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).  
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.
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 ==
== Alphatricot ==
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{{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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0102
* [[WE]]: ~2 = 1199.9762{{c}}, ~59049/40960 = 633.9998{{c}}
: [[error map]]: {{val| 0.0000 +0.0757 -0.0168 }}
: [[error map]]: {{val| -0.024 +0.044 -0.010 }}
* [[POTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0124
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0116{{c}}
: error map: {{val| 0.0000 +0.0821 +0.0454 }}
: error map: {{val| 0.000 +0.080 +0.022 }}


{{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, 15365bbc, 17632bbc }}


[[Badness]] (Smith): 0.046093
[[Badness]] (Sintel): 1.08


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


=== 2.3.5.13 subgroup ===
=== Alphatrimot (2.3.5.13 subgroup) ===
{{See also| No-fives subgroup temperaments #Threedic }}
{{See also| No-fives subgroup temperaments #Threedic }}


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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/9 = 634.0179
* WE: ~2 = 1200.2092{{c}}, ~13/9 = 634.1076{{c}}
* POTE: ~2 = 1200.000, ~13/9 = 633.9970
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0032{{c}}


{{Optimal ET sequence|legend=0| 17c, 36c, 53 }}
{{Optimal ET sequence|legend=0| 17c, 36c, 53, 335f, 388f, …, 653ff }}


Badness (Sintel): 1.262
Badness (Sintel): 1.26


=== Catatricot ===
=== Alphatrillium (2.3.5.13 subgroup) ===
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.
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
Subgroup: 2.3.5.13
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~75/52 = 634.009
* WE: ~2 = 1199.9796{{c}}, ~75/52 = 634.0000{{c}}
* POTE: ~2 = 1200.000, ~75/52 = 634.0108
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0103{{c}}


{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }}
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935, 6051f, 6986f, …, 10726bff }}


Badness (Sintel): 0.181
Badness (Sintel): 0.181


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


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.  
It can be extended to the 11-limit by tempering out [[131072/130977]], and to the 13-limit by tempering out [[2080/2079]], [[4096/4095]] and [[4225/4224]]. The optimal tunings in the 11- and 13-limit lean towards [[494edo]]; [[935edo]] and especially [[1429edo]] are recommendable tunings.  
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[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 1099511627776/1098337086315
[[Comma list]]: 4375/4374, {{monzo| 40 -22 -1 -1 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0121
* [[WE]]: ~2 = 1199.9795{{c}}, ~59049/40960 = 634.0010{{c}}
: [[error map]]: {{val| 0.0000 +0.0813 +0.0372 +0.0247 }}
: [[error map]]: {{val| -0.021 +0.048 -0.019 -0.004 }}
* [[POTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0118
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0119{{c}}
: error map: {{val| 0.0000 +0.0804 +0.0283 +0.0537 }}
: error map: {{val| 0.000 +0.081 +0.030 +0.048 }}


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


[[Badness]] (Smith): 0.030852
[[Badness]] (Sintel): 0.781


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~3888/2695 = 634.0091
* WE: ~2 = 1199.9551{{c}}, ~3888/2695 = 633.9857{{c}}
* POTE: ~2 = 1200.0000, ~3888/2695 = 634.0094
* CWE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0094{{c}}


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


Badness (Smith): 0.046758
Badness (Sintel): 1.55


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~75/52 = 634.0091
* WE: ~2 = 1199.9603{{c}}, ~75/52 = 633.9885{{c}}
* POTE: ~2 = 1200.0000, ~75/52 = 634.0095
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0094{{c}}


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


Badness (Smith): 0.019393
Badness (Sintel): 0.801


=== Pseudotrillium ===
=== Pseudotrillium ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~231/160 = 634.0195
* WE: ~2 = 1200.0692{{c}}, ~231/160 = 634.0556{{c}}
* POTE: ~2 = 1200.0000, ~231/160 = 634.0190
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0191{{c}}


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


Badness (Smith): 0.111931
Badness (Sintel): 3.70


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~75/52 = 634.0185
* WE: ~2 = 1200.0351{{c}}, ~75/52 = 634.0366{{c}}
* POTE: ~2 = 1200.0000, ~75/52 = 634.0181
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0181{{c}}


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


Badness (Smith): 0.054837
Badness (Sintel): 2.27


== Alphatrident ==
== Alphatrident ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0484
* [[WE]]: ~2 = 1199.7509{{c}}, ~4096/2835 = 633.9164{{c}}
: [[error map]]: {{val| 0.0000 +0.1901 +1.0893 +1.1421 }}
: [[error map]]: {{val| -0.249 -0.206 +0.500 +0.458 }}
* [[POTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0480
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0481{{c}}
: error map: {{val| 0.0000 +0.1890 +1.0784 +1.1579 }}
: error map: {{val| 0.000 +0.189 +1.081 +1.155 }}


{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511, 793cd }}
{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511, 793cd }}


[[Badness]] (Smith): 0.101694
[[Badness]] (Sintel): 2.57


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~231/160 = 634.0630
* WE: ~2 = 1199.8432{{c}}, ~231/160 = 633.9840{{c}}
* POTE: ~2 = 1200.0000, ~231/160 = 634.0669
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0662{{c}}


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


Badness (Smith): 0.074272
Badness (Sintel): 2.46


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~13/9 = 634.0643
* WE: ~2 = 1199.9675{{c}}, ~13/9 = 634.0480{{c}}
* POTE: ~2 = 1200.0000, ~13/9 = 634.0652
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0651{{c}}


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


Badness (Smith): 0.046593
Badness (Sintel): 1.93


== Alphatrimot ==
== Alphatrimot ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000, ~81/56 = 633.9681
* [[WE]]: ~2 = 1199.4448{{c}}, ~81/56 = 633.7326{{c}}
: [[error map]]: {{val| 0.0000 -0.0508 -1.2400 +4.8227 }}
: [[error map]]: {{val| -0.555 -0.757 +0.851 +3.898 }}
* [[POTE]]: ~2 = 1200.0000, ~81/56 = 634.0259
* [[CWE]]: ~2 = 1200.0000{{c}}, ~81/56 = 634.0071{{c}}
: error map: {{val| 0.0000 +0.1228 +0.4387 +5.4595 }}
: error map: {{val| 0.000 +0.066 -0.108 +5.252 }}


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


[[Badness]] (Smith): 0.100127
[[Badness]] (Sintel): 2.53


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~63/44 = 634.0214
* WE: ~2 = 1199.9429{{c}}, ~63/44 = 633.9971{{c}}
* POTE: ~2 = 1200.0000, ~63/44 = 634.0273
* CWE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0253{{c}}


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


Badness (Smith): 0.056134
Badness (Sintel): 1.86


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~13/9 = 634.0275
* WE: ~2 = 1200.1213{{c}}, ~13/9 = 634.0757{{c}}
* POTE: ~2 = 1200.0000, ~13/9 = 634.0115
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0154{{c}}


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


Badness (Smith): 0.032102
Badness (Sintel): 1.33


== 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


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.0000, ~100352/91125 = 165.9837
[[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 ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }}
{{Optimal ET sequence|legend=1| 159, 282, 441, 1605, 2046, 2487, 2928 }}


[[Badness]] (Smith): 0.086081
[[Badness]] (Sintel): 2.18


=== 11-limit ===
=== 11-limit ===
Line 264: Line 268:
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: {{mapping| 3 0 -39 -74 -34 | 0 3 29 52 28 }}


Optimal tuning (POTE): ~63/50 = 400.0000, ~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=0| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441 }}


Badness (Smith): 0.074002
Badness (Sintel): 2.45


==== 13-limit ====
==== 13-limit ====
Line 277: Line 283:
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: {{mapping| 3 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}


Optimal tuning (POTE): ~63/50 = 400.0000, ~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=0| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441 }}


Badness (Smith): 0.035641
Badness (Sintel): 1.47


==== 17-limit ====
==== 17-limit ====
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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: {{mapping| 3 0 -39 -74 -34 -84 -2 | 0 3 29 52 28 60 9 }}


Optimal tuning (POTE): ~34/27 = 400.0000, ~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=0| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441, 723efg, 1164eefgg }}


Badness (Smith): 0.025972
Badness (Sintel): 1.32


=== Noletaland ===
=== Noletaland ===
Line 306: Line 316:


Mapping: {{mapping| 3 6 19 30 35 | 0 -9 -87 -156 -178 }}
Mapping: {{mapping| 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 = 400.0000, ~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=0| 282, 759de, 1041, 1323, 4251e }}
{{Optimal ET sequence|legend=0| 282, 759de, 1041, 1323, 4251e }}


Badness (Smith): 0.158
Badness (Sintel): 5.23


==== 13-limit ====
==== 13-limit ====
Line 322: Line 333:
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 = 400.0000, ~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=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness (Smith): 0.0725
Badness (Sintel): 2.99


==== 17-limit ====
==== 17-limit ====
Line 335: Line 348:
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 = 400.0000, ~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=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness (Smith): 0.0380
Badness (Sintel): 1.93


==== 19-limit ====
==== 19-limit ====
Line 348: Line 363:
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 = 400.0000, ~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=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness (Smith): 0.0269
Badness (Sintel): 1.64


==== 23-limit ====
==== 23-limit ====
Line 361: Line 378:
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 = 400.0000, ~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=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness (Smith): 0.0194
Badness (Sintel): 1.39


==== 29-limit ====
==== 29-limit ====
Line 374: Line 393:
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 = 400.0000, ~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=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}


Badness (Smith): 0.0168
Badness (Sintel): 1.40


== Notes ==
== Notes ==
Line 384: Line 405:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Alphatricot family| ]] <!-- main article -->
[[Category:Alphatricot family| ]] <!-- main article -->
[[Category:Alphatricot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 05:10, 25 February 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).

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:

  • 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

Alphatrimot (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:

  • 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

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: 140628/140625, 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

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 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:

  • 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

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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