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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== Senior ==
== Senior ==
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]].
Senior (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
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[[Comma list]]: {{monzo| 554 -351 1 }}
[[Comma list]]: {{monzo| 554 -351 1 }}


[[Mapping]]: [{{val| 1 1 -203 }}, {{val| 0 1 351 }}]
[[Mapping]]: [{{val| 1 0 -554 }}, {{val| 0 1 351 }}]


[[Support]]ing [[ET]]s: 665, 53[-5], 306, 359[-5], 971, 1636, 1024[-5], 1689[-5], 1277[+5], 1383[-5]
: mapping generators: ~2, ~3


[[CTE tuning|CTE generator]]: ~[[3/2]] = 701.955
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9553
 
{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}
 
[[Badness]]: 0.372


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7


[[Subgroup]]: 2.3.5.7
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}


[[Comma list]]: {{monzo| 33 -26 12  -7 }}, {{monzo| -32  13 17 -10 }}
Mapping: [{{val| 1 0 -554 -945 }}, {{val| 0 1 351 598 }}]


[[Mapping]]: [{{val| 1 1 -203 -347 }}, {{val| 0 1 351 598 }}]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9548


[[Support]]ing [[ET]]s: 665, 306[+7], 359[-5, -7], 1024[-5, -7], 971[+7], 1689[-5, -7], 1636[+7]
{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}


[[CTE tuning|CTE generator]]: ~[[3/2]] = 701.955
Badness: 2.17


=== Nanoquectismic ===
=== Nanoquectismic ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 43046721/43025920, {{monzo|82 -19 -3 -16}}
Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}


[[Mapping]]: [{{val|1 1 -203 42}}, {{val|0 1 351 -67}}]
Mapping: [{{val| 1 0 -554 109 }}, {{val| 0 1 351 -67 }}]


[[Support]]ing [[ET]]s: 665, 306, 1636, 971, 359[-5], 1024[-5], 1277[+5]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.955


[[CTE tuning|CTE generator]]: ~[[3/2]] = 701.955
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
 
Badness: 0.604


== Raider ==
== Raider ==
{{See also| Syntonic-chromatic equivalence continuum }}
{{See also| Syntonic-chromatic equivalence continuum }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 71 -99 37 }}
[[Comma list]]: {{monzo| 71 -99 37 }}


[[Mapping]]: [{{val| 1 -9 -26 }}, {{val| 0 37 99 }}]
[[Mapping]]: [{{val| 1 28 73 }}, {{val| 0 -37 -99 }}]
 
: mapping generators: ~2, ~6561/4000


[[POTE generator]]: ~8000/6561 = 343.296098
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~8000/6561 = 343.2961


{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355 }}
{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}


[[Badness]]: 0.012867
[[Badness]]: 0.012867


== Pirate ==
== Pirate ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -90 -15 49 }}
[[Comma list]]: {{monzo| -90 -15 49 }}


[[Mapping]]: [{{val| 1 -6 0 }}, {{val| 0 49 15 }}]
[[Mapping]]: [{{val| 1 43 15 }}, {{val| 0 -49 -15 }}]


[[POTE generator]]: ~{{monzo| 24 4 -13 }} = 185.754179
: mapping generators: ~2, ~{{monzo| -23 -4 13 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 24 4 -13 }} = 185.7542


{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
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== Atomic ==
== Atomic ==
Atomic extensions discussed elsewhere include [[minutes]] and [[The Flashmob#Hafnium|hafnium]].{{see also | Kirnberger's atom }}
{{See also | Kirnberger's atom }}
 
Atomic extensions discussed elsewhere include [[minutes]] and [[The Flashmob #Hafnium|hafnium]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Mapping]]: [{{val| 12 0 161 }}, {{val| 0 1 -7 }}]
[[Mapping]]: [{{val| 12 0 161 }}, {{val| 0 1 -7 }}]


[[POTE generator]]: ~32805/32768 = 1.955162
: mapping generators: ~{{monzo| -67 35 5 }}, ~3
 
[[Optimal tuning]] ([[CTE]]): ~{{monzo| -67 35 5 }} = 1\12, ~3/2 = 701.9552 (or ~32805/32768 = 1.9552)


{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
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Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]
Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]


POTE generator: ~32805/32768 = 1.949608
Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)


{{Optimal ET sequence|legend=1| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
{{Optimal ET sequence|legend=1| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
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Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]
Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]


POTE generator: ~32805/32768 = 1.947988
Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)


{{Optimal ET sequence|legend=1| 12, …, 600e, 612, 1236, 1848 }}
{{Optimal ET sequence|legend=1| 12, …, 600e, 612, 1236, 1848 }}


Badness: 0.016029
Badness: 0.016029
==Niobium==
Can be viewed as the 41st-octave counterpart of atomic.


Subgroup: 2.3.5
== Niobium ==
Niobium can be viewed as the 41st-octave counterpart of atomic.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -875 492 41 }}


Comma list: {{Monzo|-875 492 41}}
[[Mapping]]: [{{val| 41 0 875 }}, {{val| 0 1 -12 }}]


Mapping: [{{Val|41 41 383}}, {{Val|0 -1 12}}]
: mapping generators: ~{{monzo| -64 36 3 }}, ~3


Mapping generators: ~18761829412124890125/18446744073709551616, ~3/2
[[Optimal tuning]] ([[CTE]]): ~{{monzo| -64 36 3 }} = 1\41, ~3/2 = 701.9536


Optimal tuning (CTE): ~3/2 = 701.9536
{{Optimal ET sequence|legend=1| 41, …, 2378, 2419, 2460, 4961, 7421, 17302, 24723, 66748, 91471, 116194 }}


{{Optimal ET sequence|legend=1|41, 2460, 4961, 7421, 9881}}, ...
[[Badness]]: 5.57


== Revopent ==
== Revopent ==
 
Described as the 1848 & 3125 temperament, revopent is named because the year 1848 is known for revolutions while 3125 is 5 to the 5.
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
Subgroup: 2.3.5.7


Comma list: 645700815/645657712, 2251799813685248/2251783932057135
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}


Mapping: [{{val|1 12 -155 5}}, {{val|0 -19 287 -4}}]
Mapping: [{{val| 1 12 -155 5 }}, {{val| 0 -19 287 -4 }}]


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


{{Optimal ET sequence|legend=1|135, 571, 706, 1277, 1848, 3125}}, ...
{{Optimal ET sequence|legend=1| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}
 
Badness: 0.263


== Francium ==
== Francium ==
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Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 78125000/78121827, {{monzo|152 -112 -41 43}}
Comma list: 78125000/78121827, {{monzo| 152 -112 -41 43 }}


Mapping: [[{{val|87 1 -106 -406}}, {{val|0 4 9 19}}]
Mapping: [{{val| 87 1 -106 -406 }}, {{val| 0 4 9 19 }}]


Mapping generators: ~126/125 = 1\87, ~{{monzo|37 -25 -12 11}} = 472.043
: mapping generators: ~126/125, ~{{monzo| 37 -25 -12 11 }}


Optimal tuning (CTE): ~{{monzo|37 -25 -12 11}} = 472.043
Optimal tuning (CTE): ~126/125 = 1\87, ~{{monzo| 37 -25 -12 11 }} = 472.043


[[Support]]ing [[ET]]s: {{EDOs|783, 1566, 2349, 8178, 8961, 9744, 10527, 12093, 18705}}
{{Optimal ET sequence|legend=1| 783, …, 8178, 8961, 9744, 10527, 20271, 30798 }}


TE Error: 0.003103 cents/octave
Badness: 0.762


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 78125000/78121827, 199297406/199290375, {{monzo|74 -17 -5 -4 -7}}
Comma list: 78125000/78121827, 199297406/199290375, {{monzo| 74 -17 -5 -4 -7 }}


Mapping: [[{{val|87 1 -106 -406 1225}}, {{val|0 4 9 19 -27}}]
Mapping: [{{val| 87 1 -106 -406 1225 }}, {{val| 0 4 9 19 -27 }}]


Mapping generators: ~126/125 = 1\87, ~103079215104/78466609375 = 472.043
: mapping generators: ~126/125, ~{{monzo| 35 1 -6 -3 -4 }}


Optimal tuning (CTE): ~103079215104/78466609375 = 472.043
Optimal tuning (CTE): ~126/125 = 1\87, ~{{monzo| 35 1 -6 -3 -4 }} = 472.043


[[Support]]ing [[ET]]s: {{EDOs|783, 8178, 8961, 9744, 18705}}
{{Optimal ET sequence|legend=1| 783, 8178, 8961, 9744, 28449de, 38193bcde, 47937bcdee }}


TE Error: 0.003587 cents/octave
Badness: 0.352


= Rank-3 temperaments =
= Rank-3 temperaments =

Revision as of 06:12, 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 (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 0 -554], ⟨0 1 351]]

mapping generators: ~2, ~3

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9553

Optimal ET sequence: 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551

Badness: 0.372

7-limit

Subgroup: 2.3.5.7

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

Mapping: [⟨1 0 -554 -945], ⟨0 1 351 598]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9548

Optimal ET sequence: 306d, 359cd, 665

Badness: 2.17

Nanoquectismic

Subgroup: 2.3.5.7

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

Mapping: [⟨1 0 -554 109], ⟨0 1 351 -67]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.955

Optimal ET sequence: 306, 665, 1636, 2301, 2966

Badness: 0.604

Raider

See also: Syntonic-chromatic equivalence continuum

Subgroup: 2.3.5

Comma list: [71 -99 37⟩

Mapping: [⟨1 28 73], ⟨0 -37 -99]]

mapping generators: ~2, ~6561/4000

Optimal tuning (CTE): ~2 = 1\1, ~8000/6561 = 343.2961

Optimal ET sequence: 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860

Badness: 0.012867

Pirate

Subgroup: 2.3.5

Comma list: [-90 -15 49⟩

Mapping: [⟨1 43 15], ⟨0 -49 -15]]

mapping generators: ~2, ~[-23 -4 13⟩

Optimal tuning (CTE): ~2 = 1\1, ~[24 4 -13⟩ = 185.7542

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

Badness: 0.005656

Atomic

See also: Kirnberger's atom

Atomic extensions discussed elsewhere include minutes and hafnium.

Subgroup: 2.3.5

Comma list: [161 -84 -12⟩

Mapping: [⟨12 0 161], ⟨0 1 -7]]

mapping generators: ~[-67 35 5⟩, ~3

Optimal tuning (CTE): ~[-67 35 5⟩ = 1\12, ~3/2 = 701.9552 (or ~32805/32768 = 1.9552)

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]]

Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)

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]]

Optimal tuning (CTE): ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)

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

Badness: 0.016029

Niobium

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

Subgroup: 2.3.5

Comma list: [-875 492 41⟩

Mapping: [⟨41 0 875], ⟨0 1 -12]]

mapping generators: ~[-64 36 3⟩, ~3

Optimal tuning (CTE): ~[-64 36 3⟩ = 1\41, ~3/2 = 701.9536

Optimal ET sequence: 41, …, 2378, 2419, 2460, 4961, 7421, 17302, 24723, 66748, 91471, 116194

Badness: 5.57

Revopent

Described as the 1848 & 3125 temperament, revopent is 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, [51 -13 -1 -10⟩

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

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

Optimal ET sequence: 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d

Badness: 0.263

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, ~[37 -25 -12 11⟩

Optimal tuning (CTE): ~126/125 = 1\87, ~[37 -25 -12 11⟩ = 472.043

Optimal ET sequence: 783, …, 8178, 8961, 9744, 10527, 20271, 30798

Badness: 0.762

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, ~[35 1 -6 -3 -4⟩

Optimal tuning (CTE): ~126/125 = 1\87, ~[35 1 -6 -3 -4⟩ = 472.043

Optimal ET sequence: 783, 8178, 8961, 9744, 28449de, 38193bcde, 47937bcdee

Badness: 0.352

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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