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Very high accuracy temperaments: Difference between revisions

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Update keys; normalize mappings; review ET sequences; correct the gen for gross (1/3)
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= Rank-2 temperaments =
= Rank-2 temperaments =
== Kwazy ==
== Kwazy ==
Kwazy temperament (118&612) tempers out the kwazy comma, {{Monzo|-53 10 16}} (quadla-quadquadyo). 7-limit extensions of the kwazy temperament include [[Hemimage temperaments #Bisupermajor|bisupermajor]] and [[Marvel temperaments #Gwazy|gwazy]].
Kwazy (118 & 612) tempers out the kwazy comma, {{monzo| -53 10 16 }}. 7-limit extensions of the kwazy temperament include [[Hemimage temperaments #Bisupermajor|bisupermajor]] and [[Marvel temperaments #Gwazy|gwazy]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992
[[Comma list]]: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992
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[[Mapping]]: [{{val| 2 1 6 }}, {{val| 0 8 -5 }}]
[[Mapping]]: [{{val| 2 1 6 }}, {{val| 0 8 -5 }}]


[[POTE generator]]: ~1125/1024 = 162.742626
: mapping generators: ~94921875/67108864, ~1125/1024
 
[[Optimal tuning]] ([[CTE]]): ~94921875/67108864 = 1\2, ~1125/1024 = 162.7433


{{Optimal ET sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}
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== Astro ==
== Astro ==
Astro temperament tempers out the astro comma, [91 -12 -31⟩ (tribisa-thiwegu). 7-limit extensions of the astro temperament include [[Horwell temperaments#Kastro|kastro]].
Astro temperament tempers out the astro comma, {{monzo| 91 -12 -31 }}. 7-limit extensions of the astro temperament include [[Horwell temperaments #Kastro|kastro]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 91 -12 -31 }}
[[Comma list]]: {{monzo| 91 -12 -31 }}
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[[Mapping]]: [{{val| 1 5 1 }}, {{val| 0 -31 12 }}]
[[Mapping]]: [{{val| 1 5 1 }}, {{val| 0 -31 12 }}]


[[POTE generator]]: ~296630859375/274877906944 = 132.194668
: mapping generators: ~2, ~{{monzo| -38 5 13 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| -38 5 13 }} = 132.1949


{{Optimal ET sequence|legend=1| 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}
{{Optimal ET sequence|legend=1| 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}
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== Gaster ==
== Gaster ==
{{main| Gaster temperament }}
{{Main| Gaster temperament }}
{{see also| Mirkwai clan #Gaster }}
{{See also| Mirkwai clan #Gaster }}
 
[[Subgroup]]: 2.3.5


Subgroup: 2.3.5
[[Comma list]]: {{monzo| -70 72 -19 }}


[[Comma]]: {{monzo|-70 72 -19}}
[[Mapping]]: [{{val| 1 11 38 }}, {{val| 0 -19 -72 }}]


[[Mapping]]: [{{val|1 11 38}}, {{val|0 -19 -72}}]
: mapping generators: ~2, ~{{monzo| 19 -19 5 }}


[[POTE generator]]: ~1638400000/1162261467 = 594.634565
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 19 -19 5 }} = 594.6345


{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243 }}
{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}


[[Badness]]: 0.088530
[[Badness]]: 0.088530


== Whoosh ==
== Whoosh ==
Whoosh temperament (152&289) tempers out the whoosh comma, {{Monzo|37 25 -33}} (saletrigu). 7-limit extensions of the whoosh temperament include [[Porwell temperaments #Whoops|whoops]].
Whoosh (152 & 289) tempers out the whoosh comma, {{monzo| 37 25 -33 }}. 7-limit extensions of the whoosh temperament include [[Porwell temperaments #Whoops|whoops]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 37 25 -33 }}
[[Comma list]]: {{monzo| 37 25 -33 }}


[[Mapping]]: [{{val| 1 -16 -11 }}, {{val| 0 33 25 }}]
[[Mapping]]: [{{val| 1 17 14 }}, {{val| 0 -33 -25 }}]


[[POTE generator]]: ~864/625 = 560.546826
: mapping generators: ~2, ~864/625
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~864/625 = 560.546826


{{Optimal ET sequence|legend=1| 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}
{{Optimal ET sequence|legend=1| 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}
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[[Mapping]]: [{{val| 1 0 -27 }}, {{val| 0 2 37 }}]
[[Mapping]]: [{{val| 1 0 -27 }}, {{val| 0 2 37 }}]


[[POTE generator]]: ~387420489/335544320 = 249.018406
: mapping generators: ~2, ~{{monzo| 27 -18 1 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 27 -18 1 }} = 950.9814


{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}
{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}
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== Egads ==
== Egads ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -36 -52 51 }}
[[Comma list]]: {{monzo| -36 -52 51 }}
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[[Mapping]]: [{{val| 1 15 16 }}, {{val| 0 -51 -52 }}]
[[Mapping]]: [{{val| 1 15 16 }}, {{val| 0 -51 -52 }}]


[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.647919
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.6479


{{Optimal ET sequence|legend=1| 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}
{{Optimal ET sequence|legend=1| 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}
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=== Hemiegads ===
=== Hemiegads ===
Subgroup: 2.3.5.7


Subgroup: 2.3.5.7
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}


Comma list: 78125000/78121827, 281484423828125/281474976710656
Mapping: [{{val| 1 15 16 -16 }}, {{val| 0 -102 -104 143 }}]


Mapping: [{{val|1 15 16 -16}}, {{val|0 -102 -104 143}}]
Optimal tuning (CTE): ~2 = 1\1, ~36756909/33554432 = 157.8240


Optimal tuning (CTE): ~36756909/33554432 = 157.8240
{{Optimal ET sequence|legend=1| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}


{{Optimal ET sequence|legend=1|38, 403, 441, 479, 844, 2684, 3125, 3566, 4007, 6691}}, ...
Badness: 0.0124


== Fortune ==
== Fortune ==
Fortune temperament tempers out the fortune comma, {{Monzo|-107 47 14}} (tritrila-sepbiyo). It can be described as 612&2513 temperament, which tempers out 78125000/78121827 (euzenius comma) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.
Fortune tempers out the fortune comma, {{monzo| -107 47 14 }}. It can be described as 612 & 2513 temperament, which tempers out 78125000/78121827 (euzenius comma) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -107 47 14 }}
[[Comma list]]: {{monzo| -107 47 14 }}


[[Mapping]]: [{{val| 1 -1 11 }}, {{val| 0 14 -47 }}]
[[Mapping]]: [{{val| 1 13 -36 }}, {{val| 0 -14 47 }}]


[[POTE generator]]: ~8388608/7381125 = 221.567899
: mapping generators: ~2, 7381125/4194304


{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151 }}
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5679
 
{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}


[[Badness]]: 0.027876
[[Badness]]: 0.027876
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Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}


Mapping: [{{val| 1 -1 11 63 }}, {{val| 0 14 -47 -326 }}]
Mapping: [{{val| 1 13 -36 -263 }}, {{val| 0 -14 47 326 }}]


POTE generator: ~8388608/7381125 = 221.568031
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5680


{{Optimal ET sequence|legend=1| 612, 1901, 2513, 3125, 6862, 9987 }}
{{Optimal ET sequence|legend=1| 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}


Badness: 0.023704
Badness: 0.023704
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Comma list: 151263/151250, 21437500/21434787, 369140625/369098752
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752


Mapping: [{{val| 1 -1 11 63 134 }}, {{val| 0 14 -47 -326 -707 }}]
Mapping: [{{val| 1 13 -36 -263 -573 }}, {{val| 0 -14 47 326 707 }}]


POTE generator: ~8388608/7381125 = 221.568139
Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5681


{{Optimal ET sequence|legend=1| 612, 2513, 3125, 3737, 8086 }}
{{Optimal ET sequence|legend=1| 612, 2513, 3125, 3737, 8086, 19909d }}


Badness: 0.036641
Badness: 0.036641


== Senior ==
== Senior ==
Senior temperament (171&1171) tempers out the senior comma, {{Monzo|-17 62 -35}} (quadla-sepquingu). 7-limit extensions of the senior temperament include [[Ragismic microtemperaments #Seniority|seniority]].
Senior temperament (171 & 1171) tempers out the senior comma, {{monzo| -17 62 -35 }}. 7-limit extensions of the senior temperament include [[Ragismic microtemperaments #Seniority|seniority]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -17 62 -35 }}
[[Comma list]]: {{monzo| -17 62 -35 }}
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[[Mapping]]: [{{val| 1 11 19 }}, {{val| 0 -35 -62 }}]
[[Mapping]]: [{{val| 1 11 19 }}, {{val| 0 -35 -62 }}]


[[POTE generator]]: ~94143178827/78125000000 = 322.801358
: mapping generators: ~2, ~{{monzo| -6 23 -13 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| -6 23 -13 }} = 322.8013


{{Optimal ET sequence|legend=1| 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
{{Optimal ET sequence|legend=1| 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
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== Gross ==
== Gross ==
Gross temperament (118&1783) tempers out the gross comma, {{Monzo|144 -22 -47}}, by which a stack of 22 [[tritave]]s (third harmonic) and 47 [[5/1|pentave]]s (fifth harmonic) falls short of 144 (gross) octaves.
Gross (118 & 1783) tempers out the gross comma, {{monzo| 144 -22 -47 }}, by which a stack of 22 [[3/1|3rd harmonics]] and 47 [[5/1|5th harmonics]] falls short of 144 (gross) octaves.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 144 -22 -47 }}
[[Comma list]]: {{monzo| 144 -22 -47 }}


[[Mapping]]: [{{val| 1 -2 4 }}, {{val| 0 47 -22 }}]
[[Mapping]]: [{{val| 1 45 -18 }}, {{val| 0 -47 22 }}]
 
: mapping generators: ~2, ~{{monzo| -45 7 15 }}


[[POTE generator]]: ~1953125000000000/1853020188851841 = 91.531001
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 46 -7 -15 }} = 91.5310


{{Optimal ET sequence|legend=1| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467 }}
{{Optimal ET sequence|legend=1| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c }}


[[Badness]]: 0.046315
[[Badness]]: 0.046315
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