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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 ({{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 ==
Line 21: Line 19:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0116{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0116{{c}}
: error map: {{val| 0.000 +0.080 +0.022 }}
: error map: {{val| 0.000 +0.080 +0.022 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0102{{c}}
: [[error map]]: {{val| 0.0000 +0.0757 -0.0168 }}
* [[POTE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0124{{c}}
: error map: {{val| 0.0000 +0.0821 +0.0454 }} -->


{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc }}
{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc }}
Line 34: Line 28:
* [[Alphatricot19]] – improper [[17L 2s]]
* [[Alphatricot19]] – improper [[17L 2s]]


=== Alphatrimot (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:
* WE: ~2 = 1200.2092{{c}}, ~13/9 = 634.1076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0032{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0179{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~13/9 = 633.9970{{c}} -->
 
{{Optimal ET sequence|legend=0| 17c, 36c, 53, 335f, 388f, …, 653ff }}


Badness (Sintel): 1.26
==== 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 (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: {{mapping| 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}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~75/52 = 634.009{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0108{{c}} -->
 
{{Optimal ET sequence|legend=1| 17cff, 36cff, 53, 282, 335, 388, 441, 494, 935, 6051f, 6986f, …, 10726bff }}
 
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.  
Line 81: 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 }}
Line 90: Line 51:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0119{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~59049/40960 = 634.0119{{c}}
: error map: {{val| 0.000 +0.081 +0.030 +0.048 }}
: error map: {{val| 0.000 +0.081 +0.030 +0.048 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~23625/16384 = 634.0121{{c}}
: [[error map]]: {{val| 0.0000 +0.0813 +0.0372 +0.0247 }}
* [[POTE]]: ~2 = 1200.0000{{c}}, ~23625/16384 = 634.0118{{c}}
: error map: {{val| 0.0000 +0.0804 +0.0283 +0.0537 }} -->


{{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 }}
Line 104: 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:  
* WE: ~2 = 1199.9551{{c}}, ~3888/2695 = 633.9857{{c}}
* WE: ~2 = 1199.9551{{c}}, ~3888/2695 = 633.9857{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0094{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3888/2695 = 634.0091{{c}}
* POTE: ~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 }}
Line 121: 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:  
* WE: ~2 = 1199.9603{{c}}, ~75/52 = 633.9885{{c}}
* WE: ~2 = 1199.9603{{c}}, ~75/52 = 633.9885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0094{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0091{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0095{{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 }}
Line 138: 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:  
* WE: ~2 = 1200.0692{{c}}, ~231/160 = 634.0556{{c}}
* WE: ~2 = 1200.0692{{c}}, ~231/160 = 634.0556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0191{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0191{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0195{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0190{{c}} -->


{{Optimal ET sequence|legend=0| 53, 335, 388 }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}
Line 155: 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:  
* WE: ~2 = 1200.0351{{c}}, ~75/52 = 634.0366{{c}}
* WE: ~2 = 1200.0351{{c}}, ~75/52 = 634.0366{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0181{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~75/52 = 634.0185{{c}}
* POTE: ~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 }}
Line 168: Line 117:


== 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 181: Line 130:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0481{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0481{{c}}
: error map: {{val| 0.000 +0.189 +1.081 +1.155 }}
: error map: {{val| 0.000 +0.189 +1.081 +1.155 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0484{{c}}
: [[error map]]: {{val| 0.0000 +0.1901 +1.0893 +1.1421 }}
* [[POTE]]: ~2 = 1200.0000{{c}}, ~4096/2835 = 634.0480{{c}}
: error map: {{val| 0.0000 +0.1890 +1.0784 +1.1579 }} -->


{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511, 793cd }}
{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511, 793cd }}
Line 195: 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:  
* WE: ~2 = 1199.8432{{c}}, ~231/160 = 633.9840{{c}}
* WE: ~2 = 1199.8432{{c}}, ~231/160 = 633.9840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0662{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0662{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0630{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~231/160 = 634.0669{{c}} -->


{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}
Line 212: 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:  
* WE: ~2 = 1199.9675{{c}}, ~13/9 = 634.0480{{c}}
* WE: ~2 = 1199.9675{{c}}, ~13/9 = 634.0480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0651{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0651{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0643{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0652{{c}} -->


{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}
{{Optimal ET sequence|legend=0| 53, 123, 176, 229 }}
Line 238: Line 179:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~81/56 = 634.0071{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~81/56 = 634.0071{{c}}
: error map: {{val| 0.000 +0.066 -0.108 +5.252 }}
: error map: {{val| 0.000 +0.066 -0.108 +5.252 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~81/56 = 633.9681{{c}}
: [[error map]]: {{val| 0.0000 -0.0508 -1.2400 +4.8227 }}
* [[POTE]]: ~2 = 1200.0000{{c}}, ~81/56 = 634.0259{{c}}
: error map: {{val| 0.0000 +0.1228 +0.4387 +5.4595 }} -->


{{Optimal ET sequence|legend=1| 17c, 36c, 53, 229dd, 282dd }}
{{Optimal ET sequence|legend=1| 17c, 36c, 53, 229dd, 282dd }}
Line 252: 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:  
* WE: ~2 = 1199.9429{{c}}, ~63/44 = 633.9971{{c}}
* WE: ~2 = 1199.9429{{c}}, ~63/44 = 633.9971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0253{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0253{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0214{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~63/44 = 634.0273{{c}} -->


{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}
Line 269: 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:  
* WE: ~2 = 1200.1213{{c}}, ~13/9 = 634.0757{{c}}
* WE: ~2 = 1200.1213{{c}}, ~13/9 = 634.0757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0154{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0154{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0275{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~13/9 = 634.0115{{c}} -->


{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}
{{Optimal ET sequence|legend=0| 17c, 36ce, 53 }}
Line 284: Line 217:
[[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 0 -39 -74 | 0 3 29 52 }}
{{Mapping|legend=1| 3 0 -39 -74 | 0 3 29 52 }}
Line 294: Line 227:
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~59049/40960 = 634.0155{{c}} (~100352/91125 = 165.9845{{c}})
* [[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 }}
: error map: {{val| 0.000 +0.092 -0.137 -0.018 }}
<!-- * [[POTE]]: ~63/50 = 400.0000{{c}}, ~59049/40960 = 634.0163{{c}} (~100352/91125 = 165.9837{{c}}) -->


{{Optimal ET sequence|legend=1| 159, 282, 441, 1605, 2046, 2487, 2928 }}
{{Optimal ET sequence|legend=1| 159, 282, 441, 1605, 2046, 2487, 2928 }}
Line 305: Line 237:
Comma list: 4000/3993, 166698/166375, 200704/200475
Comma list: 4000/3993, 166698/166375, 200704/200475


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


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9686{{c}}, ~3969/2750 = 633.9667{{c}} (~11/10 = 165.9705{{c}})
* 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}})
* CWE: ~63/50 = 400.0000{{c}}, ~3969/2750 = 634.0142{{c}} (~11/10 = 165.9858{{c}})
<!-- * POTE: ~63/50 = 400.0000{{c}}, ~3969/2750 = 634.0165{{c}} (~11/10 = 165.9835{{c}}) -->


{{Optimal ET sequence|legend=0| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441 }}
Line 321: 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 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}
{{Mapping|legend=0| 3 0 -39 -74 -34 -84 | 0 3 29 52 28 60 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9692{{c}}, ~75/52 = 633.9669{{c}} (~11/10 = 165.9714{{c}})
* 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}})
* CWE: ~63/50 = 400.0000{{c}}, ~75/52 = 634.0137{{c}} (~11/10 = 165.9863{{c}})
<!-- * POTE: ~63/50 = 400.0000{{c}}, ~75/52 = 634.0158{{c}} (~11/10 = 165.9842{{c}}) -->


{{Optimal ET sequence|legend=0| 159, 282, 441 }}
{{Optimal ET sequence|legend=0| 159, 282, 441 }}
Line 337: 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 0 -39 -74 -34 -84 -2 | 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 tunings:  
Optimal tunings:  
* WE: ~34/27 = 399.9491{{c}}, ~75/52 = 633.9389{{c}} (~11/10 = 165.9594{{c}})
* 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}})
* CWE: ~34/27 = 400.0000{{c}}, ~75/52 = 634.0166{{c}} (~11/10 = 165.9834{{c}})
<!-- * POTE: ~34/27 = 400.0000{{c}}, ~75/52 = 634.0195{{c}} (~11/10 = 165.9805{{c}}) -->


{{Optimal ET sequence|legend=0| 159, 282, 441, 723efg, 1164eefgg }}
{{Optimal ET sequence|legend=0| 159, 282, 441, 723efg, 1164eefgg }}
Line 355: 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


Line 361: Line 290:
* WE: ~63/50 = 399.9895{{c}}, ~1936/1875 = 55.3269{{c}}
* WE: ~63/50 = 399.9895{{c}}, ~1936/1875 = 55.3269{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3286{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3286{{c}}
<!-- * CTE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3290{{c}} -->


{{Optimal ET sequence|legend=0| 282, 759de, 1041, 1323, 4251e }}
{{Optimal ET sequence|legend=0| 282, 759de, 1041, 1323, 4251e }}
Line 372: 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 tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9896{{c}}, ~1936/1875 = 55.3273{{c}}
* WE: ~63/50 = 399.9896{{c}}, ~1936/1875 = 55.3273{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3289{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3289{{c}}
<!-- * CTE: ~63/50 = 400.0000{{c}}, ~1936/1875 = 55.3294{{c}} -->


{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
Line 388: 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 tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9876{{c}}, ~351/340 = 55.3270{{c}}
* WE: ~63/50 = 399.9876{{c}}, ~351/340 = 55.3270{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3290{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3290{{c}}
<!-- * CTE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3295{{c}} -->


{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
Line 404: 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 tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9914{{c}}, ~351/340 = 55.3277{{c}}
* WE: ~63/50 = 399.9914{{c}}, ~351/340 = 55.3277{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}
<!-- * CTE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3295{{c}} -->


{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
Line 420: 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 tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9899{{c}}, ~351/340 = 55.3276{{c}}
* WE: ~63/50 = 399.9899{{c}}, ~351/340 = 55.3276{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3291{{c}}
<!-- * CTE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3296{{c}} -->


{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
Line 436: 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 tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9940{{c}}, ~351/340 = 55.3283{{c}}
* WE: ~63/50 = 399.9940{{c}}, ~351/340 = 55.3283{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3293{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3293{{c}}
<!-- * CTE: ~63/50 = 400.0000{{c}}, ~351/340 = 55.3296{{c}} -->


{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
{{Optimal ET sequence|legend=0| 282, 759def, 1041, 1323 }}
Line 447: Line 370:
Badness (Sintel): 1.40
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:Pages with mostly numerical content]]
[[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]]
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