Very high accuracy temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
Below are listed some very high accuracy temperaments ([[Tenney-Euclidean temperament measures|TE error]] &lt; 0.005 cents/octave).
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 ({{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]].
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 29: 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 49: 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 68: 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 104: 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 121: 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 }}
 
Badness (Sintel): 0.313


{{Optimal ET sequence|legend=1| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}
=== Catabolic ===
Subgroup: 2.3.5.13


Badness: 0.0124
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 137: 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 154: 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 169: 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 186: 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 205: 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|legend=1| 7 1 7 | 0 12 11 }}
 
: mapping generators: ~{{monzo| 21 22 -24 }}, ~{{monzo| -20 -21 23 }}
: mapping generators: ~2, ~3


[[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
Subgroup: 2.3.5.7


Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}
Comma list: 78125000/78121827, 140737488355328/140710042265625


Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}
Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.955
* 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=0| 133, 308, 441, 1631, 2072, 2513, 2954 }}
 
Badness (Sintel): 0.777
 
== Raider ==
{{See also| Syntonic–chromatic equivalence continuum }}
 
[[Subgroup]]: 2.3.5


{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
[[Comma list]]: {{monzo| 71 -99 37 }}
 
{{Mapping|legend=1| 1 -9 -26 | 0 37 99 }}
: mapping generators: ~2, ~8000/6561
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.999880537{{c}}, ~8000/6561 = 343.296063356{{c}}
: [[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| 7, …, 381, 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
 
[[Badness]] (Sintel): 0.302


Badness: 0.604
== Pirate ==
{{See also| Hemfiness temperaments #Piracy }}


=== Conanoquectismic ===
[[Subgroup]]: 2.3.5
Subgroup: 2.3.5.7


Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
[[Comma list]]: {{monzo| -90 -15 49 }}


Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}
{{Mapping|legend=1| 1 -6 0 | 0 49 15 }}
: mapping generators: ~2, ~{{monzo| 24 4 -13 }}


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~3/2 = 701.9548
* [[WE]]: ~2 = 1200.000195604{{c}}, ~{{monzo| 24 4 -13 }} = 185.754209314{{c}}
: [[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| 306d, 359cd, 665 }}
{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}


Badness: 2.17
[[Badness]] (Sintel): 0.133


== Counterquectismic ==
== Counterquectismic ==
Line 274: Line 327:


{{Mapping|legend=1| 1 0 500 | 0 1 -314 }}
{{Mapping|legend=1| 1 0 500 | 0 1 -314 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 701.9544
* [[WE]]: ~2 = 1200.000184840{{c}}, ~3/2 = 701.954524421{{c}}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 701.9544
: [[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 }}


{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}
{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}


[[Badness]]: 0.510
[[Badness]] (Sintel): 11.971


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


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


[[Comma list]]: {{monzo| 71 -99 37 }}
[[Comma list]]: {{monzo| 554 -351 1 }}
 
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.999902567{{c}}, ~3/2 = 701.955253373{{c}}
: [[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| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}


{{Mapping|legend=1| 1 28 73 | 0 -37 -99 }}
[[Badness]] (Sintel): 8.736


: mapping generators: ~2, ~6561/4000
=== Nanoquectismic ===
Subgroup: 2.3.5.7


[[Optimal tuning]]s:  
Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}
* [[CTE]]: ~2 = 1\1, ~8000/6561 = 343.2961
* [[POTE]]: ~2 = 1\1, ~8000/6561 = 343.296098


{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}


[[Badness]]: 0.012867
Optimal tunings:  
* WE: ~2 = 1199.979407{{c}}, ~3/2 = 701.943437{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955457{{c}}


== Pirate ==
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
{{see also|Hemfiness temperaments #Piracy}}


[[Subgroup]]: 2.3.5
Badness (Sintel): 15.285


[[Comma list]]: {{monzo| -90 -15 49 }}
=== Conanoquectismic ===
Subgroup: 2.3.5.7


{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}


: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}
Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}


[[Optimal tuning]]s:  
Optimal tunings:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 24 4 -13 }} = 185.754186
* WE: ~2 = 1200.010750{{c}}, ~3/2 = 701.961041{{c}}
* [[POTE]]: ~2 = 1\1, ~{{monzo| 24 4 -13 }} = 185.754179
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954771{{c}}


{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}


[[Badness]]: 0.005656
Badness (Sintel): 54.974


== Atomic ==
== Atomic ==
{{See also | Kirnberger's atom }}
: ''For extensions, see [[Landscape microtemperaments #Atomic]].''
 
Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].


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.
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.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 335: Line 399:


{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}
{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}
: mapping generators: ~{{monzo| -67 35 5 }}, ~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]]: ~{{monzo| -67 35 5 }} = 99.999995361{{c}}, ~3/2 = 701.955129505{{c}} (~32805/32768 = 1.955161980{{c}})
* [[POTE]]: ~{{monzo| -67 35 5 }} = 1\12, ~3/2 = 701.955162 (~32805/32768 = 1.955162)
: [[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 }}
{{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 ===
== Revopentic ==
Subgroup: 2.3.5.7
 
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
 
Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}
 
Optimal tunings:
* CTE: ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)
* POTE: ~30375/28672 = 1\12, ~3/2 = 701.949608 (or ~32805/32768 = 1.949608)
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}
 
Optimal tunings:
* CTE: ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)
* POTE: ~30375/28672 = 1\12, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)
 
{{Optimal ET sequence|legend=1| 12, …, 600e, 612, 1236, 1848 }}
 
Badness: 0.016029
 
== Revopentic (2.3.7) ==
{{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 401: 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 418: Line 456:


{{Mapping|legend=2| 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:  
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 701.9578
* [[WE]]: ~2 = 1199.999111{{c}}, ~3/2 = 701.957264{{c}}
* [[POTE]]: ~2 = 1200.0000, ~3/2 = 701.9578
: [[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 436: 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 505: 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 533: 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 548: 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 561: 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 574: 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 587: 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]]

Latest revision as of 08:59, 30 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

Below are listed some very high accuracy temperaments (Breed's 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 gwazy, bisupermajor, and crazy.

Subgroup: 2.3.5

Comma list: [-53 10 16⟩

Mapping: [⟨2 1 6], ⟨0 8 -5]]

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

Optimal tunings:

  • WE: ~94921875/67108864 = 600.003398 ¢, ~1125/1024 = 162.743548 ¢
error map: ⟨+0.0068 -0.0032 -0.0111]
  • CWE: ~94921875/67108864 = 600.000000 ¢, ~1125/1024 = 162.742801 ¢
error map: ⟨0.0000 -0.0126 -0.0277]

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

Badness (Sintel): 0.331

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 -26 13], ⟨0 31 -12]]

mapping generators: ~2, ~[39 -5 -13⟩

Optimal tunings:

  • WE: ~2 = 1199.994636 ¢, ~[39 -5 -13⟩ = 1067.800558 ¢
error map: ⟨-0.0054 +0.0018 +0.0099]
  • CWE: ~2 = 1200.000000 ¢, ~[39 -5 -13⟩ = 1067.805278 ¢
error map: ⟨0.0000 +0.0086 +0.0230]

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 (Sintel): 1.473

Gaster

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

Subgroup: 2.3.5

Comma list: [-70 72 -19⟩

Mapping: [⟨1 -8 -34], ⟨0 19 72]]

mapping generators: ~2, ~[-18 19 -5⟩

Optimal tunings:

  • WE: ~2 = 1200.002817 ¢, ~[-18 19 -5⟩ = 605.366856 ¢
error map: ⟨+0.0028 -0.0073 +0.0041]
  • CWE: ~2 = 1200.000000 ¢, ~[-18 19 -5⟩ = 605.365482 ¢
error map: ⟨0.0000 -0.0108 +0.0010]

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

Badness (Sintel): 2.077

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 -16 -11], ⟨0 33 25]]

mapping generators: ~2, ~625/432

Optimal tunings:

  • WE: ~2 = 1199.997806 ¢, ~625/432 = 639.452005 ¢
error map: ⟨-0.0022 -0.0037 +0.0106]
  • CWE: ~2 = 1200.000000 ¢, ~625/432 = 639.453117 ¢
error map: ⟨0.0000 -0.0022 +0.0142]

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

Badness (Sintel): 0.602

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⟩

Mapping: [⟨1 0 -27], ⟨0 2 37]]

mapping generators: ~2, ~[27 -18 1⟩

Optimal tunings:

  • WE: ~2 = 1199.997524 ¢, ~[27 -18 1⟩ = 950.979631 ¢
error map: ⟨-0.0025 +0.0043 -0.0005]
  • CWE: ~2 = 1200.000000 ¢, ~[27 -18 1⟩ = 950.981535 ¢
error map: ⟨0.0000 +0.0081 +0.0031]

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

Badness (Sintel): 0.244

Egads

Subgroup: 2.3.5

Comma list: [-36 -52 51⟩

Mapping: [⟨1 -36 -36], ⟨0 51 52]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1200.000553 ¢, ~5/3 = 884.352488 ¢
error map: ⟨+0.0006 +0.0020 -0.0042]
  • CWE: ~2 = 1200.000000 ¢, ~5/3 = 844.352090 ¢
error map: ⟨0.0000 +0.0016 -0.0050]

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 (Sintel): 0.981

Hemiegads

Subgroup: 2.3.5.7

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

Mapping: [⟨1 -87 -88 127], ⟨0 102 104 -143]]

mapping generators: ~2, ~67108864/36756909

Optimal tunings:

  • WE: ~2 = 1200.000308 ¢, ~67108864/36756909 = 1042.176313 ¢
  • CWE: ~2 = 1200.000000 ¢, ~67108864/36756909 = 1042.176046 ¢

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

Badness (Sintel): 0.313

Catabolic

Subgroup: 2.3.5.13

Comma list: 140625/140608, [-4 20 -9 0 0 -4⟩

Mapping: [⟨1 -36 -36 -98], ⟨0 51 52 138]]

Optimal tunings:

  • WE: ~2 = 1200.000308 ¢, ~5/3 = 884.354064 ¢
  • CWE: ~2 = 1200.000000 ¢, ~5/3 = 884.351835 ¢

Optimal ET sequence: 19, …, 441, 460, 901, 1342

Badness (Sintel): 0.222

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 -1 11], ⟨0 14 -47]]

mapping generators: ~2, ~8388608/7381125

Optimal tunings:

  • WE: ~2 = 1200.000553 ¢, ~8388608/7381125 = 221.568202 ¢
error map: ⟨+0.0016 -0.0018 -0.0012]
  • CWE: ~2 = 1200.000000 ¢, ~8388608/7381125 = 221.567889 ¢
error map: ⟨0.0000 -0.0046 -0.0045]

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

Badness (Sintel): 0.654

7-limit

Subgroup: 2.3.5.7

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

Mapping: [⟨1 -1 11 63], ⟨0 14 -47 -326]]

Optimal tunings:

  • WE: ~2 = 1200.001970 ¢, ~8388608/7381125 = 221.568395 ¢
  • CWE: ~2 = 1200.000000 ¢, ~8388608/7381125 = 221.568026 ¢

Optimal ET sequence: 65d, …, 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323

Badness (Sintel): 0.600

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 11 63 134], ⟨0 14 -47 -326 -707]]

Optimal tunings:

  • WE: ~2 = 1200.003854 ¢, ~8388608/7381125 = 221.568850 ¢
  • CWE: ~2 = 1200.000000 ¢, ~8388608/7381125 = 221.568133 ¢

Optimal ET sequence: 65dee, …, 612, 2513, 3125, 3737, 8086, 19909d

Badness (Sintel): 1.211

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 -24 -43], ⟨0 35 62]]

mapping generators: ~2, ~[7 -23 13⟩

Optimal tunings:

  • WE: ~2 = 1200.000236 ¢, ~[7 -23 13⟩ = 877.198814 ¢
error map: ⟨+0.0002 -0.0022 +0.0026]
  • CWE: ~2 = 1200.000000 ¢, ~[7 -23 13⟩ = 877.198647 ¢
error map: ⟨0.0000 -0.0024 -0.0024]

Optimal ET sequence: 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680

Badness (Sintel): 0.498

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 -2 4], ⟨0 47 -22]]

mapping generators: ~2, ~[46 -7 -15⟩

Optimal tunings:

  • WE: ~2 = 1199.998956 ¢, ~[46 -7 -15⟩ = 91.530922 ¢
error map: ⟨-0.0010 +0.0004 +0.0018]
  • CWE: ~2 = 1200.000000 ¢, ~[46 -7 -15⟩ = 91.530995 ¢
error map: ⟨0.0000 +0.0018 +0.0044]

Optimal ET sequence: 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c

Badness (Sintel): 1.086

Music

Septarium

Septarium splits the octave in seven, and tempers out the septarium comma, [73 77 -84⟩. It can be described as 441 & 2513 temperament, which tempers out 78125000/78121827 and [47 -7 -7 -7⟩ (akjaysma) in the 7-limit.

Subgroup: 2.3.5

Comma list: [73 77 -84⟩

Mapping: [⟨7 1 7], ⟨0 12 11]]

mapping generators: ~[21 22 -24⟩, ~[-20 -21 23⟩

Optimal tunings:

  • WE: ~[21 22 -24⟩ = 171.429 ¢, ~[-20 -21 23⟩ = 144.210 ¢
  • CWE: ~[21 22 -24⟩ = 171.429 ¢, ~[-20 -21 23⟩ = 144.210 ¢

Optimal ET sequence: 133, 308, 441, 1631, 2072, 2513, 5467, 7980

Badness (Sintel): 1.395

7-limit

Subgroup: 2.3.5.7

Comma list: 78125000/78121827, 140737488355328/140710042265625

Mapping: [⟨7 1 7 39], ⟨0 12 11 -23]]

Optimal tunings:

  • WE: ~1157625/1048576 = 171.428 ¢, ~2097152/1929375 = 144.211 ¢
  • CWE: ~1157625/1048576 = 171.429 ¢, ~2097152/1929375 = 144.212 ¢

Optimal ET sequence: 133, 308, 441, 1631, 2072, 2513, 2954

Badness (Sintel): 0.777

Raider

See also: Syntonic–chromatic equivalence continuum

Subgroup: 2.3.5

Comma list: [71 -99 37⟩

Mapping: [⟨1 -9 -26], ⟨0 37 99]]

mapping generators: ~2, ~8000/6561

Optimal tunings:

  • WE: ~2 = 1199.999880537 ¢, ~8000/6561 = 343.296063356 ¢
error map: ⟨-0.000119 +0.000418 -0.000336]
  • CWE: ~2 = 1200.000000000 ¢, ~8000/6561 = 343.296095985 ¢
error map: ⟨0.000000 +0.000551 -0.000211]

Optimal ET sequence: 7, …, 381, 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860

Badness (Sintel): 0.302

Pirate

See also: Hemfiness temperaments § Piracy

Subgroup: 2.3.5

Comma list: [-90 -15 49⟩

Mapping: [⟨1 -6 0], ⟨0 49 15]]

mapping generators: ~2, ~[24 4 -13⟩

Optimal tunings:

  • WE: ~2 = 1200.000195604 ¢, ~[24 4 -13⟩ = 185.754209314 ¢
error map: ⟨-0.000196 +0.000418 -0.000336]
  • CWE: ~2 = 1200.000000000 ¢, ~[24 4 -13⟩ = 185.754209314 ¢
error map: ⟨0.000000 +0.000082 -0.000574]

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

Badness (Sintel): 0.133

Counterquectismic

See also: Counterquectisma

Subgroup: 2.3.5

Comma list: [-500 314 1⟩

Mapping: [⟨1 0 500], ⟨0 1 -314]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1200.000184840 ¢, ~3/2 = 701.954524421 ¢
error map: ⟨+0.000185 -0.000292 -0.000002]
  • CWE: ~2 = 1200.000000000 ¢, ~3/2 = 701.954415845 ¢
error map: ⟨0.000000 -0.000585 -0.000289]

Optimal ET sequence: 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486

Badness (Sintel): 11.971

Quectismic

See also: Quectisma

Subgroup: 2.3.5

Comma list: [554 -351 1⟩

Mapping: [⟨1 0 -554], ⟨0 1 351]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.999902567 ¢, ~3/2 = 701.955253373 ¢
error map: ⟨-0.000097 +0.000155 -0.000001]
  • CWE: ~2 = 1200.000000000 ¢, ~3/2 = 701.955310152 ¢
error map: ⟨0.000000 +0.000309 +0.000149]

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

Badness (Sintel): 8.736

Nanoquectismic

Subgroup: 2.3.5.7

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

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

Optimal tunings:

  • WE: ~2 = 1199.979407 ¢, ~3/2 = 701.943437 ¢
  • CWE: ~2 = 1200.000000 ¢, ~3/2 = 701.955457 ¢

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

Badness (Sintel): 15.285

Conanoquectismic

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 tunings:

  • WE: ~2 = 1200.010750 ¢, ~3/2 = 701.961041 ¢
  • CWE: ~2 = 1200.000000 ¢, ~3/2 = 701.954771 ¢

Optimal ET sequence: 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 schismas and a syntonic comma, or twelfth schismas and a Pythagorean comma. It is a member of the schismic–commatic equivalence continuum with n = 12. In this extremely accurate temperament, the perfect fifth is one schisma sharp of its 12edo value.

Subgroup: 2.3.5

Comma list: [161 -84 -12⟩

Mapping: [⟨12 0 161], ⟨0 1 -7]]

mapping generators: ~[-67 35 5⟩, ~3

Optimal tunings:

  • WE: ~[-67 35 5⟩ = 99.999995361 ¢, ~3/2 = 701.955129505 ¢ (~32805/32768 = 1.955161980 ¢)
error map: ⟨-0.000056 +0.000072 +0.000022]
  • CWE: ~[-67 35 5⟩ = 100.000000000 ¢, ~3/2 = 701.955166184 ¢ (~32805/32768 = 1.955166184 ¢)
error map: ⟨0.000000 +0.000165 +0.000123]

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

Badness (Sintel): 0.0892

Revopentic

See also: Revopentisma

Subgroup: 2.3.7

Comma list: [47 4 -19⟩

Subgroup-val mapping: [⟨1 -7 1], ⟨0 19 4]]

mapping generators: ~2, ~16807/12288

Optimal tunings:

  • WE: ~2 = 1199.998821 ¢, ~16807/12288 = 542.207710 ¢
error map: ⟨-0.0012 -0.0003 +0.0038]
  • CWE: ~2 = 1200.000000 ¢, ~16807/12288 = 542.208204 ¢
error map: ⟨0.0000 +0.0009 +0.0069]

Optimal ET sequence: 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, …

Badness (Sintel): 0.0704

Revopent

Revopent can be described as the 1848 & 3125 temperament.

Subgroup: 2.3.5.7

Comma list: 645700815/645657712, [51 -13 -1 -10⟩

Mapping: [⟨1 -7 132 1], ⟨0 19 -287 4]]

Optimal tunings:

  • WE: ~2 = 1200.000110 ¢, ~16807/12288 = 542.208018 ¢
  • CWE: ~2 = 1200.000000 ¢, ~16807/12288 = 542.207968 ¢

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

Badness (Sintel): 0.665

Nanic

See also: Nanisma

Subgroup: 2.3.7

Comma list: [109 -67 -1⟩

Subgroup-val mapping: [⟨1 0 109], ⟨0 1 -67]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1199.999111 ¢, ~3/2 = 701.957264 ¢
error map: ⟨-0.0009 +0.0014 +0.0001]
  • CWE: ~2 = 1200.000000 ¢, ~3/2 = 701.957796 ¢
error map: ⟨0.0000 +0.0028 +0.0018]

Optimal ET sequence: 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733

Badness (Sintel): TBD

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: [-32 -64 71⟩

Subgroup-val mapping: [⟨1 -36 -32], ⟨0 71 64]]

mapping generators: ~3, ~[-4 -9 10⟩

Optimal tunings:

  • WE: ~3 = 1901.955003 ¢, ~[-4 -9 10⟩ = 1003.615406 ¢
error map: ⟨+0.0000 +0.0000 -0.0000]
  • CWE: ~3 = 1901.955001 ¢, ~[-4 -9 10⟩ = 1003.615405 ¢
error map: ⟨0.0000 -0.0000 -0.0000]

Optimal ET sequence: b163cd, b199c, b235, b271, b4643, b4914, …, b8708, b8979, b18229, b27208, b99853, b127061, b154269, b181477, b208685, b390162

Badness (Sintel): TBD

Rank-3 temperaments

Akjaysmic

See also: Akjaysma

Subgroup: 2.3.5.7

Comma list: [47 -7 -7 -7⟩

Mapping: [⟨7 0 0 47], ⟨0 1 0 -1], ⟨0 0 1 -1]]

mapping generators: ~1157625/1048576, ~3, ~5

Optimal tunings:

  • WE: ~1157625/1048576 = 171.427811 ¢, ~3/2 = 701.962313 ¢, ~5/4 = 386.328628 ¢
error map: ⟨-0.0053 +0.0020 +0.0043 +0.0062]
  • CWE: ~1157625/1048576 = 171.428571 ¢, ~3/2 = 701.964859 ¢, ~5/4 = 386.330310 ¢
error map: ⟨0.0000 +0.0099 +0.0166 +0.0218]

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 701.964561 ¢, ~5/4 = 386.329837 ¢
  • CWE: ~29160/26411 = 171.428571 ¢, ~3/2 = 701.967291 ¢, ~5/4 = 386.331624 ¢

Optimal ET sequence: 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: [3 -24 3 10⟩ (282475249000/282429536481)

Mapping: [⟨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: 171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069

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 41 & 49 & 130 temperament, and tempers out the izar comma, [0 -11 -7 12⟩.

Subgroup: 2.3.5.7

Comma list: 13841287201/13839609375

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

mapping generators: ~2, ~3, ~16807/10125

Optimal tunings:

  • TE: ~2 = 1200.0000 ¢, ~3/2 = 701.9584 ¢, ~16807/10125 = 877.2825 ¢
  • WE: ~2 = 1200.0000 ¢, ~3/2 = 701.9584 ¢, ~16807/10125 = 877.2825 ¢

Optimal ET sequence: 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d

Badness (Sintel): 0.595

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 tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 701.9501 ¢, ~250/189 = 484.2956 ¢

Optimal ET sequence: 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 (Sintel): 1.31

Laleruyo (171 & 1547 & 3125)

Subgroup: 2.3.5.7

Comma list: [-1 4 11 -11⟩ (3955078125/3954653486)

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~375/343 = 154.3678 ¢, ~5/4 = 386.3070 ¢

Optimal ET sequence: 171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987

Badness (Sintel): 0.501

Starscape

Subgroup: 2.3.5.7

Comma list: 645700815/645657712

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 701.9514 ¢, ~9/7 = 435.0709 ¢

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 (Sintel): 0.240

Sepbizo-asegu (171 & 3566 & 4973)

Subgroup: 2.3.5.7

Comma list: [-3 2 -17 14⟩ (6104007655641/6103515625000)

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 701.9548 ¢, ~31250/16807 = 1073.8021 ¢

Optimal ET sequence: 171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653

Badness (Sintel): 0.982

Euzenius

See also: Euzenius comma

Subgroup: 2.3.5.7

Comma list: 78125000/78121827

Mapping: [⟨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: 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

Nommismic

See also: Nommisma

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 tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 701.9516 ¢, ~5/4 = 386.3132 ¢

Optimal ET sequence: 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: [1 -15 -18 23⟩

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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 701.9551 ¢, ~56953125/40353607 = 596.5022 ¢

Optimal ET sequence: 171, 863, 1034, 1205, 1376, 1547, 1718, 3255, 3426, 3597, 4460, 4631, 4802, 4973, 5144, 6520, 6691, 11664, 18355, 48374, 48545, 66900, 145464, …

Badness (Sintel): 0.505

Technologismic

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 tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 701.9553 ¢, ~63/32 = 1172.7374 ¢

Optimal ET sequence: 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, …

Badness (Sintel): 0.116

Quinbisa-quinbizo-alegu (1848 & 3737 & 6079)

Subgroup: 2.3.5.7

Comma list: [110 -71 -11 10⟩

Mapping: [⟨1 0 0 -11], ⟨0 1 9 17], ⟨0 0 -10 -11]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~3/2 = 701.955 ¢, ~32805/28672 = 233.128 ¢

Optimal ET sequence: 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, …

Badness (Sintel): 0.538

Septrongismic

See also: Septrongisma

Subgroup: 2.3.5.7

Comma list: [-7 30 -9 -7⟩ = 205891132094649/205885750000000

Mapping: [⟨1 0 0 -1], ⟨0 1 1 3], ⟨0 0 7 -9]]

Optimal tuning (CWE): ~2 = 1200.000 ¢, ~3/2 = 701.954 ¢, ~1715000/1594323 = 126.337 ¢

Optimal ET sequence: 171, 2783, 2954, 3125, 8539, 11664, 14789, …

Badness (Sintel): 0.320

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