Alphatricot family: Difference between revisions

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{{Technical data page}}
{{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).  
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 ==
Line 14: 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


[[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
Line 31: Line 28:
* [[Alphatricot19]] – improper [[17L 2s]]
* [[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.
This extension identifies the generator with [[13/9]] by tempering out the threedie, [[2197/2187]], providing a relatively low-complexity mapping for 13.
 
Subgroup: 2.3.5.13
 
Comma list: 2197/2187, 41067/40960
 
Mapping: {{mapping| 1 0 -13 0 | 0 3 29 7 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/9 = 634.0179
* POTE: ~2 = 1200.000, ~13/9 = 633.9970
 
{{Optimal ET sequence|legend=0| 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: {{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|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935 }}


Badness (Sintel): 0.181
==== Subgroup extensions ====
This family has two notable [[2.3.5.13 subgroup|2.3.5.13-subgroup]] extensions, discussed in [[#Subgroup extenions_2|#Subgroup extensions]].  


== Alphatrillium ==
== 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.  
Line 74: Line 42:
[[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 ===
Line 93: Line 61:
Comma list: 4375/4374, 131072/130977, 759375/758912
Comma list: 4375/4374, 131072/130977, 759375/758912


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


Optimal tunings:  
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 ====
Line 108: Line 76:
Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078
Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078


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


Optimal tunings:  
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 ===
Line 123: Line 91:
Comma list: 4375/4374, 5632/5625, 4108797/4096000
Comma list: 4375/4374, 5632/5625, 4108797/4096000


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


Optimal tunings:  
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 ====
Line 138: Line 106:
Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374
Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374


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


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 158: Line 126:


[[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 ===
Line 172: 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 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 ===
Line 187: 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 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 ==
Line 207: Line 175:


[[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 ===
Line 221: Line 189:
Comma list: 99/98, 121/120, 5120/5103
Comma list: 99/98, 121/120, 5120/5103


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


Optimal tunings:  
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 ===
Line 236: Line 204:
Comma list: 99/98, 121/120, 169/168, 352/351
Comma list: 99/98, 121/120, 169/168, 352/351


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


Optimal tunings:  
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 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 = 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 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 = 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 ====
Line 290: 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 = 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 305: 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 = 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 320: 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 = 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 333: 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 = 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 346: 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 = 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 359: 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 = 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 372: 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 = 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
 
== 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


== Notes ==
{{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
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