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Very high accuracy temperaments: Difference between revisions

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This page is a collection of microtemperaments of different ranks which all have a [[Tenney-Euclidean temperament measures|TE error]] of less than 0.005 cents/octave.
{{Technical data page}}
Below are listed some very high accuracy temperaments (Breed's [[Tenney-Euclidean temperament measures|TE error]] < 0.005 cents/octave).


= Rank-2 temperaments =
== Kwazy ==
== Kwazy ==
Kwazy temperament (118&amp;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 ({{nowrap| 118 & 612 }}) tempers out the [[kwazy comma]], {{monzo| -53 10 16 }}. 7-limit extensions of the kwazy temperament include [[marvel temperaments #Gwazy|gwazy]],  [[hemimage temperaments #Bisupermajor|bisupermajor]], and [[ragismic microtemperaments #Crazy|crazy]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -53 10 16 }}


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


[[Mapping]]: [{{val| 2 1 6 }}, {{val| 0 8 -5 }}]
: mapping generators: ~94921875/67108864, ~1125/1024


[[POTE generator]]: ~1125/1024 = 162.742626
[[Optimal tuning]]s:
* [[CTE]]: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.743279
* [[POTE]]: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.742626


{{Optimal GPV 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 }}


[[Badness]]: 0.014091
[[Badness]] (Sintel): 0.331


== 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 }}


[[Mapping]]: [{{val| 1 5 1 }}, {{val| 0 -31 12 }}]
{{Mapping|legend=1| 1 5 1 | 0 -31 12 }}
 
: mapping generators: ~2, ~{{monzo| -38 5 13 }}


[[POTE generator]]: ~296630859375/274877906944 = 132.194668
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194857
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194668


{{Optimal GPV 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 }}


[[Badness]]: 0.062778
[[Badness]] (Sintel): 1.47


== 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|legend=1| 1 11 38 | 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]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634456
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634565


{{Optimal GPV 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]] (Sintel): 2.08


== Whoosh ==
== Whoosh ==
Whoosh temperament (152&amp;289) tempers out the whoosh comma, {{Monzo|37 25 -33}} (saletrigu). 7-limit extensions of the whoosh temperament include [[Porwell temperaments #Whoops|whoops]].
Whoosh ({{nowrap| 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|legend=1| 1 17 14 | 0 -33 -25 }}


[[POTE generator]]: ~864/625 = 560.546826
: mapping generators: ~2, ~864/625


{{Optimal GPV sequence|legend=1| 15, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~864/625 = 560.546952
* [[POTE]]: ~2 = 1200.000000, ~864/625 = 560.546826


[[Badness]]: 0.025653
{{Optimal ET sequence|legend=1| 15, …, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}
 
[[Badness]] (Sintel): 0.602


== Monzismic ==
== Monzismic ==
Monzismic (53 &amp; 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}. For its 7-limit extension, see [[Ragismic microtemperaments #Monzismic]].
Monzismic ({{nowrap| 53 & 612 }}) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}. For its 7-limit extension, see [[Ragismic microtemperaments #Monzismic]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 54 -37 2 }} = 450359962737049600 / 450283905890997363
[[Comma list]]: {{monzo| 54 -37 2 }}


[[Mapping]]: [{{val| 1 0 -27 }}, {{val| 0 2 37 }}]
{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}


[[POTE generator]]: ~387420489/335544320 = 249.018406
: mapping generators: ~2, ~{{monzo| 27 -18 1 }}


{{Optimal GPV sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981427
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981594


[[Badness]]: 0.010391
{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}
 
[[Badness]] (Sintel): 0.244


== Egads ==
== Egads ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -36 -52 51 }}
[[Comma list]]: {{monzo| -36 -52 51 }}


[[Mapping]]: [{{val| 1 15 16 }}, {{val| 0 -51 -52 }}]
{{Mapping|legend=1| 1 15 16 | 0 -51 -52 }}
 
: mapping generators: ~2, ~6/5


[[Optimal tuning]] ([[CTE]]): ~6/5 = 315.647919
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~6/5 = 315.647899
* [[POTE]]: ~2 = 1200.000000, ~6/5 = 315.647919


{{Optimal GPV 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 }}


[[Badness]]: 0.041819
[[Badness]] (Sintel): 0.981


=== 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: {{mapping| 1 15 16 -16 | 0 -102 -104 143 }}


Mapping: [{{val|1 15 16 -16}}, {{val|0 -102 -104 143}}]
Optimal tunings:  
* CTE: ~2 = 1200.000000, ~36756909/33554432 = 157.823952
* CWE: ~2 = 1200.000000, ~36756909/33554432 = 157.823954


Optimal tuning (CTE): ~36756909/33554432 = 157.8240
{{Optimal ET sequence|legend=0| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}


Vals: {{EDOs|38, 403, 441, 479, 844, 2684, 3125, 3566, 4007, 6691}}, ...
Badness (Sintel): 0.313


== Fortune ==
== Fortune ==
Fortune temperament tempers out the fortune comma, {{Monzo|-107 47 14}} (tritrila-sepbiyo). It can be described as 612&amp;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 {{nowrap| 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|legend=1| 1 13 -36 | 0 -14 47 }}


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


{{Optimal GPV sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567861
* [[POTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567899


[[Badness]]: 0.027876
{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}
 
[[Badness]] (Sintel): 0.654


=== 7-limit ===
=== 7-limit ===
Line 122: Line 155:
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: {{mapping| 1 13 -36 -263 | 0 -14 47 326 }}


POTE generator: ~8388608/7381125 = 221.568031
Optimal tunings:
* CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568013
* POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568031


Optimal GPV sequence: {{Val list| 612, 1901, 2513, 3125, 6862, 9987 }}
{{Optimal ET sequence|legend=0| 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}


Badness: 0.023704
Badness (Sintel): 0.600


=== 11-limit ===
=== 11-limit ===
Line 135: Line 170:
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: {{mapping| 1 13 -36 -263 -573 | 0 -14 47 326 707 }}


POTE generator: ~8388608/7381125 = 221.568139
Optimal tunings:
* CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568116
* POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568139


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


Badness: 0.036641
Badness (Sintel): 1.21


== Senior ==
== Senior ==
Senior temperament (171&amp;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 ({{nowrap| 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 }}


[[Mapping]]: [{{val| 1 11 19 }}, {{val| 0 -35 -62 }}]
{{Mapping|legend=1| 1 11 19 | 0 -35 -62 }}
 
: mapping generators: ~2, ~{{monzo| -6 23 -13 }}


[[POTE generator]]: ~94143178827/78125000000 = 322.801358
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801349
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801358


{{Optimal GPV sequence|legend=1| 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
{{Optimal ET sequence|legend=1| 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}


[[Badness]]: 0.021220
[[Badness]] (Sintel): 0.498


== Gross ==
== Gross ==
Gross temperament (118&amp;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 ({{nowrap| 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|legend=1| 1 45 -18 | 0 -47 22 }}
 
: mapping generators: ~2, ~{{monzo| -45 7 15 }}
 
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.530979
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.531001
 
{{Optimal ET sequence|legend=1| 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c }}
 
[[Badness]] (Sintel): 1.09
 
; Music
* ''Gross temperament EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/02Fz6IIJ2x9k7bEvFylZ2A Spotify] | [https://francium223.bandcamp.com/album/gross-temperament-ep Bandcamp] | [https://youtube.com/playlist?list=PLLZE7hMjEXRZy_JWYbZxbsxPsXahiEJA2&si=RttlCFpkeN4Rl7Tj YouTube] – 2-piece extended play
 
== Quectismic ==
{{See also| Quectisma }}
 
[[Subgroup]]: 2.3.5


[[POTE generator]]: ~1953125000000000/1853020188851841 = 91.531001
[[Comma list]]: {{monzo| 554 -351 1 }}


{{Optimal GPV sequence|legend=1| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467 }}
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}


[[Badness]]: 0.046315
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.955310
* [[CWE]]: ~2 = 1200.000000, ~3/2 = 701.955310
 
{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}
 
[[Badness]] (Sintel): 8.74
 
=== Nanoquectismic ===
Subgroup: 2.3.5.7
 
Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}
 
Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000000, ~3/2 = 701.955371
* CWE: ~2 = 1200.000000, ~3/2 = 701.955457
 
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
 
Badness (Sintel): 15.3
 
=== Conanoquectismic ===
Subgroup: 2.3.5.7
 
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
 
Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000000, ~3/2 = 701.954810
* CWE: ~2 = 1200.000000, ~3/2 = 701.954771
 
{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}
 
Badness (Sintel): 55.0
 
== Counterquectismic ==
{{See also| Counterquectisma }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -500 314 1 }}
 
{{Mapping|legend=1| 1 0 500 | 0 1 -314 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.954415
* [[CWE]]: ~2 = 1200.000000, ~3/2 = 701.954416
 
{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}
 
[[Badness]] (Sintel): 12.0


== 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|legend=1| 1 28 73 | 0 -37 -99 }}


[[POTE generator]]: ~8000/6561 = 343.296098
: mapping generators: ~2, ~6561/4000


{{Optimal GPV sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~8000/6561 = 343.296094
* [[POTE]]: ~2 = 1200.000000, ~8000/6561 = 343.296098


[[Badness]]: 0.012867
{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
 
[[Badness]] (Sintel): 0.302


== Pirate ==
== Pirate ==
Subgroup: 2.3.5
{{See also| Hemfiness temperaments #Piracy }}
 
[[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|legend=1| 1 43 15 | 0 -49 -15 }}
 
: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}


[[POTE generator]]: ~{{monzo| 24 4 -13 }} = 185.754179
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 24 4 -13 }} = 185.754186
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 24 4 -13 }} = 185.754179


{{Optimal GPV 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 … }}


[[Badness]]: 0.005656
[[Badness]] (Sintel): 0.133


== Atomic ==
== Atomic ==
Atomic extensions discussed elsewhere include [[minutes]] and [[The Flashmob#Hafnium|hafnium]].{{see also | Kirnberger's atom }}
{{See also | Kirnberger's atom }}
 
Atomic observes the [[schisma]], and the 3/2 is one schisma sharp of its [[12edo]] value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the [[Pythagorean comma]] is twelve schismas, and hence [[81/80]], the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.
 
Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 208: Line 338:
[[Comma list]]: {{monzo| 161 -84 -12 }}
[[Comma list]]: {{monzo| 161 -84 -12 }}


[[Mapping]]: [{{val| 12 0 161 }}, {{val| 0 1 -7 }}]
{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}
 
: mapping generators: ~{{monzo| -67 35 5 }}, ~3


[[POTE generator]]: ~32805/32768 = 1.955162
[[Optimal tuning]]s:
* [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000, ~3/2 = 701.955176 (~32805/32768 = 1.955176)
* [[POTE]]: ~{{monzo| -67 35 5 }} = 100.000000, ~3/2 = 701.955162 (~32805/32768 = 1.955162)


{{Optimal GPV 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 }}


[[Badness]]: 0.003803
[[Badness]] (Sintel): 0.0892


=== 7-limit ===
=== 7-limit ===
Line 221: Line 355:
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}


Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]
Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}


POTE generator: ~32805/32768 = 1.949608
Optimal tunings:
* CTE: ~30375/28672 = 100.000000, ~3/2 = 701.949917 (or ~32805/32768 = 1.949917)
* POTE: ~30375/28672 = 100.000000, ~3/2 = 701.949608 (or ~32805/32768 = 1.949608)


Optimal GPV sequence: {{Val list| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
{{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}


Badness: 0.045818
Badness (Sintel): 1.16


=== 11-limit ===
=== 11-limit ===
Line 234: Line 370:
Comma list: 9801/9800, 151263/151250, 184549376/184528125
Comma list: 9801/9800, 151263/151250, 184549376/184528125


Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]
Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}
 
Optimal tunings:
* CTE: ~30375/28672 = 100.000000, ~3/2 = 701.948456 (or ~32805/32768 = 1.948456)
* POTE: ~30375/28672 = 100.000000, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)
 
{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}
 
Badness (Sintel): 0.530
 
== Revopentic (2.3.7) ==
{{See also| Revopentisma }}


POTE generator: ~32805/32768 = 1.947988
Subgroup: 2.3.7


Optimal GPV sequence: {{Val list| 12, …, 600e, 612, 1236, 1848 }}
Comma list: 11399736556781568/11398895185373143


Badness: 0.016029
{{Mapping|legend=2| 1 12 5 | 0 -19 -4 }}
==Niobium==
Can be viewed as the 41st-octave counterpart of atomic.


Subgroup: 2.3.5
: mapping generators: ~2, ~24576/16807


Comma list: {{Monzo|-875 492 41}}
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~24576/16807 = 657.791865
* [[CWE]]: ~2 = 1200.000000, ~24576/16807 = 657.791796


Mapping: [{{Val|41 41 383}}, {{Val|0 -1 12}}]
{{Optimal ET sequence|legend=1| 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, … }}


Mapping generators: ~18761829412124890125/18446744073709551616, ~3/2
[[Badness]] (Sintel): 0.0704


Optimal tuning (CTE): ~3/2 = 701.9536
=== Revopent ===
Revopent can be described as the {{nowrap| 1848 & 3125 }} temperament.


Vals: {{EDOs|41, 2460, 4961, 7421, 9881}}, ...
== Sasaquinbizo-atriyo (171 &amp; 1106 &amp; 3566) (rank-3) ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 282475249000/282429536481
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}


Mapping: [{{val| 1 0 9 -3 }}, {{val| 0 1 8 0 }}, {{val| 0 0 10 -3 }}]
Mapping: {{mapping| 1 12 -155 5 | 0 -19 287 -4 }}


Vals: {{Val list| 171, 935, 1106, 1277, 2289, 2460, 3395, 3566, 3737}}
Optimal tunings:  
* CTE: ~2 = 1200.000000, ~24576/16807 = 657.792032
* CWE: ~2 = 1200.000000, ~24576/16807 = 657.792032


Badness: 0.000281
{{Optimal ET sequence|legend=0| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}


== Saquadtrizo-asepgu (171 &amp; 2019 &amp; 3566) (rank-3) ==
Badness (Sintel): 0.665
Subgroup: 2.3.5.7
 
== Nanic ==
{{See also| Nanisma }}
 
[[Subgroup]]: 2.3.7
 
[[Comma list]]: {{monzo| 109 -67 -1 }}
 
{{Mapping|legend=2| 1 0 109 | 0 1 -67 }}
 
: sval mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.957820
* [[POTE]]: ~2 = 1200.000000, ~3/2 = 701.957784
 
{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}
 
[[Badness]] (Sintel): 0.474
 
== Icarus ==
This temperament is named after a very distant star. Curiously, the names of two even more distant stars coincide with named temperaments, those being [[godzilla]] and [[mothra]].
 
[[Subgroup]]: 3.5.7
 
[[Comma list]]: {{monzo| -32 -64 71 }}
 
{{Mapping|legend=1| 1 35 32 | 0 -71 -64 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~3 = 1901.955, ~{{monzo| 5 9 -10 }} = 898.340
* [[CWE]]: ~3 = 1901.955, ~{{monzo| 5 9 -10 }} = 898.340
 
[[Optimal ET sequence]]: [[163edt|b163cd]], [[199edt|b199c]], [[235edt|b235]], [[271edt|b271]], [[4643edt|b4643]], [[4914edt|b4914]], …, [[8708edt|b8708]], [[8979edt|b8979]], [[18229edt|b18229]], [[27208edt|b27208]], [[99853edt|b99853]], [[127061edt|b127061]], [[154269edt|b154269]], [[181477edt|b181477]], [[208685edt|b208685]], [[390162edt|b390162]]
 
[[Badness]] (Sintel): 0.00493
 
= Rank-3 temperaments =
== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 3 -24 3 10 }} (282475249000/282429536481)
 
{{Mapping|legend=1| 1 0 9 -3 | 0 1 8 0 | 0 0 10 -3 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9624, ~6561/3430 = 1122.9387
 
{{Optimal ET sequence|legend=1| 171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069 }}
 
[[Badness]] (Sintel): 1.24
 
== Saquadtrizo-asepgu (171 & 2019 & 3566) ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 0 -11 -7 12 }} (13841287201/13839609375)
 
{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9584, ~16807/10125 = 877.2825


Comma list: 13841287201/13839609375
{{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}


Mapping: [{{val| 1 0 0 0 }}, {{val| 0 1 7 5 }}, {{val| 0 0 12 7 }}]
[[Badness]] (Sintel): 0.595


Vals: {{Val list| 171, 1547, 1677, 1718, 1848, 2019, 3395, 3566, 3737}}
== Laquinzo-aquadquadgu (171 & 441 & 4973) ==
[[Subgroup]]: 2.3.5.7


Badness: 0.000135
[[Comma list]]: {{monzo| -7 19 -16 5 }} (19534128475869/19531250000000)


== Laquinzo-aquadquadgu (171 &amp; 441 &amp; 4973) (rank-3) ==
{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}
Subgroup: 2.3.5.7


Comma list: {{monzo| -7 19 -16 5 }} = 19534128475869/19531250000000
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9501, ~250/189 = 484.2956


Mapping: [{{val| 1 0 3 11 }}, {{val| 0 1 4 9 }}, {{val| 0 0 5 16 }}]
{{Optimal ET sequence|legend=1| 171, 441, 612, 2696, 2795, 2867, 2966, 3137, 3308, 3407, 3479, 3578, 3749, 3920, 4091, 4190, 4361, 4532, 4703, 4802, 4973, 5144, 10117, 10558, 15531, 15702, 52079, 62637, 67781, 78339 }}


Vals: {{Val list| 171, 270, 441, 612, 783, 4802, 4973, 5144}}
[[Badness]] (Sintel): 1.31


Badness: 0.000297
== Laleruyo (171 & 1547 & 3125) ==
[[Subgroup]]: 2.3.5.7


== Laleruyo (171 &amp; 1547 &amp; 3125) (rank-3) ==
[[Comma list]]: {{monzo| -1 4 11 -11 }} (3955078125/3954653486)
Subgroup: 2.3.5.7


Comma list: 3955078125/3954653486
{{Mapping|legend=1| 1 3 0 1 | 0 -11 0 -4 | 0 0 1 1 }}


Mapping: [{{val| 1 3 0 1 }}, {{val| 0 11 0 4 }}, {{val| 0 0 1 1 }}]
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~375/343 = 154.3678, ~5/4 = 386.3070


Vals: {{Val list| 171, 1407, 1547, 1578, 1718, 2783, 2954, 3125, 3296}}
{{Optimal ET sequence|legend=1| 171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987 }}


Badness: 0.000114
[[Badness]] (Sintel): 0.501


== Starscape or Latritriru-ayo (rank-3) ==
== Starscape or Latritriru-ayo ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 645700815/645657712
[[Comma list]]: 645700815/645657712


[[Mapping]]: [{{val| 1 0 4 0 }}, {{val| 0 1 1 2 }}, {{val| 0 0 9 1 }}]
{{Mapping|legend=1| 1 0 4 0 | 0 1 1 2 | 0 0 -9 -1 }}


{{Val list|legend=1| 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634, }} …
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9514, ~9/7 = 435.0709
 
{{Optimal ET sequence|legend=1| 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634 }}
 
[[Badness]] (Sintel): 0.240
 
== Sepbizo-asegu (171 & 3566 & 4973) ==
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.000545
[[Comma list]]: {{monzo| -3 2 -17 14 }} (6104007655641/6103515625000)


== Sepbizo-asegu (171 &amp; 3566 &amp; 4973) (rank-3) ==
{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 -14 -17 }}
Subgroup: 2.3.5.7


Comma list: 6104007655641/6103515625000
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9548, ~31250/16807 = 1073.8021


Mapping: [{{val| 1 0 13 16 }}, {{val| 0 1 10 12 }}, {{val| 0 0 14 17 }}]
{{Optimal ET sequence|legend=1| 171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653 }}


Vals: {{Val list| 171, 1236, 1407, 3566, 3737, 3908, 4802, 4973, 8539, 8710}}
[[Badness]] (Sintel): 0.982


Badness: 0.000223
== Euzenius or Sabiruquinyo ==
{{See also| Euzenius comma }}


== Euzenius or Sabiruquinyo (rank-3) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 78125000/78121827
[[Comma list]]: 78125000/78121827


[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 2 0 -13 }}, {{val| 0 0 1 5 }}]
{{Mapping|legend=1| 1 1 0 -5 | 0 2 0 -13 | 0 0 1 5 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~6250/5103 = 350.9787, ~5/4 = 386.3099
 
{{Optimal ET sequence|legend=1| 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 }}


{{Val list|legend=1| 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]] (Sintel): 0.0902


[[Badness]]: 0.000205
== Nommismic or Quinla-zoyoyo ==
{{See also| Nommisma }}


== Quinla-zoyoyo (612 &amp; 1848 &amp; 3125) (rank-3) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| -55 30 2 1 }}
[[Comma list]]: {{monzo| -55 30 2 1 }}


[[Mapping]]: [{{val| 1 0 0 55 }}, {{val| 0 1 0 -30 }}, {{val| 0 0 1 -2 }}]
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9516, ~5/4 = 386.3132
 
{{Optimal ET sequence|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, }} …
 
[[Badness]] (Sintel): 1.62
 
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 1 -15 -18 23 }}


{{Val list|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795 }}
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}


[[Badness]]: 0.000366
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9551, ~56953125/40353607 = 596.5022


== Satwethezo-atritribigu (171 &amp; 1547 &amp; 4973) (rank-3) ==
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 3255, 3426, 3597, 4460, 4631, 4802, 4973, 5144, 6520, 6691, 11664, 18355, 48374, 48545, 66900, 145464, }} …
Subgroup: 2.3.5.7


Comma list: {{monzo| 1 -15 -18 23 }}
[[Badness]] (Sintel): 0.505


Mapping: [{{val| 1 0 9 7 }}, {{val| 0 1 3 3 }}, {{val| 0 0 23 18 }}]
== Technologismic or Trisa-quinbiru-agu ==
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 171, 1547, 4802, 4973, 5144, 6520, 6691, 6862, 11493, 11664, 11835, 18184, 18355, 18526, 25046, 30019, 30190, 48545}}
[[Comma list]]: {{monzo| 51 -13 -1 -10 }}


Badness: 0.000114
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}


== Trisa-quinbiru-agu (441 &amp; 1848 &amp; 3125) (rank-3) ==
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9553, ~63/32 = 1172.7374
Subgroup: 2.3.5.7


Comma list: {{monzo| 51 -13 -1 -10 }}
{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …


Mapping: [{{val| 1 0 1 5 }}, {{val| 0 1 7 -2 }}, {{val| 0 0 10 -1 }}]
[[Badness]] (Sintel): 0.116


Vals: {{Val list| 441, 1848, 3125, 3566, 4973, 6691, 8098, 8539, 9816, 11664, 14789, 15230, 18355, 21480, 30019, 33144, 39835, 51499}}
== Quinbisa-quinbizo-alegu (1848 & 11664 & 21480) ==
[[Subgroup]]: 2.3.5.7


Badness: 0.000263
[[Comma list]]: {{monzo| 110 -71 -11 10 }}


== Quinbisa-quinbizo-alegu (21480 &amp; 33144 &amp; 103169) (rank-3) ==
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}
Subgroup: 2.3.5.7


Comma list: {{monzo| 110 -71 -11 10 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.955, ~32805/28672 = 233.128


Mapping: [{{val| 1 0 0 -11 }}, {{val| 0 1 -1 6 }}, {{val| 0 0 10 11 }}]
{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …


Vals: {{Val list| 11664, 21480, 33144, 36881, 48545, 54624, 70025, 81689, 91505, 103169}}
[[Badness]] (Sintel): 0.538


[[Category:Temperament collections]]
[[Category:Temperament collections]]
{{Todo| add definition | intro }}
[[Category:Pages with mostly numerical content]]
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