Xenharmonic Wiki

Very high accuracy temperaments: Difference between revisions

← Older edit
VisualWikitext
m →Kwazy: +link
→Rank-3 temperaments: move akjaysmic here
 
(48 intermediate revisions by 9 users not shown)
Line 1: Line 1:
Below are listed some very high accuracy temperaments ([[Tenney-Euclidean temperament measures|TE error]] < 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 =
= Rank-2 temperaments =
== Kwazy ==
== Kwazy ==
Kwazy (118 &amp; 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]].
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 }} = 9010162353515625 / 9007199254740992
[[Comma list]]: {{monzo| -53 10 16 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~94921875/67108864 = 1\2, ~1125/1024 = 162.7433
* [[WE]]: ~94921875/67108864 = 600.003398{{c}}, ~1125/1024 = 162.743548{{c}}
* [[POTE]]: ~94921875/67108864 = 1\2, ~1125/1024 = 162.742626
: [[error map]]: {{val| +0.0068 -0.0032 -0.0111 }}
* [[CWE]]: ~94921875/67108864 = 600.000000{{c}}, ~1125/1024 = 162.742801{{c}}
: error map: {{val| 0.0000 -0.0126 -0.0277 }}


{{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 }}


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


== Astro ==
== Astro ==
Astro temperament tempers out the astro comma, {{monzo| 91 -12 -31 }}. 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
Line 28: Line 30:
[[Comma list]]: {{monzo| 91 -12 -31 }}
[[Comma list]]: {{monzo| 91 -12 -31 }}


{{Mapping|legend=1| 1 5 1 | 0 -31 12 }}
{{Mapping|legend=1| 1 -26 13 | 0 31 -12 }}
 
: mapping generators: ~2, ~{{monzo| 39 -5 -13 }}
: mapping generators: ~2, ~{{monzo| -38 5 13 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| -38 5 13 }} = 132.1949
* [[WE]]: ~2 = 1199.994636{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.800558{{c}}
* [[POTE]]: ~2 = 1\1, ~{{monzo| -38 5 13 }} = 132.194668
: [[error map]]: {{val| -0.0054 +0.0018 +0.0099 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.805278{{c}}
: error map: {{val| 0.0000 +0.0086 +0.0230 }}


{{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 }}


[[Badness]]: 0.062778
[[Badness]] (Sintel): 1.473


== Gaster ==
== Gaster ==
Line 48: Line 51:
[[Comma list]]: {{monzo| -70 72 -19 }}
[[Comma list]]: {{monzo| -70 72 -19 }}


{{Mapping|legend=1| 1 11 38 | 0 -19 -72 }}
{{Mapping|legend=1| 1 -8 -34 | 0 19 72 }}
 
: mapping generators: ~2, ~{{monzo| -18 19 -5 }}
: mapping generators: ~2, ~{{monzo| 19 -19 5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 19 -19 5 }} = 594.6345
* [[WE]]: ~2 = 1200.002817{{c}}, ~{{monzo| -18 19 -5 }} = 605.366856{{c}}
* [[POTE]]: ~2 = 1\1, ~{{monzo| 19 -19 5 }} = 594.634565
: [[error map]]: {{val| +0.0028 -0.0073 +0.0041 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| -18 19 -5 }} = 605.365482{{c}}
: error map: {{val| 0.0000 -0.0108 +0.0010 }}


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


[[Badness]]: 0.088530
[[Badness]] (Sintel): 2.077


== Whoosh ==
== Whoosh ==
Whoosh (152 &amp; 289) tempers out the whoosh comma, {{monzo| 37 25 -33 }}. 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
Line 67: Line 71:
[[Comma list]]: {{monzo| 37 25 -33 }}
[[Comma list]]: {{monzo| 37 25 -33 }}


{{Mapping|legend=1| 1 17 14 | 0 -33 -25 }}
{{Mapping|legend=1| 1 -16 -11 | 0 33 25 }}
 
: mapping generators: ~2, ~625/432
: mapping generators: ~2, ~864/625


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~864/625 = 560.5470
* [[WE]]: ~2 = 1199.997806{{c}}, ~625/432 = 639.452005{{c}}
* [[POTE]]: ~2 = 1\1, ~864/625 = 560.546826
: [[error map]]: {{val| -0.0022 -0.0037 +0.0106 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~625/432 = 639.453117{{c}}
: error map: {{val| 0.0000 -0.0022 +0.0142 }}


{{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 }}


[[Badness]]: 0.025653
[[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|legend=1| 1 0 -27 | 0 2 37 }}
{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}
: mapping generators: ~2, ~{{monzo| 27 -18 1 }}
: mapping generators: ~2, ~{{monzo| 27 -18 1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 27 -18 1 }} = 950.9814
* [[WE]]: ~2 = 1199.997524{{c}}, ~{{monzo| 27 -18 1 }} = 950.979631{{c}}
* [[POTE]]: ~2 = 1\1, ~{{monzo| 27 -18 1 }} = 950.981594
: [[error map]]: {{val| -0.0025 +0.0043 -0.0005 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 27 -18 1 }} = 950.981535{{c}}
: error map: {{val| 0.0000 +0.0081 +0.0031 }}


{{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 }}


[[Badness]]: 0.010391
[[Badness]] (Sintel): 0.244


== Egads ==
== Egads ==
Line 103: Line 109:
[[Comma list]]: {{monzo| -36 -52 51 }}
[[Comma list]]: {{monzo| -36 -52 51 }}


{{Mapping|legend=1| 1 15 16 | 0 -51 -52 }}
{{Mapping|legend=1| 1 -36 -36 | 0 51 52 }}
 
: mapping generators: ~2, ~5/3
: mapping generators: ~2, ~6/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 315.6479
* [[WE]]: ~2 = 1200.000553{{c}}, ~5/3 = 884.352488{{c}}
* [[POTE]]: ~2 = 1\1, ~6/5 = 315.647919
: [[error map]]: {{val| +0.0006 +0.0020 -0.0042 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~5/3 = 844.352090{{c}}
: error map: {{val| 0.0000 +0.0016 -0.0050 }}


{{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 }}


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


=== Hemiegads ===
=== Hemiegads ===
Line 120: Line 127:
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}


Mapping: {{mapping| 1 15 16 -16 | 0 -102 -104 143 }}
Mapping: {{mapping| 1 -87 -88 127 | 0 102 104 -143 }}
: mapping generators: ~2, ~67108864/36756909


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~36756909/33554432 = 157.8240
* WE: ~2 = 1200.000308{{c}}, ~67108864/36756909 = 1042.176313{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~67108864/36756909 = 1042.176046{{c}}
 
{{Optimal ET sequence|legend=0| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}


{{Optimal ET sequence|legend=1| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}
Badness (Sintel): 0.313


Badness: 0.0124
=== Catabolic ===
Subgroup: 2.3.5.13
 
Comma list: [[140625/140608]], {{monzo| -4 20 -9 0 0 -4 }}
 
Mapping: {{mapping| 1 -36 -36 -98 | 0 51 52 138 }}
 
Optimal tunings:
* WE: ~2 = 1200.000308{{c}}, ~5/3 = 884.354064{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~5/3 = 884.351835{{c}}
 
{{Optimal ET sequence|legend=0| 19, …, 441, 460, 901, 1342 }}
 
Badness (Sintel): 0.222


== Fortune ==
== Fortune ==
Fortune tempers out the fortune comma, {{monzo| -107 47 14 }}. 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
Line 136: Line 160:
[[Comma list]]: {{monzo| -107 47 14 }}
[[Comma list]]: {{monzo| -107 47 14 }}


{{Mapping|legend=1| 1 13 -36 | 0 -14 47 }}
{{Mapping|legend=1| 1 -1 11 | 0 14 -47 }}
 
: mapping generators: ~2, ~8388608/7381125
: mapping generators: ~2, 7381125/4194304


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~8388608/7381125 = 221.5679
* [[WE]]: ~2 = 1200.000553{{c}}, ~8388608/7381125 = 221.568202{{c}}
* [[POTE]]: ~2 = 1\1, ~8388608/7381125 = 221.567899
: [[error map]]: {{val| +0.0016 -0.0018 -0.0012 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.567889{{c}}
: error map: {{val| 0.0000 -0.0046 -0.0045 }}


{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}
{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}


[[Badness]]: 0.027876
[[Badness]] (Sintel): 0.654


=== 7-limit ===
=== 7-limit ===
Line 153: Line 178:
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}


Mapping: {{mapping| 1 13 -36 -263 | 0 -14 47 326 }}
Mapping: {{mapping| 1 -1 11 63 | 0 14 -47 -326 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~8388608/7381125 = 221.5680
* WE: ~2 = 1200.001970{{c}}, ~8388608/7381125 = 221.568395{{c}}
* POTE: ~2 = 1\1, ~8388608/7381125 = 221.568031
* CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568026{{c}}


{{Optimal ET sequence|legend=1| 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}
{{Optimal ET sequence|legend=0| 65d, …, 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}


Badness: 0.023704
Badness (Sintel): 0.600


=== 11-limit ===
=== 11-limit ===
Line 168: Line 193:
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752


Mapping: {{mapping| 1 13 -36 -263 -573 | 0 -14 47 326 707 }}
Mapping: {{mapping| 1 -1 11 63 134 | 0 14 -47 -326 -707 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~8388608/7381125 = 221.5681
* WE: ~2 = 1200.003854{{c}}, ~8388608/7381125 = 221.568850{{c}}
* POTE: ~2 = 1\1, ~8388608/7381125 = 221.568139
* CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568133{{c}}


{{Optimal ET sequence|legend=1| 612, 2513, 3125, 3737, 8086, 19909d }}
{{Optimal ET sequence|legend=0| 65dee, …, 612, 2513, 3125, 3737, 8086, 19909d }}


Badness: 0.036641
Badness (Sintel): 1.211


== Senior ==
== Senior ==
Senior (171 &amp; 1171) tempers out the senior comma, {{monzo| -17 62 -35 }}. 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
Line 185: Line 210:
[[Comma list]]: {{monzo| -17 62 -35 }}
[[Comma list]]: {{monzo| -17 62 -35 }}


{{Mapping|legend=1| 1 11 19 | 0 -35 -62 }}
{{Mapping|legend=1| 1 -24 -43 | 0 35 62 }}
 
: mapping generators: ~2, ~{{monzo| 7 -23 13 }}
: mapping generators: ~2, ~{{monzo| -6 23 -13 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| -6 23 -13 }} = 322.8013
* [[WE]]: ~2 = 1200.000236{{c}}, ~{{monzo| 7 -23 13 }} = 877.198814{{c}}
* [[POTE]]: ~2 = 1\1, ~{{monzo| -6 23 -13 }} = 322.801358
: [[error map]]: {{val| +0.0002 -0.0022 +0.0026 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 7 -23 13 }} = 877.198647{{c}}
: error map: {{val| 0.0000 -0.0024 -0.0024 }}


{{Optimal ET 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 (118 &amp; 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.
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
Line 204: Line 230:
[[Comma list]]: {{monzo| 144 -22 -47 }}
[[Comma list]]: {{monzo| 144 -22 -47 }}


{{Mapping|legend=1| 1 45 -18 | 0 -47 22 }}
{{Mapping|legend=1| 1 -2 4 | 0 47 -22 }}
 
: mapping generators: ~2, ~{{monzo| 46 -7 -15 }}
: mapping generators: ~2, ~{{monzo| -45 7 15 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 46 -7 -15 }} = 91.5310
* [[WE]]: ~2 = 1199.998956{{c}}, ~{{monzo| 46 -7 -15 }} = 91.530922{{c}}
* [[POTE]]: ~2 = 1\1, ~{{monzo| 46 -7 -15 }} = 91.531001
: [[error map]]: {{val| -0.0010 +0.0004 +0.0018 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~{{monzo| 46 -7 -15 }} = 91.530995{{c}}
: error map: {{val| 0.0000 +0.0018 +0.0044 }}


{{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 }}
{{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]]: 0.046315
[[Badness]] (Sintel): 1.086


; Music
; 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
* ''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 ==
== Septarium ==
{{See also| Quectisma }}
Septarium splits the octave in seven, and tempers out the [[septarium comma]], {{monzo| 73 77 -84 }}. It can be described as {{nowrap| 441 & 2513 }} temperament, which tempers out 78125000/78121827 and {{monzo| 47 -7 -7 -7 }} ([[akjaysma]]) in the 7-limit.


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


[[Comma list]]: {{monzo| 554 -351 1 }}
[[Comma list]]: {{monzo| 73 77 -84 }}
 
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}


: mapping generators: ~2, ~3
{{Mapping|legend=1| 7 1 7 | 0 12 11 }}
: mapping generators: ~{{monzo| 21 22 -24 }}, ~{{monzo| -20 -21 23 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9553
* [[WE]]: ~{{monzo| 21 22 -24 }} = 171.429{{c}}, ~{{monzo| -20 -21 23 }} = 144.210{{c}}
* [[CWE]]: ~{{monzo| 21 22 -24 }} = 171.429{{c}}, ~{{monzo| -20 -21 23 }} = 144.210{{c}}


{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}
{{Optimal ET sequence|legend=1| 133, 308, 441, 1631, 2072, 2513, 5467, 7980 }}


[[Badness]]: 0.372
[[Badness]] (Sintel): 1.395


=== Nanoquectismic ===
=== 7-limit ===
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 = 1\1, ~3/2 = 701.955
 
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
 
Badness: 0.604
 
=== Conanoquectismic ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
Comma list: 78125000/78121827, 140737488355328/140710042265625


Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}
Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.9548
* WE: ~1157625/1048576 = 171.428{{c}}, ~2097152/1929375 = 144.211{{c}}
* CWE: ~1157625/1048576 = 171.429{{c}}, ~2097152/1929375 = 144.212{{c}}


{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}
{{Optimal ET sequence|legend=0| 133, 308, 441, 1631, 2072, 2513, 2954 }}


Badness: 2.17
Badness (Sintel): 0.777


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 272: Line 286:
[[Comma list]]: {{monzo| 71 -99 37 }}
[[Comma list]]: {{monzo| 71 -99 37 }}


{{Mapping|legend=1| 1 28 73 | 0 -37 -99 }}
{{Mapping|legend=1| 1 -9 -26 | 0 37 99 }}
 
: mapping generators: ~2, ~8000/6561
: mapping generators: ~2, ~6561/4000


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~8000/6561 = 343.2961
* [[WE]]: ~2 = 1199.999880537{{c}}, ~8000/6561 = 343.296063356{{c}}
* [[POTE]]: ~2 = 1\1, ~8000/6561 = 343.296098
: [[error map]]: {{val| -0.000119 +0.000418 -0.000336 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~8000/6561 = 343.296095985{{c}}
: error map: {{val| 0.000000 +0.000551 -0.000211 }}


{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
{{Optimal ET sequence|legend=1| 7, …, 381, 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}


[[Badness]]: 0.012867
[[Badness]] (Sintel): 0.302


== Pirate ==
== Pirate ==
{{see also|Hemfiness temperaments #Piracy}}
{{See also| Hemfiness temperaments #Piracy }}


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 291: Line 306:
[[Comma list]]: {{monzo| -90 -15 49 }}
[[Comma list]]: {{monzo| -90 -15 49 }}


{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}
{{Mapping|legend=1| 1 -6 0 | 0 49 15 }}
 
: mapping generators: ~2, ~{{monzo| 24 4 -13 }}
: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 24 4 -13 }} = 185.754186
* [[WE]]: ~2 = 1200.000195604{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}
* [[POTE]]: ~2 = 1\1, ~{{monzo| 24 4 -13 }} = 185.754179
: [[error map]]: {{val| -0.000196 +0.000418 -0.000336 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}
: error map: {{val| 0.000000 +0.000082 -0.000574 }}


{{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 … }}


[[Badness]]: 0.005656
[[Badness]] (Sintel): 0.133
 
== 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:
* [[WE]]: ~2 = 1200.000184840{{c}}, ~3/2 = 701.954524421{{c}}
: [[error map]]: {{val| +0.000185 -0.000292 -0.000002 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.954415845{{c}}
: error map: {{val| 0.000000 -0.000585 -0.000289 }}


== Atomic ==
{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}
{{See also | Kirnberger's atom }}


Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].
[[Badness]] (Sintel): 11.971


Atomic does not temper out the [[schisma]], so 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.
== Quectismic ==
{{See also| Quectisma }}


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


[[Comma list]]: {{monzo| 161 -84 -12 }}
[[Comma list]]: {{monzo| 554 -351 1 }}


{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}
 
: mapping generators: ~2, ~3
: mapping generators: ~{{monzo| -67 35 5 }}, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~{{monzo| -67 35 5 }} = 1\12, ~3/2 = 701.9552 (~32805/32768 = 1.9552)
* [[WE]]: ~2 = 1199.999902567{{c}}, ~3/2 = 701.955253373{{c}}
* [[POTE]]: ~{{monzo| -67 35 5 }} = 1\12, ~3/2 = 701.955162 (~32805/32768 = 1.955162)
: [[error map]]: {{val| -0.000097 +0.000155 -0.000001 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.955310152{{c}}
: error map: {{val| 0.000000 +0.000309 +0.000149 }}


{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}


[[Badness]]: 0.003803
[[Badness]] (Sintel): 8.736


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


Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}


Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}
Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)
* WE: ~2 = 1199.979407{{c}}, ~3/2 = 701.943437{{c}}
* POTE: ~30375/28672 = 1\12, ~3/2 = 701.949608 (or ~32805/32768 = 1.949608)
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955457{{c}}


{{Optimal ET sequence|legend=1| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}


Badness: 0.045818
Badness (Sintel): 15.285


=== 11-limit ===
=== Conanoquectismic ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7


Comma list: 9801/9800, 151263/151250, 184549376/184528125
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}


Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}
Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)
* WE: ~2 = 1200.010750{{c}}, ~3/2 = 701.961041{{c}}
* POTE: ~30375/28672 = 1\12, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954771{{c}}
 
{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}
 
Badness (Sintel): 54.974
 
== Atomic ==
: ''For extensions, see [[Landscape microtemperaments #Atomic]].''
 
Atomic tempers out [[Kirnberger's atom]], the microscopic interval separating eleven [[schisma]]s and a [[syntonic comma]], or twelfth schismas and a [[Pythagorean comma]]. It is a member of the [[schismic–commatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 12 }}. In this extremely accurate temperament, the [[3/2|perfect fifth]] is one [[schisma]] sharp of its [[12edo]] value.  


{{Optimal ET sequence|legend=1| 12, …, 600e, 612, 1236, 1848 }}
[[Subgroup]]: 2.3.5


Badness: 0.016029
[[Comma list]]: {{monzo| 161 -84 -12 }}


== Revopentic (2.3.7) ==
{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}
: mapping generators: ~{{monzo| -67 35 5 }}, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~{{monzo| -67 35 5 }} = 99.999995361{{c}}, ~3/2 = 701.955129505{{c}} (~32805/32768 = 1.955161980{{c}})
: [[error map]]: {{val| -0.000056 +0.000072 +0.000022 }}
* [[CWE]]: ~{{monzo| -67 35 5 }} = 100.000000000{{c}}, ~3/2 = 701.955166184{{c}} (~32805/32768 = 1.955166184{{c}})
: error map: {{val| 0.000000 +0.000165 +0.000123 }}
 
{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
 
[[Badness]] (Sintel): 0.0892
 
== Revopentic ==
{{See also| Revopentisma }}
{{See also| Revopentisma }}


Subgroup: 2.3.7
Subgroup: 2.3.7


Comma list: 11399736556781568/11398895185373143
Comma list: {{monzo| 47 4 -19 }}


{{Mapping|legend=2| 1 12 5 | 0 -19 -4 }}
{{Mapping|legend=2| 1 -7 1 | 0 19 4 }}
 
: mapping generators: ~2, ~16807/12288
: mapping generators: ~2 = 1\1, 24576/16807


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~24576/16807 = 657.792
* [[WE]]: ~2 = 1199.998821{{c}}, ~16807/12288 = 542.207710{{c}}
: [[error map]]: {{val| -0.0012 -0.0003 +0.0038 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.208204{{c}}
: error map: {{val| 0.0000 +0.0009 +0.0069 }}


[[Support]]ing [[ET]]s: {{EDOs| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794 }}
{{Optimal ET sequence|legend=1| 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, … }}


Badness (Sintel's TE logflat): 0.070
[[Badness]] (Sintel): 0.0704


=== Revopent ===
=== Revopent ===
Revopent can be described as the 1848 & 3125 temperament.
Revopent can be described as the {{nowrap| 1848 & 3125 }} temperament.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 381: Line 438:
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}


Mapping: {{mapping| 1 12 -155 5 | 0 -19 287 -4 }}
Mapping: {{mapping| 1 -7 132 1 | 0 19 -287 4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~24576/16807 = 657.792
* WE: ~2 = 1200.000110{{c}}, ~16807/12288 = 542.208018{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.207968{{c}}


{{Optimal ET sequence|legend=1| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}
{{Optimal ET sequence|legend=0| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}


Badness: 0.263
Badness (Sintel): 0.665


== Nanic ==
== Nanic ==
Line 397: Line 455:
[[Comma list]]: {{monzo| 109 -67 -1 }}
[[Comma list]]: {{monzo| 109 -67 -1 }}


{{Mapping|legend=1| 1 0 109 | 0 1 -67 }}
{{Mapping|legend=2| 1 0 109 | 0 1 -67 }}
 
: mapping generators: ~2, ~3
: sval mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1\1, ~3/2 = 701.9578
* [[WE]]: ~2 = 1199.999111{{c}}, ~3/2 = 701.957264{{c}}
: [[error map]]: {{val| -0.0009 +0.0014 +0.0001 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.957796{{c}}
: error map: {{val| 0.0000 +0.0028 +0.0018 }}


{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}
{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}


[[Badness]]: 0.0138
[[Badness]] (Sintel): TBD


== Icarus ==
== Icarus ==
Named after a very distant star. Curiously, the names of two even more distant stars coincide with named temperaments, those being [[godzilla]] and [[mothra]].
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
[[Subgroup]]: 3.5.7
Line 415: Line 475:
[[Comma list]]: {{monzo| -32 -64 71 }}
[[Comma list]]: {{monzo| -32 -64 71 }}


{{Mapping|legend=1| 1 35 32 | 0 -71 -64 }}
{{Mapping|legend=2| 1 -36 -32 | 0 71 64 }}
: mapping generators: ~3, ~{{monzo| -4 -9 10 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3 = 1\1<3/1>, ~{{monzo| 5 9 -10 }} = 898.340
* [[WE]]: ~3 = 1901.955003{{c}}, ~{{monzo| -4 -9 10 }} = 1003.615406{{c}}
: [[error map]]: {{val| +0.0000 +0.0000 -0.0000 }}
* [[CWE]]: ~3 = 1901.955001{{c}}, ~{{monzo| -4 -9 10 }} = 1003.615405{{c}}
: error map: {{val| 0.0000 -0.0000 -0.0000 }}


[[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]]
[[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]]: 0.493 × 10<sup>-3</sup>
[[Badness]] (Sintel): TBD


= Rank-3 temperaments =
= Rank-3 temperaments =
== Sasaquinbizo-atriyo (171 &amp; 1106 &amp; 3566) ==
== Akjaysmic ==
{{See also| Akjaysma }}
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| 3 -24 3 10 }} = 282475249000/282429536481
[[Comma list]]: {{monzo| 47 -7 -7 -7 }}
 
{{Mapping|legend=1| 7 0 0 47 | 0 1 0 -1 | 0 0 1 -1 }}
: mapping generators: ~1157625/1048576, ~3, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}
: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}
* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}
: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}
 
{{Optimal ET sequence|legend=1| 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}
 
[[Badness]] (Sintel): 2.22
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 184549376/184528125, 199297406/199290375
 
Mapping: {{mapping| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}
: mapping generators: ~29160/26411, ~3, ~5
 
Optimal tunings:
* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}
* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}
 
{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}
 
Badness (Sintel): 1.32
 
== 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 }}
{{Mapping|legend=1| 1 0 9 -3 | 0 1 8 0 | 0 0 10 -3 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9624, ~6561/3430 = 1122.9387
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9624{{c}}, ~6561/3430 = 1122.9387{{c}}


{{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 }}
{{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]]: 0.281 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.24
 
== Izarismic ==
This temperament is identical to the [[izar]] in the no-twos subgroup, but has an independent generator for prime 2. It can be described as {{nowrap| 41 & 49 & 130 }} temperament, and tempers out the [[izar comma]], {{monzo| 0 -11 -7 12 }}.


== Saquadtrizo-asepgu (171 &amp; 2019 &amp; 3566) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| 0 -11 -7 12 }} = 13841287201/13839609375
[[Comma list]]: 13841287201/13839609375


{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
: mapping generators: ~2, ~3, ~16807/10125


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9584, ~16807/10125 = 877.2825
[[Optimal tuning]]s:
* [[TE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}
* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9584{{c}}, ~16807/10125 = 877.2825{{c}}


{{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 }}
{{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 }}


[[Badness]]: 0.135 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.595


== Laquinzo-aquadquadgu (171 &amp; 441 &amp; 4973) ==
== Laquinzo-aquadquadgu (171 & 441 & 4973) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}
{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9501, ~250/189 = 484.2956
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9501{{c}}, ~250/189 = 484.2956{{c}}


{{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 }}
{{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 }}


[[Badness]]: 0.297 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.31


== Laleruyo (171 &amp; 1547 &amp; 3125) ==
== Laleruyo (171 & 1547 & 3125) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| -1 4 11 -11 }} = 3955078125/3954653486
[[Comma list]]: {{monzo| -1 4 11 -11 }} (3955078125/3954653486)


{{Mapping|legend=1| 1 3 0 1 | 0 -11 0 -4 | 0 0 1 1 }}
{{Mapping|legend=1| 1 3 0 1 | 0 -11 0 -4 | 0 0 1 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~375/343 = 154.3678, ~5/4 = 386.3070
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~375/343 = 154.3678{{c}}, ~5/4 = 386.3070{{c}}


{{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 }}
{{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]]: 0.114 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.501


== Starscape or Latritriru-ayo ==
== Starscape ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 484: Line 589:
{{Mapping|legend=1| 1 0 4 0 | 0 1 1 2 | 0 0 -9 -1 }}
{{Mapping|legend=1| 1 0 4 0 | 0 1 1 2 | 0 0 -9 -1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9514, ~9/7 = 435.0709
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9514{{c}}, ~9/7 = 435.0709{{c}}


{{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 }}
{{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]]: 0.545 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.240


== Sepbizo-asegu (171 &amp; 3566 &amp; 4973) ==
== Sepbizo-asegu (171 & 3566 & 4973) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 -14 -17 }}
{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 -14 -17 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9548, ~31250/16807 = 1073.8021
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9548{{c}}, ~31250/16807 = 1073.8021{{c}}


{{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 }}
{{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 }}


[[Badness]]: 0.223 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.982


== Euzenius or Sabiruquinyo ==
== Euzenius ==
{{See also| Euzenius comma }}
{{See also| Euzenius comma }}


Line 512: Line 617:
{{Mapping|legend=1| 1 1 0 -5 | 0 2 0 -13 | 0 0 1 5 }}
{{Mapping|legend=1| 1 1 0 -5 | 0 2 0 -13 | 0 0 1 5 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6250/5103 = 350.9787, ~5/4 = 386.3099
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6250/5103 = 350.9787{{c}}, ~5/4 = 386.3099{{c}}


{{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 }}
{{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 }}


[[Badness]]: 0.205 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.0902


== Nommismic or Quinla-zoyoyo ==
== Nommismic ==
{{See also| Nommisma }}
{{See also| Nommisma }}


Line 527: Line 632:
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9516, ~5/4 = 386.3132
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9516{{c}}, ~5/4 = 386.3132{{c}}


{{Optimal ET sequence|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, }} …
{{Optimal ET sequence|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, }} …


[[Badness]]: 0.366 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.62


== Satwethezo-atritribigu (171 &amp; 1547 &amp; 4973)  ==
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 540: Line 645:
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9551, ~56953125/40353607 = 596.5022
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9551{{c}}, ~56953125/40353607 = 596.5022{{c}}


{{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, }} …
{{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, }} …


[[Badness]]: 0.114 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.505


== Technologismic or Trisa-quinbiru-agu (441 &amp; 1848 &amp; 3125)  ==
== Technologismic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 553: Line 658:
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9553, ~63/32 = 1172.7374
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9553{{c}}, ~63/32 = 1172.7374{{c}}


{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …
{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …


[[Badness]]: 0.263 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.116


== Quinbisa-quinbizo-alegu (1848 &amp; 11664 &amp; 21480) ==
== Quinbisa-quinbizo-alegu (1848 & 3737 & 6079) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 566: Line 671:
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.955, ~32805/28672 = 233.128
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~32805/28672 = 233.128{{c}}


{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …
{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …


[[Badness]]: 0.122 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.538
 
== Septrongismic ==
{{See also| Septrongisma }}
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -7 30 -9 -7 }} = 205891132094649/205885750000000
 
{{Mapping|legend=1| 1 0 0 -1 | 0 1 1 3 | 0 0 7 -9 }}
 
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.954{{c}}, ~1715000/1594323 = 126.337{{c}}
 
{{Optimal ET sequence|legend=1| 171, 2783, 2954, 3125, 8539, 11664, 14789, }} …
 
[[Badness]] (Sintel): 0.320


[[Category:Temperament collections]]
[[Category:Temperament collections]]
Retrieved from "https://en.xen.wiki/w/Very_high_accuracy_temperaments"