/52 = 634.0091
* POTE: ~2 = 1200.0000, ~75/52 = 634.0095


{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511 }}
{{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e, 3352de }}


[[Badness]]: 0.101694
Badness (Smith): 0.019393


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


Comma list: 3388/3375, 6144/6125, 8019/8000
Comma list: 4375/4374, 5632/5625, 4108797/4096000


Mapping: {{mapping| 1 0 -13 25 -33 | 0 3 29 -42 69 }}
Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}


Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0669
Optimal tunings:
* CTE: ~2 = 1200.0000, ~231/160 = 634.0195
* POTE: ~2 = 1200.0000, ~231/160 = 634.0190


{{Optimal ET sequence|legend=1| 53, 176, 229 }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}


Badness: 0.074272
Badness (Smith): 0.111931


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


Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125
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: {{mapping| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~75/52 = 634.0185
* POTE: ~2 = 1200.0000, ~75/52 = 634.0181


Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0652
{{Optimal ET sequence|legend=0| 53, 335, 388 }}


{{Optimal ET sequence|legend=1| 53, 176, 229 }}
Badness (Smith): 0.054837


Badness: 0.046593
== 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.  


== Trillium ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 1099511627776/1098337086315
[[Comma list]]: 6144/6125, 14348907/14336000


{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}
{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}


{{Multival|legend=1| 3 29 -95 39 -159 -302 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0484
: [[error map]]: {{val| 0.0000 +0.1901 +1.0893 +1.1421 }}
* [[POTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0480
: error map: {{val| 0.0000 +0.1890 +1.0784 +1.1579 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~23625/16384 = 634.0118
{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511, 793cd }}


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1376, 3193, 4569 }}
[[Badness]] (Smith): 0.101694
 
[[Badness]]: 0.030852


=== 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: 3388/3375, 6144/6125, 8019/8000


Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}
Mapping: {{mapping| 1 0 -13 25 -33 | 0 3 29 -42 69 }}


Optimal tuning (POTE): ~2 = 1\1, ~3888/2695 = 634.0094
Optimal tunings:
* CTE: ~2 = 1200.0000, ~231/160 = 634.0630
* POTE: ~2 = 1200.0000, ~231/160 = 634.0669


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}


Badness: 0.046758
Badness (Smith): 0.074272


==== 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: 351/350, 2080/2079, 2197/2187, 3146/3125
 
Mapping: {{mapping| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~13/9 = 634.0643
* POTE: ~2 = 1200.0000, ~13/9 = 634.0652
 
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}
 
Badness (Smith): 0.046593
 
== 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 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
 
[[Comma list]]: 2430/2401, 5120/5103


Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}
{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0095
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~81/56 = 633.9681
: [[error map]]: {{val| 0.0000 -0.0508 -1.2400 +4.8227 }}
* [[POTE]]: ~2 = 1200.0000, ~81/56 = 634.0259
: error map: {{val| 0.0000 +0.1228 +0.4387 +5.4595 }}


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
{{Optimal ET sequence|legend=1| 17c, 36c, 53, 229dd, 282dd }}


Badness: 0.019393
[[Badness]] (Smith): 0.100127


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


Comma list: 4375/4374, 5632/5625, 4108797/4096000
Comma list: 99/98, 121/120, 5120/5103


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


Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0190
Optimal tunings:
* CTE: ~2 = 1200.0000, ~63/44 = 634.0214
* POTE: ~2 = 1200.0000, ~63/44 = 634.0273


{{Optimal ET sequence|legend=1| 53, 335, 388 }}
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}


Badness: 0.111931
Badness (Smith): 0.056134


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


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


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


Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0181
Optimal tunings:
* CTE: ~2 = 1200.0000, ~13/9 = 634.0275
* POTE: ~2 = 1200.0000, ~13/9 = 634.0115


{{Optimal ET sequence|legend=1| 53, 335, 388 }}
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}


Badness: 0.054837
Badness (Smith): 0.032102


== Tritricot ==
== Tritricot ==
Line 174: Line 253:
{{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 = 1\3, ~100352/91125 = 165.9837


{{Optimal ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }}
{{Optimal ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }}


[[Badness]]: 0.086081
[[Badness]] (Smith): 0.086081


=== 11-limit ===
=== 11-limit ===
Line 189: Line 266:
Mapping: {{mapping| 3 6 19 30 22 | 0 -3 -29 -52 -28 }}
Mapping: {{mapping| 3 6 19 30 22 | 0 -3 -29 -52 -28 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9835
Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9835


{{Optimal ET sequence|legend=1| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441 }}


Badness: 0.074002
Badness (Smith): 0.074002


==== 13-limit ====
==== 13-limit ====
Line 202: Line 279:
Mapping: {{mapping| 3 6 19 30 22 36 | 0 -3 -29 -52 -28 -60 }}
Mapping: {{mapping| 3 6 19 30 22 36 | 0 -3 -29 -52 -28 -60 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9842
Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9842


{{Optimal ET sequence|legend=1| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441 }}


Badness: 0.035641
Badness (Smith): 0.035641


==== 17-limit ====
==== 17-limit ====
Line 215: Line 292:
Mapping: {{mapping| 3 6 19 30 22 36 16 | 0 -3 -29 -52 -28 -60 -9 }}
Mapping: {{mapping| 3 6 19 30 22 36 16 | 0 -3 -29 -52 -28 -60 -9 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~11/10 = 165.9805
Optimal tuning (POTE): ~34/27 = 400.0000, ~11/10 = 165.9805


{{Optimal ET sequence|legend=1| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441 }}


Badness: 0.025972
Badness (Smith): 0.025972


=== 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
Line 232: Line 309:
: mappin generators: ~63/50, ~1936/1875
: mappin generators: ~63/50, ~1936/1875


Optimal tuning (CTE): ~63/50 = 1\3, ~1936/1875 = 55.3290
Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3290


{{Optimal ET sequence|legend=1| 282, 759de, 1041, 1323, 4251e }}
{{Optimal ET sequence|legend=0| 282, 759de, 1041, 1323, 4251e }}


Badness: 0.158
Badness (Smith): 0.158


==== 13-limit ====
==== 13-limit ====
Line 245: Line 322:
Mapping: {{mapping| 3 6 19 30 35 36 | 0 -9 -87 -156 -178 -180 }}
Mapping: {{mapping| 3 6 19 30 35 36 | 0 -9 -87 -156 -178 -180 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~1936/1875 = 55.3294
Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3294


{{Optimal ET sequence|legend=1| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness: 0.0725
Badness (Smith): 0.0725


==== 17-limit ====
==== 17-limit ====
Line 258: Line 335:
Mapping: {{mapping| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }}
Mapping: {{mapping| 3 6 19 30 35 36 29 | 0 -9 -87 -156 -178 -180 -121 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3295
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295


{{Optimal ET sequence|legend=1| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness: 0.0380
Badness (Smith): 0.0380


==== 19-limit ====
==== 19-limit ====
Line 271: Line 348:
Mapping: {{mapping| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }}
Mapping: {{mapping| 3 6 19 30 35 36 29 18 | 0 -9 -87 -156 -178 -180 -121 -38 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3295
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295


{{Optimal ET sequence|legend=1| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness: 0.0269
Badness (Smith): 0.0269


==== 23-limit ====
==== 23-limit ====
Line 284: Line 361:
Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }}
Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 | 0 -9 -87 -156 -178 -180 -121 -38 -126 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3296
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296


{{Optimal ET sequence|legend=1| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness: 0.0194
Badness (Smith): 0.0194


==== 29-limit ====
==== 29-limit ====
Line 297: Line 374:
Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }}
Mapping: {{mapping| 3 6 19 30 35 36 29 18 31 19 | 0 -9 -87 -156 -178 -180 -121 -38 -126 -32 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3296
Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296
 
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


{{Optimal ET sequence|legend=1| 282, 759def, 1041, 1323 }}
Badness (Smith): 0.0168


Badness: 0.0168
== Notes ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Tricot family| ]] <!-- main article -->
[[Category:Pages with mostly numerical content]]
[[Category:Tricot| ]] <!-- key article -->
[[Category:Alphatricot family| ]] <!-- main article -->
[[Category:Alphatricot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:30, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The alphatricot family of temperaments tempers out the alphatricot comma (monzo: [39 -29 3⟩, ratio: 68 719 476 736 000 / 68 630 377 364 883).

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

Alphatricot

Alphatricot is a microtemperament whose generator is the real cube root of the 3rd harmonic, 31/3, tuned between 63/44 and 13/9 and representing the acute augmented fourth of 59049/40960, that is, a Pythagorean augmented fourth plus a syntonic comma. Its ploidacot is alpha-tricot. It is a member of the schismic–Mercator equivalence continuum with 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[1][2], but renamed in 2025 following the specifications of ploidacot.

Subgroup: 2.3.5

Comma list: [39 -29 3⟩

Mapping: [⟨1 0 -13], ⟨0 3 29]]

mapping generators: ~2, ~59049/40960

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~59049/40960 = 634.0102
error map: ⟨0.0000 +0.0757 -0.0168]
  • POTE: ~2 = 1200.0000, ~59049/40960 = 634.0124
error map: ⟨0.0000 +0.0821 +0.0454]

Optimal ET sequence: 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc

Badness (Smith): 0.046093

Scales

2.3.5.13 subgroup

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: [⟨1 0 -13 0], ⟨0 3 29 7]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/9 = 634.0179
  • POTE: ~2 = 1200.000, ~13/9 = 633.9970

Optimal ET sequence: 17c, 36c, 53

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

Comma list: 140628/140625, 256000/255879

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: 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935

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 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 and 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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 1099511627776/1098337086315

Mapping: [⟨1 0 -13 53], ⟨0 3 29 -95]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~23625/16384 = 634.0121
error map: ⟨0.0000 +0.0813 +0.0372 +0.0247]
  • POTE: ~2 = 1200.0000, ~23625/16384 = 634.0118
error map: ⟨0.0000 +0.0804 +0.0283 +0.0537]

Optimal ET sequence: 53, …, 335, 388, 441, 935, 1376, 3193, 4569, 5945, 10514b

Badness (Smith): 0.030852

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -13 53 -89], ⟨0 3 29 -95 175]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~3888/2695 = 634.0091
  • POTE: ~2 = 1200.0000, ~3888/2695 = 634.0094

Optimal ET sequence: 53, 388e, 441, 494, 935, 1429, 1923e

Badness (Smith): 0.046758

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -13 53 -89 -28], ⟨0 3 29 -95 175 60]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~75/52 = 634.0091
  • POTE: ~2 = 1200.0000, ~75/52 = 634.0095

Optimal ET sequence: 53, 388e, 441, 494, 935, 1429, 1923e, 3352de

Badness (Smith): 0.019393

Pseudotrillium

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -13 53 -61], ⟨0 3 29 -95 122]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~231/160 = 634.0195
  • POTE: ~2 = 1200.0000, ~231/160 = 634.0190

Optimal ET sequence: 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: [⟨1 0 -13 53 -61 -28], ⟨0 3 29 -95 122 60]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~75/52 = 634.0185
  • POTE: ~2 = 1200.0000, ~75/52 = 634.0181

Optimal ET sequence: 53, 335, 388

Badness (Smith): 0.054837

Alphatrident

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

Subgroup: 2.3.5.7

Comma list: 6144/6125, 14348907/14336000

Mapping: [⟨1 0 -13 25], ⟨0 3 29 -42]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~4096/2835 = 634.0484
error map: ⟨0.0000 +0.1901 +1.0893 +1.1421]
  • POTE: ~2 = 1200.0000, ~4096/2835 = 634.0480
error map: ⟨0.0000 +0.1890 +1.0784 +1.1579]

Optimal ET sequence: 53, 176, 229, 282, 511, 793cd

Badness (Smith): 0.101694

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -13 25 -33], ⟨0 3 29 -42 69]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~231/160 = 634.0630
  • POTE: ~2 = 1200.0000, ~231/160 = 634.0669

Optimal ET sequence: 53, 123, 176, 229

Badness (Smith): 0.074272

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -13 25 -33 0], ⟨0 3 29 -42 69 7]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~13/9 = 634.0643
  • POTE: ~2 = 1200.0000, ~13/9 = 634.0652

Optimal ET sequence: 53, 123, 176, 229

Badness (Smith): 0.046593

Alphatrimot

Alphatrimot, named by Petr Pařízek in 2011[3] but renamed following the specifications of ploidacot, can be described as the 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

Comma list: 2430/2401, 5120/5103

Mapping: [⟨1 0 -13 -3], ⟨0 3 29 11]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~81/56 = 633.9681
error map: ⟨0.0000 -0.0508 -1.2400 +4.8227]
  • POTE: ~2 = 1200.0000, ~81/56 = 634.0259
error map: ⟨0.0000 +0.1228 +0.4387 +5.4595]

Optimal ET sequence: 17c, 36c, 53, 229dd, 282dd

Badness (Smith): 0.100127

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -13 -3 -5], ⟨0 3 29 11 16]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~63/44 = 634.0214
  • POTE: ~2 = 1200.0000, ~63/44 = 634.0273

Optimal ET sequence: 17c, 36ce, 53

Badness (Smith): 0.056134

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -13 -3 -5 0], ⟨0 3 29 11 16 7]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~13/9 = 634.0275
  • POTE: ~2 = 1200.0000, ~13/9 = 634.0115

Optimal ET sequence: 17c, 36ce, 53

Badness (Smith): 0.032102

Tritricot

Subgroup: 2.3.5.7

Comma list: 250047/250000, 11785390260224/11767897353375

Mapping: [⟨3 6 19 30], ⟨0 -3 -29 -52]]

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

Optimal ET sequence: 159, 282, 441, 2487, 2928, 3369

Badness (Smith): 0.086081

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 6 19 30 22], ⟨0 -3 -29 -52 -28]]

Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9835

Optimal ET sequence: 159, 282, 441

Badness (Smith): 0.074002

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 6 19 30 22 36], ⟨0 -3 -29 -52 -28 -60]]

Optimal tuning (POTE): ~63/50 = 400.0000, ~11/10 = 165.9842

Optimal ET sequence: 159, 282, 441

Badness (Smith): 0.035641

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨3 6 19 30 22 36 16], ⟨0 -3 -29 -52 -28 -60 -9]]

Optimal tuning (POTE): ~34/27 = 400.0000, ~11/10 = 165.9805

Optimal ET sequence: 159, 282, 441

Badness (Smith): 0.025972

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.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 6 19 30 35], ⟨0 -9 -87 -156 -178]]

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

Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3290

Optimal ET sequence: 282, 759de, 1041, 1323, 4251e

Badness (Smith): 0.158

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 6 19 30 35 36], ⟨0 -9 -87 -156 -178 -180]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~1936/1875 = 55.3294

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0725

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨3 6 19 30 35 36 29], ⟨0 -9 -87 -156 -178 -180 -121]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0380

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨3 6 19 30 35 36 29 18], ⟨0 -9 -87 -156 -178 -180 -121 -38]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3295

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0269

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨3 6 19 30 35 36 29 18 31], ⟨0 -9 -87 -156 -178 -180 -121 -38 -126]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0194

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

Mapping: [⟨3 6 19 30 35 36 29 18 31 19], ⟨0 -9 -87 -156 -178 -180 -121 -38 -126 -32]]

Optimal tuning (CTE): ~63/50 = 400.0000, ~351/340 = 55.3296

Optimal ET sequence: 282, 759def, 1041, 1323

Badness (Smith): 0.0168

Notes

  1. ↑ Yahoo! Tuning Group | Paul's new names
  2. ↑ Yahoo! Tuning Group | Ultimate 5-limit comma list
  3. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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