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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)
Update keys; normalize mappings; review ET sequences (2/3)
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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 =
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