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Alphatricot family: Difference between revisions

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{{Multival|legend=1| 3 29 39 }}
{{Multival|legend=1| 3 29 39 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~59049/40960 = 634.012
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0102
: [[error map]]: {{val| 0.0000 +0.0757 -0.0168 }}
* [[POTE]]: ~2 = 1200.0000, ~59049/40960 = 634.0124
: error map: {{val| 0.0000 +0.0821 +0.0454 }}


{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258, 15365bbc, 17632bbc }}


[[Badness]] (Smith): 0.046093
[[Badness]] (Smith): 0.046093
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Comma list: 2197/2187, 41067/40960
Comma list: 2197/2187, 41067/40960


[[Sval]] [[mapping]]: {{mapping| 1 0 -13 0 | 0 3 29 7 }}
Sval mapping: {{mapping| 1 0 -13 0 | 0 3 29 7 }}


[[Gencom]] [[mapping]]: {{mapping| 1 0 -13 0 0 0 | 0 3 29 0 0 7 }}
Gencom mapping]: {{mapping| 1 0 -13 0 0 0 | 0 3 29 0 0 7 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~13/9 = 633.997
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 }}
{{Optimal ET sequence|legend=0| 17c, 36c, 53 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2342 cents
RMS error: 0.2342 cents


; Scales
; Scales
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== 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]] 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]].  
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
[[Subgroup]]: 2.3.5.7
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{{Multival|legend=1| 3 29 -95 39 -159 -302 }}
{{Multival|legend=1| 3 29 -95 39 -159 -302 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000, ~23625/16384 = 634.0118
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0121
: [[error map]]: {{val| 0.0000 +0.0813 +0.0372 +0.0247 }}
* [[POTE]]: ~2 = 1200.0000, ~23625/16384 = 634.0118
: error map: {{val| 0.0000 +0.0804 +0.0283 +0.0537 }}


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1376, 3193, 4569 }}
{{Optimal ET sequence|legend=1| 53, …, 335, 388, 441, 935, 1376, 3193, 4569, 5945, 10514b }}


[[Badness]] (Smith): 0.030852
[[Badness]] (Smith): 0.030852
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Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}
Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~3888/2695 = 634.0094
Optimal tunings:
* CTE: ~2 = 1200.0000, ~3888/2695 = 634.0091
* POTE: ~2 = 1200.0000, ~3888/2695 = 634.0094


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


Badness (Smith): 0.046758
Badness (Smith): 0.046758
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Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}
Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~75/52 = 634.0095
Optimal tunings:
* CTE: ~2 = 1200.0000, ~75/52 = 634.0091
* POTE: ~2 = 1200.0000, ~75/52 = 634.0095


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


Badness (Smith): 0.019393
Badness (Smith): 0.019393
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Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}
Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}


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


{{Optimal ET sequence|legend=0| 53, 335, 388 }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}
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Mapping: {{mapping| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}
Mapping: {{mapping| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~75/52 = 634.0181
Optimal tunings:
* CTE: ~2 = 1200.0000, ~75/52 = 634.0185
* POTE: ~2 = 1200.0000, ~75/52 = 634.0181


{{Optimal ET sequence|legend=0| 53, 335, 388 }}
{{Optimal ET sequence|legend=0| 53, 335, 388 }}
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{{Multival|legend=1| 3 29 -42 39 -75 -179 }}
{{Multival|legend=1| 3 29 -42 39 -75 -179 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000, ~4096/2835 = 634.0480
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0484
: [[error map]]: {{val| 0.0000 +0.1901 +1.0893 +1.1421 }}
* [[POTE]]: ~2 = 1200.0000, ~4096/2835 = 634.0480
: error map: {{val| 0.0000 +0.1890 +1.0784 +1.1579 }}


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


[[Badness]] (Smith): 0.101694
[[Badness]] (Smith): 0.101694
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Mapping: {{mapping| 1 0 -13 25 -33 | 0 3 29 -42 69 }}
Mapping: {{mapping| 1 0 -13 25 -33 | 0 3 29 -42 69 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~231/160 = 634.0669
Optimal tunings:
* CTE: ~2 = 1200.0000, ~231/160 = 634.0630
* POTE: ~2 = 1200.0000, ~231/160 = 634.0669


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


Badness (Smith): 0.074272
Badness (Smith): 0.074272
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Mapping: {{mapping| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}
Mapping: {{mapping| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~13/9 = 634.0652
Optimal tunings:
* CTE: ~2 = 1200.0000, ~13/9 = 634.0643
* POTE: ~2 = 1200.0000, ~13/9 = 634.0652


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


Badness (Smith): 0.046593
Badness (Smith): 0.046593
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{{Multival|legend=1| 3 29 11 39 9 -56 }}
{{Multival|legend=1| 3 29 11 39 9 -56 }}


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


{{Optimal ET sequence|legend=1| 17c, 36c, 53, 70, 229dd, 282dd }}
{{Optimal ET sequence|legend=1| 17c, 36c, 53, 229dd, 282dd }}


[[Badness]] (Smith): 0.100127
[[Badness]] (Smith): 0.100127
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Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}
Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~63/44 = 634.0273
Optimal tunings:
* CTE: ~2 = 1200.0000, ~63/44 = 634.0214
* POTE: ~2 = 1200.0000, ~63/44 = 634.0273


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


Badness (Smith): 0.056134
Badness (Smith): 0.056134
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Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}
Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}


Optimal tuning (POTE): ~2 = 1200.0000, ~13/9 = 634.0115
Optimal tunings:
* CTE: ~2 = 1200.0000, ~13/9 = 634.0275
* POTE: ~2 = 1200.0000, ~13/9 = 634.0115


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


Badness (Smith): 0.032102
Badness (Smith): 0.032102
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