Alphatricot family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
m FloraC moved page Tricot family to Alphatricot family: Follow the specifications of ploidacot
m Restore the category of the corresponding temp
 
(29 intermediate revisions by 7 users not shown)
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).  
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).  
There are some mappings for 7-limit extension of this temperament: trimot (53 & 70), trident (53 & 229) and trillium (53 & 441). Tempering out [[5120/5103|hemifamity comma]] (5120/5103) leads to trimot, [[6144/6125|porwell comma]] (6144/6125) leads to trident, and [[4375/4374|ragisma]] (4375/4374) leads to trillium.


== 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.  
Line 13: Line 12:


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


{{Multival|legend=1| 3 29 39 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9762{{c}}, ~59049/40960 = 633.9998{{c}}
: [[error map]]: {{val| -0.024 +0.044 -0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0116{{c}}
: error map: {{val| 0.000 +0.080 +0.022 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~59049/40960 = 634.012
{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc }}


{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
[[Badness]] (Sintel): 1.08


[[Badness]]: 0.046093
; Scales
* [[Alphatricot17]] – proper [[2L 15s]]
* [[Alphatricot19]] – improper [[17L 2s]]


=== 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.


Subgroup: 2.3.5.13
==== 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]].


[[Comma list]]: 2197/2187, 41067/40960
== 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]] that contains it is the 123-note one, though otonal and utonal tetrads don't occur until the 176-note mos due to 7/5 being mapped to -124 generators. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].


[[Gencom]]: [2 13/9; 2197/2187, 41067/40960]
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.


[[Gencom|Gencom mapping]]: [{{val|1 0 -13 0 0 0}}, {{val|0 3 29 0 0 7}}]
[[Subgroup]]: 2.3.5.7


[[Mapping|Sval mapping]]: [{{val|1 0 -13 0}}, {{val|0 3 29 7}}]
[[Comma list]]: 4375/4374, {{monzo| 40 -22 -1 -1 }}


[[Tp tuning|POL2 generator]]: ~13/9 = 633.997
{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}


{{Optimal ET sequence|legend=1| 17c, 36c, 53 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9795{{c}}, ~59049/40960 = 634.0010{{c}}
: [[error map]]: {{val| -0.021 +0.048 -0.019 -0.004 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0119{{c}}
: error map: {{val| 0.000 +0.081 +0.030 +0.048 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2342 cents
{{Optimal ET sequence|legend=1| 53, …, 335, 388, 441, 935, 1376, 3193, 4569, 5945, 10514b }}


=== Scales ===
[[Badness]] (Sintel): 0.781
* [[Tricot17]] – proper [[2L 15s]]
* [[Tricot19]] – improper [[17L 2s]]
* [[Tricot36]] – improper [[17L 19s]]


== Alphatrimot ==
=== 11-limit ===
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 53 & 70 temperament.
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 131072/130977, 759375/758912
 
{{Mapping|legend=0| 1 0 -13 53 -89 | 0 3 29 -95 175 }}
 
Optimal tunings:  
* WE: ~2 = 1199.9551{{c}}, ~3888/2695 = 633.9857{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0094{{c}}
 
{{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e }}


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 1.55


[[Comma list]]: 2430/2401, 5120/5103
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}
Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078


{{Multival|legend=1| 3 29 11 39 9 -56 }}
{{Mapping|legend=0| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~81/56 = 634.0259
Optimal tunings:
* WE: ~2 = 1199.9603{{c}}, ~75/52 = 633.9885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0094{{c}}


{{Optimal ET sequence|legend=1| 17c, 36c, 53, 70, 229dd, 282dd }}
{{Optimal ET sequence|legend=0| 53, 388e, 441, 494, 935, 1429, 1923e, 3352de }}


[[Badness]]: 0.100127
Badness (Sintel): 0.801


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


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


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


Optimal tuning (POTE): ~2 = 1\1, ~63/44 = 634.0273
Optimal tunings:
* WE: ~2 = 1200.0692{{c}}, ~231/160 = 634.0556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0191{{c}}


{{Optimal ET sequence|legend=1| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}


Badness: 0.056134
Badness (Sintel): 3.70


=== 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|legend=0| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0115
Optimal tunings:
* WE: ~2 = 1200.0351{{c}}, ~75/52 = 634.0366{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0181{{c}}


{{Optimal ET sequence|legend=1| 17c, 36ce, 53, 70, 123de }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}


Badness: 0.032102
Badness (Sintel): 2.27


== Alphatrident ==
== Alphatrident ==
Alphatrident, named by [[Xenllium]] in 2021, can be described as the 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 [[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
Line 100: Line 125:
{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}
{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}


{{Multival|legend=1| 3 29 -42 39 -75 -179 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7509{{c}}, ~4096/2835 = 633.9164{{c}}
: [[error map]]: {{val| -0.249 -0.206 +0.500 +0.458 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0481{{c}}
: error map: {{val| 0.000 +0.189 +1.081 +1.155 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4096/2835 = 634.0480
{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511, 793cd }}


{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511 }}
[[Badness]] (Sintel): 2.57
 
[[Badness]]: 0.101694


=== 11-limit ===
=== 11-limit ===
Line 113: Line 140:
Comma list: 3388/3375, 6144/6125, 8019/8000
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 tuning (POTE): ~2 = 1\1, ~231/160 = 634.0669
Optimal tunings:
* WE: ~2 = 1199.8432{{c}}, ~231/160 = 633.9840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0662{{c}}


{{Optimal ET sequence|legend=1| 53, 176, 229 }}
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}


Badness: 0.074272
Badness (Sintel): 2.46


=== 13-limit ===
=== 13-limit ===
Line 126: Line 155:
Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125
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 tuning (POTE): ~2 = 1\1, ~13/9 = 634.0652
Optimal tunings:
* WE: ~2 = 1199.9675{{c}}, ~13/9 = 634.0480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0651{{c}}


{{Optimal ET sequence|legend=1| 53, 176, 229 }}
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}


Badness: 0.046593
Badness (Sintel): 1.93


== Alphatrillium ==
== Alphatrimot ==
Alphatrillium, also named by [[Xenllium]] in 2021, can be described as the 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 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
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~23625/16384 = 634.0118
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4448{{c}}, ~81/56 = 633.7326{{c}}
: [[error map]]: {{val| -0.555 -0.757 +0.851 +3.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~81/56 = 634.0071{{c}}
: error map: {{val| 0.000 +0.066 -0.108 +5.252 }}


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


[[Badness]]: 0.030852
[[Badness]] (Sintel): 2.53


=== 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|legend=0| 1 0 -13 -3 -5 | 0 3 29 11 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~3888/2695 = 634.0094
Optimal tunings:
* WE: ~2 = 1199.9429{{c}}, ~63/44 = 633.9971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0253{{c}}


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}


Badness: 0.046758
Badness (Sintel): 1.86


==== 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 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0095


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
{{Mapping|legend=0| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}


Badness: 0.019393
Optimal tunings:  
* WE: ~2 = 1200.1213{{c}}, ~13/9 = 634.0757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0154{{c}}


=== Pseudotrillium ===
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 5632/5625, 4108797/4096000
Badness (Sintel): 1.33
 
Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0190
 
{{Optimal ET sequence|legend=1| 53, 335, 388 }}
 
Badness: 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 = 1\1, ~75/52 = 634.0181
 
{{Optimal ET sequence|legend=1| 53, 335, 388 }}
 
Badness: 0.054837


== Tritricot ==
== Tritricot ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 11785390260224/11767897353375
[[Comma list]]: 250047/250000, {{monzo| 35 -23 -3 3 }}


{{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }}
{{Mapping|legend=1| 3 0 -39 -74 | 0 3 29 52 }}
: mapping generators: ~63/50, ~59049/40960


{{Multival|legend=1| 9 87 156 117 222 118 }}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9887{{c}}, ~59049/40960 = 633.7326{{c}} (~100352/91125 = 165.9790{{c}})
: [[error map]]: {{val| -0.034 +0.040 +0.081 -0.073 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~59049/40960 = 634.0155{{c}} (~100352/91125 = 165.9845{{c}})
: error map: {{val| 0.000 +0.092 -0.137 -0.018 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~100352/91125 = 165.9837
{{Optimal ET sequence|legend=1| 159, 282, 441, 1605, 2046, 2487, 2928 }}


{{Optimal ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }}
[[Badness]] (Sintel): 2.18
 
[[Badness]]: 0.086081


=== 11-limit ===
=== 11-limit ===
Line 223: Line 237:
Comma list: 4000/3993, 166698/166375, 200704/200475
Comma list: 4000/3993, 166698/166375, 200704/200475


Mapping: {{mapping| 3 6 19 30 22 | 0 -3 -29 -52 -28 }}
{{Mapping|legend=0| 3 0 -39 -74 -34 | 0 3 29 52 28 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9835
Optimal tunings:
* WE: ~63/50 = 399.9686{{c}}, ~3969/2750 = 633.9667{{c}} (~11/10 = 165.9705{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~3969/2750 = 634.0142{{c}} (~11/10 = 165.9858{{c}})


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


Badness: 0.074002
Badness (Sintel): 2.45


==== 13-limit ====
==== 13-limit ====
Line 236: Line 252:
Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875
Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875


Mapping: {{mapping| 3 6 19 30 22 36 | 0 -3 -29 -52 -28 -60 }}
{{Mapping|legend=0| 3 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9842
Optimal tunings:
* WE: ~63/50 = 399.9692{{c}}, ~75/52 = 633.9669{{c}} (~11/10 = 165.9714{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~75/52 = 634.0137{{c}} (~11/10 = 165.9863{{c}})


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


Badness: 0.035641
Badness (Sintel): 1.47


==== 17-limit ====
==== 17-limit ====
Line 249: Line 267:
Comma list: 936/935, 1575/1573, 1701/1700, 2025/2023, 8624/8619
Comma list: 936/935, 1575/1573, 1701/1700, 2025/2023, 8624/8619


Mapping: {{mapping| 3 6 19 30 22 36 16 | 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 tuning (POTE): ~34/27 = 1\3, ~11/10 = 165.9805
Optimal tunings:
* WE: ~34/27 = 399.9491{{c}}, ~75/52 = 633.9389{{c}} (~11/10 = 165.9594{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~75/52 = 634.0166{{c}} (~11/10 = 165.9834{{c}})


{{Optimal ET sequence|legend=1| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441, 723efg, 1164eefgg }}


Badness: 0.025972
Badness (Sintel): 1.32


=== 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 264: Line 284:
Comma list: 250047/250000, 56723625/56689952, 78675968/78594219
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
: mappin generators: ~63/50, ~1936/1875


Optimal tuning (CTE): ~63/50 = 1\3, ~1936/1875 = 55.3290
Optimal tunings:
* WE: ~63/50 = 399.9895{{c}}, ~1936/1875 = 55.3269{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3286{{c}}


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


Badness: 0.158
Badness (Sintel): 5.23


==== 13-limit ====
==== 13-limit ====
Line 279: Line 300:
Comma list: 10648/10647, 43904/43875, 85750/85683, 250047/250000
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 tuning (CTE): ~63/50 = 1\3, ~1936/1875 = 55.3294
Optimal tunings:
* WE: ~63/50 = 399.9896{{c}}, ~1936/1875 = 55.3273{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3289{{c}}


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


Badness: 0.0725
Badness (Sintel): 2.99


==== 17-limit ====
==== 17-limit ====
Line 292: Line 315:
Comma list: 2058/2057, 4914/4913, 8624/8619, 12376/12375, 250047/250000
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 tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3295
Optimal tunings:
* WE: ~63/50 = 399.9876{{c}}, ~351/340 = 55.3270{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3290{{c}}


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


Badness: 0.0380
Badness (Sintel): 1.93


==== 19-limit ====
==== 19-limit ====
Line 305: Line 330:
Comma list: 2058/2057, 2926/2925, 3136/3135, 4200/4199, 4914/4913, 250047/250000
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 tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3295
Optimal tunings:
* WE: ~63/50 = 399.9914{{c}}, ~351/340 = 55.3277{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}


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


Badness: 0.0269
Badness (Sintel): 1.64


==== 23-limit ====
==== 23-limit ====
Line 318: Line 345:
Comma list: 2058/2057, 2926/2925, 3136/3135, 3381/3380, 3520/3519, 4914/4913, 18515/18513
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 tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3296
Optimal tunings:
* WE: ~63/50 = 399.9899{{c}}, ~351/340 = 55.3276{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}


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


Badness: 0.0194
Badness (Sintel): 1.39


==== 29-limit ====
==== 29-limit ====
Line 331: Line 360:
Comma list: 2058/2057, 2755/2754, 2926/2925, 3136/3135, 3381/3380, 3451/3450, 3520/3519, 4914/4913
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 tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3296
Optimal tunings:
* WE: ~63/50 = 399.9940{{c}}, ~351/340 = 55.3283{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3293{{c}}


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


Badness: 0.0168
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:Temperament families]]
[[Category:Alphatricot family| ]] <!-- main article -->
[[Category:Alphatricot family| ]] <!-- main article -->
[[Category:Alphatricot| ]] <!-- key article -->
[[Category:Alphatricot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 17:22, 7 October 2026

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).

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:

  • WE: ~2 = 1199.9762 ¢, ~59049/40960 = 633.9998 ¢
error map: ⟨-0.024 +0.044 -0.010]
  • CWE: ~2 = 1200.0000 ¢, ~59049/40960 = 634.0116 ¢
error map: ⟨0.000 +0.080 +0.022]

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

Badness (Sintel): 1.08

Scales

Overview to extensions

7-limit extensions

Strong 7-limit extensions of this temperament include alphatrillium (53 & 441), alphatrident (53 & 229), and alphatrimot (53 & 70), which temper out the ragisma (4375/4374), porwell comma (6144/6125), and aberschisma (5120/5103), respectively.

Subgroup extensions

This family has two notable 2.3.5.13-subgroup extensions, discussed in #Subgroup extensions.

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 that contains it is the 123-note one, though otonal and utonal tetrads don't occur until the 176-note mos due to 7/5 being mapped to -124 generators. 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, [40 -22 -1 -1⟩

Mapping: [⟨1 0 -13 53], ⟨0 3 29 -95]]

Optimal tunings:

  • WE: ~2 = 1199.9795 ¢, ~59049/40960 = 634.0010 ¢
error map: ⟨-0.021 +0.048 -0.019 -0.004]
  • CWE: ~2 = 1200.0000 ¢, ~59049/40960 = 634.0119 ¢
error map: ⟨0.000 +0.081 +0.030 +0.048]

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

Badness (Sintel): 0.781

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:

  • WE: ~2 = 1199.9551 ¢, ~3888/2695 = 633.9857 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3888/2695 = 634.0094 ¢

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

Badness (Sintel): 1.55

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:

  • WE: ~2 = 1199.9603 ¢, ~75/52 = 633.9885 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/52 = 634.0094 ¢

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

Badness (Sintel): 0.801

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:

  • WE: ~2 = 1200.0692 ¢, ~231/160 = 634.0556 ¢
  • CWE: ~2 = 1200.0000 ¢, ~231/160 = 634.0191 ¢

Optimal ET sequence: 53, 335, 388

Badness (Sintel): 3.70

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:

  • WE: ~2 = 1200.0351 ¢, ~75/52 = 634.0366 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/52 = 634.0181 ¢

Optimal ET sequence: 53, 335, 388

Badness (Sintel): 2.27

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 septischisma, 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:

  • WE: ~2 = 1199.7509 ¢, ~4096/2835 = 633.9164 ¢
error map: ⟨-0.249 -0.206 +0.500 +0.458]
  • CWE: ~2 = 1200.0000 ¢, ~4096/2835 = 634.0481 ¢
error map: ⟨0.000 +0.189 +1.081 +1.155]

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

Badness (Sintel): 2.57

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:

  • WE: ~2 = 1199.8432 ¢, ~231/160 = 633.9840 ¢
  • CWE: ~2 = 1200.0000 ¢, ~231/160 = 634.0662 ¢

Optimal ET sequence: 53, 123, 176, 229

Badness (Sintel): 2.46

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:

  • WE: ~2 = 1199.9675 ¢, ~13/9 = 634.0480 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/9 = 634.0651 ¢

Optimal ET sequence: 53, 123, 176, 229

Badness (Sintel): 1.93

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:

  • WE: ~2 = 1199.4448 ¢, ~81/56 = 633.7326 ¢
error map: ⟨-0.555 -0.757 +0.851 +3.898]
  • CWE: ~2 = 1200.0000 ¢, ~81/56 = 634.0071 ¢
error map: ⟨0.000 +0.066 -0.108 +5.252]

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

Badness (Sintel): 2.53

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:

  • WE: ~2 = 1199.9429 ¢, ~63/44 = 633.9971 ¢
  • CWE: ~2 = 1200.0000 ¢, ~63/44 = 634.0253 ¢

Optimal ET sequence: 17c, 36ce, 53

Badness (Sintel): 1.86

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:

  • WE: ~2 = 1200.1213 ¢, ~13/9 = 634.0757 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/9 = 634.0154 ¢

Optimal ET sequence: 17c, 36ce, 53

Badness (Sintel): 1.33

Tritricot

Subgroup: 2.3.5.7

Comma list: 250047/250000, [35 -23 -3 3⟩

Mapping: [⟨3 0 -39 -74], ⟨0 3 29 52]]

mapping generators: ~63/50, ~59049/40960

Optimal tunings:

  • WE: ~63/50 = 399.9887 ¢, ~59049/40960 = 633.7326 ¢ (~100352/91125 = 165.9790 ¢)
error map: ⟨-0.034 +0.040 +0.081 -0.073]
  • CWE: ~63/50 = 400.0000 ¢, ~59049/40960 = 634.0155 ¢ (~100352/91125 = 165.9845 ¢)
error map: ⟨0.000 +0.092 -0.137 -0.018]

Optimal ET sequence: 159, 282, 441, 1605, 2046, 2487, 2928

Badness (Sintel): 2.18

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 0 -39 -74 -34], ⟨0 3 29 52 28]]

Optimal tunings:

  • WE: ~63/50 = 399.9686 ¢, ~3969/2750 = 633.9667 ¢ (~11/10 = 165.9705 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~3969/2750 = 634.0142 ¢ (~11/10 = 165.9858 ¢)

Optimal ET sequence: 159, 282, 441

Badness (Sintel): 2.45

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 0 -39 -74 -34 -84], ⟨0 3 29 52 28 60]]

Optimal tunings:

  • WE: ~63/50 = 399.9692 ¢, ~75/52 = 633.9669 ¢ (~11/10 = 165.9714 ¢)
  • CWE: ~63/50 = 400.0000 ¢, ~75/52 = 634.0137 ¢ (~11/10 = 165.9863 ¢)

Optimal ET sequence: 159, 282, 441

Badness (Sintel): 1.47

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨3 0 -39 -74 -34 -84 -2], ⟨0 3 29 52 28 60 9]]

Optimal tunings:

  • WE: ~34/27 = 399.9491 ¢, ~75/52 = 633.9389 ¢ (~11/10 = 165.9594 ¢)
  • CWE: ~34/27 = 400.0000 ¢, ~75/52 = 634.0166 ¢ (~11/10 = 165.9834 ¢)

Optimal ET sequence: 159, 282, 441, 723efg, 1164eefgg

Badness (Sintel): 1.32

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 tunings:

  • WE: ~63/50 = 399.9895 ¢, ~1936/1875 = 55.3269 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~1936/1875 = 55.3286 ¢

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

Badness (Sintel): 5.23

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 tunings:

  • WE: ~63/50 = 399.9896 ¢, ~1936/1875 = 55.3273 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~1936/1875 = 55.3289 ¢

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

Badness (Sintel): 2.99

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 tunings:

  • WE: ~63/50 = 399.9876 ¢, ~351/340 = 55.3270 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~351/340 = 55.3290 ¢

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

Badness (Sintel): 1.93

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 tunings:

  • WE: ~63/50 = 399.9914 ¢, ~351/340 = 55.3277 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~351/340 = 55.3291 ¢

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

Badness (Sintel): 1.64

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 tunings:

  • WE: ~63/50 = 399.9899 ¢, ~351/340 = 55.3276 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~351/340 = 55.3291 ¢

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

Badness (Sintel): 1.39

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 tunings:

  • WE: ~63/50 = 399.9940 ¢, ~351/340 = 55.3283 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~351/340 = 55.3293 ¢

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

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: [⟨1 0 -13 -28], ⟨0 3 29 60]]

Optimal tunings:

  • WE: ~2 = 1199.9796 ¢, ~75/52 = 634.0000 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/52 = 634.0103 ¢

Optimal ET sequence: 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: [⟨1 0 -13 0], ⟨0 3 29 7]]

Optimal tunings:

  • WE: ~2 = 1200.2092 ¢, ~13/9 = 634.1076 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/9 = 634.0032 ¢

Optimal ET sequence: 17c, 36c, 53, 335f, 388f, …, 653ff

Badness (Sintel): 1.26

References

  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
Retrieved from "https://en.xen.wiki/index.php?title=Alphatricot_family&oldid=238800"