Very high accuracy temperaments: Difference between revisions

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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

Revision as of 05:22, 24 May 2023

Below are listed some very high accuracy temperaments (TE error < 0.005 cents/octave).

Rank-2 temperaments

Kwazy

Kwazy (118 & 612) tempers out the kwazy comma, [-53 10 16⟩. 7-limit extensions of the kwazy temperament include bisupermajor and gwazy.

Subgroup: 2.3.5

Comma list: [-53 10 16⟩ = 9010162353515625 / 9007199254740992

Mapping: [⟨2 1 6], ⟨0 8 -5]]

mapping generators: ~94921875/67108864, ~1125/1024

Optimal tuning (CTE): ~94921875/67108864 = 1\2, ~1125/1024 = 162.7433

Optimal ET sequence: 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934

Badness: 0.014091

Astro

Astro temperament tempers out the astro comma, [91 -12 -31⟩. 7-limit extensions of the astro temperament include kastro.

Subgroup: 2.3.5

Comma list: [91 -12 -31⟩

Mapping: [⟨1 5 1], ⟨0 -31 12]]

mapping generators: ~2, ~[-38 5 13⟩

Optimal tuning (CTE): ~2 = 1\1, ~[-38 5 13⟩ = 132.1949

Optimal ET sequence: 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c

Badness: 0.062778

Gaster

Main article: Gaster temperament
See also: Mirkwai clan § Gaster

Subgroup: 2.3.5

Comma list: [-70 72 -19⟩

Mapping: [⟨1 11 38], ⟨0 -19 -72]]

mapping generators: ~2, ~[19 -19 5⟩

Optimal tuning (CTE): ~2 = 1\1, ~[19 -19 5⟩ = 594.6345

Optimal ET sequence: 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630

Badness: 0.088530

Whoosh

Whoosh (152 & 289) tempers out the whoosh comma, [37 25 -33⟩. 7-limit extensions of the whoosh temperament include whoops.

Subgroup: 2.3.5

Comma list: [37 25 -33⟩

Mapping: [⟨1 17 14], ⟨0 -33 -25]]

mapping generators: ~2, ~864/625

Optimal tuning (CTE): ~2 = 1\1, ~864/625 = 560.546826

Optimal ET sequence: 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c

Badness: 0.025653

Monzismic

Monzismic (53 & 612) tempers out the monzisma, [54 -37 2⟩. For its 7-limit extension, see Ragismic microtemperaments #Monzismic.

Subgroup: 2.3.5

Comma list: [54 -37 2⟩ = 450359962737049600 / 450283905890997363

Mapping: [⟨1 0 -27], ⟨0 2 37]]

mapping generators: ~2, ~[27 -18 1⟩

Optimal tuning (CTE): ~2 = 1\1, ~[27 -18 1⟩ = 950.9814

Optimal ET sequence: 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520

Badness: 0.010391

Egads

Subgroup: 2.3.5

Comma list: [-36 -52 51⟩

Mapping: [⟨1 15 16], ⟨0 -51 -52]]

mapping generators: ~2, ~6/5

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 315.6479

Optimal ET sequence: 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127

Badness: 0.041819

Hemiegads

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [-48 0 11 8⟩

Mapping: [⟨1 15 16 -16], ⟨0 -102 -104 143]]

Optimal tuning (CTE): ~2 = 1\1, ~36756909/33554432 = 157.8240

Optimal ET sequence: 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c

Badness: 0.0124

Fortune

Fortune tempers out the fortune comma, [-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

Comma list: [-107 47 14⟩

Mapping: [⟨1 13 -36], ⟨0 -14 47]]

mapping generators: ~2, 7381125/4194304

Optimal tuning (CTE): ~2 = 1\1, ~8388608/7381125 = 221.5679

Optimal ET sequence: 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc

Badness: 0.027876

7-limit

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [-52 17 12 -1⟩

Mapping: [⟨1 13 -36 -263], ⟨0 -14 47 326]]

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

Optimal ET sequence: 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323

Badness: 0.023704

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 13 -36 -263 -573], ⟨0 -14 47 326 707]]

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

Optimal ET sequence: 612, 2513, 3125, 3737, 8086, 19909d

Badness: 0.036641

Senior

Senior temperament (171 & 1171) tempers out the senior comma, [-17 62 -35⟩. 7-limit extensions of the senior temperament include seniority.

Subgroup: 2.3.5

Comma list: [-17 62 -35⟩

Mapping: [⟨1 11 19], ⟨0 -35 -62]]

mapping generators: ~2, ~[-6 23 -13⟩

Optimal tuning (CTE): ~2 = 1\1, ~[-6 23 -13⟩ = 322.8013

Optimal ET sequence: 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680

Badness: 0.021220

Gross

Gross (118 & 1783) tempers out the gross comma, [144 -22 -47⟩, by which a stack of 22 3rd harmonics and 47 5th harmonics falls short of 144 (gross) octaves.

Subgroup: 2.3.5

Comma list: [144 -22 -47⟩

Mapping: [⟨1 45 -18], ⟨0 -47 22]]

mapping generators: ~2, ~[-45 7 15⟩

Optimal tuning (CTE): ~2 = 1\1, ~[46 -7 -15⟩ = 91.5310

Optimal ET sequence: 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

Quectismic

Subgroup: 2.3.5

Comma list: [554 -351 1⟩

Mapping: [⟨1 1 -203], ⟨0 1 351]]

Supporting ETs: 665, 53[-5], 306, 359[-5], 971, 1636, 1024[-5], 1689[-5], 1277[+5], 1383[-5]

CTE generator: ~3/2 = 701.955

7-limit

Subgroup: 2.3.5.7

Comma list: [33 -26 12 -7⟩, [-32 13 17 -10⟩

Mapping: [⟨1 1 -203 -347], ⟨0 1 351 598]]

Supporting ETs: 665, 306[+7], 359[-5, -7], 1024[-5, -7], 971[+7], 1689[-5, -7], 1636[+7]

CTE generator: ~3/2 = 701.955

Nanoquectismic

Subgroup: 2.3.5.7

Comma list: 43046721/43025920, [82 -19 -3 -16⟩

Mapping: [⟨1 1 -203 42], ⟨0 1 351 -67]]

Supporting ETs: 665, 306, 1636, 971, 359[-5], 1024[-5], 1277[+5]

CTE generator: ~3/2 = 701.955

Raider

See also: Syntonic-chromatic equivalence continuum

Subgroup: 2.3.5

Comma list: [71 -99 37⟩

Mapping: [⟨1 -9 -26], ⟨0 37 99]]

POTE generator: ~8000/6561 = 343.296098

Optimal ET sequence: 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355

Badness: 0.012867

Pirate

Subgroup: 2.3.5

Comma list: [-90 -15 49⟩

Mapping: [⟨1 -6 0], ⟨0 49 15]]

POTE generator: ~[24 4 -13⟩ = 185.754179

Optimal ET sequence: 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 …

Badness: 0.005656

Atomic

Atomic extensions discussed elsewhere include minutes and hafnium.

See also: Kirnberger's atom

Subgroup: 2.3.5

Comma list: [161 -84 -12⟩

Mapping: [⟨12 0 161], ⟨0 1 -7]]

POTE generator: ~32805/32768 = 1.955162

Optimal ET sequence: 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032

Badness: 0.003803

7-limit

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-55 30 2 1⟩

Mapping: [⟨12 0 161 338], ⟨0 1 -7 -16]]

POTE generator: ~32805/32768 = 1.949608

Optimal ET sequence: 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd

Badness: 0.045818

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250, 184549376/184528125

Mapping: [⟨12 0 161 338 517], ⟨0 1 -7 -16 -25]]

POTE generator: ~32805/32768 = 1.947988

Optimal ET sequence: 12, …, 600e, 612, 1236, 1848

Badness: 0.016029

Niobium

Can be viewed as the 41st-octave counterpart of atomic.

Subgroup: 2.3.5

Comma list: [-875 492 41⟩

Mapping: [⟨41 41 383], ⟨0 -1 12]]

Mapping generators: ~18761829412124890125/18446744073709551616, ~3/2

Optimal tuning (CTE): ~3/2 = 701.9536

Optimal ET sequence: 41, 2460, 4961, 7421, 9881, ...

Revopent

Described as the 1848 & 3125 temperament, and named because the year 1848 is known for revolutions while 3125 is 5 to the 5.

Subgroup: 2.3.5.7

Comma list: 645700815/645657712, 2251799813685248/2251783932057135

Mapping: [⟨1 12 -155 5], ⟨0 -19 287 -4]]

Optimal tuning (CTE): ~24576/16807 = 657.792

Optimal ET sequence: 135, 571, 706, 1277, 1848, 3125, ...

Francium

Francium tempers out the euzenius comma and the 87-126/125 comma in the 7-limit, hereby dividing the octave into 87 parts.

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, [152 -112 -41 43⟩

Mapping: [[⟨87 1 -106 -406], ⟨0 4 9 19]]

Mapping generators: ~126/125 = 1\87, ~[37 -25 -12 11⟩ = 472.043

Optimal tuning (CTE): ~[37 -25 -12 11⟩ = 472.043

Supporting ETs: 783, 1566, 2349, 8178, 8961, 9744, 10527, 12093, 18705

TE Error: 0.003103 cents/octave

11-limit

Subgroup: 2.3.5.7.11

Comma list: 78125000/78121827, 199297406/199290375, [74 -17 -5 -4 -7⟩

Mapping: [[⟨87 1 -106 -406 1225], ⟨0 4 9 19 -27]]

Mapping generators: ~126/125 = 1\87, ~103079215104/78466609375 = 472.043

Optimal tuning (CTE): ~103079215104/78466609375 = 472.043

Supporting ETs: 783, 8178, 8961, 9744, 18705

TE Error: 0.003587 cents/octave

Rank-3 temperaments

Sasaquinbizo-atriyo (171 & 1106 & 3566)

Subgroup: 2.3.5.7

Comma list: 282475249000/282429536481

Mapping: [⟨1 0 9 -3], ⟨0 1 8 0], ⟨0 0 10 -3]]

Optimal ET sequence: 171, 935, 1106, 1277, 2289, 2460, 3395, 3566, 3737

Badness: 0.000281

Saquadtrizo-asepgu (171 & 2019 & 3566)

Subgroup: 2.3.5.7

Comma list: 13841287201/13839609375

Mapping: [⟨1 0 0 0], ⟨0 1 7 5], ⟨0 0 12 7]]

Optimal ET sequence: 171, 1547, 1677, 1718, 1848, 2019, 3395, 3566, 3737

Badness: 0.000135

Laquinzo-aquadquadgu (171 & 441 & 4973)

Subgroup: 2.3.5.7

Comma list: [-7 19 -16 5⟩ = 19534128475869/19531250000000

Mapping: [⟨1 0 3 11], ⟨0 1 4 9], ⟨0 0 5 16]]

Optimal ET sequence: 171, 270, 441, 612, 783, 4802, 4973, 5144

Badness: 0.000297

Laleruyo (171 & 1547 & 3125)

Subgroup: 2.3.5.7

Comma list: 3955078125/3954653486

Mapping: [⟨1 3 0 1], ⟨0 11 0 4], ⟨0 0 1 1]]

Optimal ET sequence: 171, 1407, 1547, 1578, 1718, 2783, 2954, 3125, 3296

Badness: 0.000114

Starscape or Latritriru-ayo

Subgroup: 2.3.5.7

Comma list: 645700815/645657712

Mapping: [⟨1 0 4 0], ⟨0 1 1 2], ⟨0 0 9 1]]

Optimal ET sequence: 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634, …

Badness: 0.000545

Sepbizo-asegu (171 & 3566 & 4973)

Subgroup: 2.3.5.7

Comma list: 6104007655641/6103515625000

Mapping: [⟨1 0 13 16], ⟨0 1 10 12], ⟨0 0 14 17]]

Optimal ET sequence: 171, 1236, 1407, 3566, 3737, 3908, 4802, 4973, 8539, 8710

Badness: 0.000223

Euzenius or Sabiruquinyo

Subgroup: 2.3.5.7

Comma list: 78125000/78121827

Mapping: [⟨1 0 0 -5], ⟨0 2 0 -13], ⟨0 0 1 5]]

Optimal ET sequence: 171, 441, 612, 1730, 1901, 2072, 2171, 2342, 2513, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 16336, 16507, 23027, 26152, 32843, 42659, 58995, 68811, 101654, …

Badness: 0.000205

Quinla-zoyoyo (612 & 1848 & 3125)

Subgroup: 2.3.5.7

Comma list: [-55 30 2 1⟩

Mapping: [⟨1 0 0 55], ⟨0 1 0 -30], ⟨0 0 1 -2]]

Optimal ET sequence: 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795

Badness: 0.000366

Satwethezo-atritribigu (171 & 1547 & 4973)

Subgroup: 2.3.5.7

Comma list: [1 -15 -18 23⟩

Mapping: [⟨1 0 9 7], ⟨0 1 3 3], ⟨0 0 23 18]]

Optimal ET sequence: 171, 1547, 4802, 4973, 5144, 6520, 6691, 6862, 11493, 11664, 11835, 18184, 18355, 18526, 25046, 30019, 30190, 48545

Badness: 0.000114

Trisa-quinbiru-agu (441 & 1848 & 3125)

Subgroup: 2.3.5.7

Comma list: [51 -13 -1 -10⟩

Mapping: [⟨1 0 1 5], ⟨0 1 7 -2], ⟨0 0 10 -1]]

Optimal ET sequence: 441, 1848, 3125, 3566, 4973, 6691, 8098, 8539, 9816, 11664, 14789, 15230, 18355, 21480, 30019, 33144, 39835, 51499

Badness: 0.000263

Quinbisa-quinbizo-alegu (21480 & 33144 & 103169)

Subgroup: 2.3.5.7

Comma list: [110 -71 -11 10⟩

Mapping: [⟨1 0 0 -11], ⟨0 1 -1 6], ⟨0 0 10 11]]

Optimal ET sequence: 11664, 21480, 33144, 36881, 48545, 54624, 70025, 81689, 91505, 103169

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