Very high accuracy temperaments: Difference between revisions

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


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992
[[Comma list]]: {{monzo| -53 10 16 }}


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


[[POTE generator]]: ~1125/1024 = 162.742626
[[Optimal tuning]]s:
* [[WE]]: ~94921875/67108864 = 600.003398{{c}}, ~1125/1024 = 162.743548{{c}}
: [[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 GPV sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}
{{Optimal ET sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}


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


== Astro ==
== Astro ==
Subgroup: 2.3.5
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


[[Comma list]]: {{monzo| 91 -12 -31 }}
[[Comma list]]: {{monzo| 91 -12 -31 }}


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


[[POTE generator]]: ~296630859375/274877906944 = 132.194668
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.994636{{c}}, ~{{monzo| 39 -5 -13 }} = 1067.800558{{c}}
: [[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 GPV sequence|legend=1| 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}
{{Optimal ET sequence|legend=1| 9, …, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}


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


== Gaster ==
== Gaster ==
{{main| Gaster temperament }}
{{Main| Gaster temperament }}
{{see also| Mirkwai clan #Gaster }}
{{See also| Mirkwai clan #Gaster }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma]]: {{monzo|-70 72 -19}}
[[Comma list]]: {{monzo| -70 72 -19 }}


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


[[POTE generator]]: ~1638400000/1162261467 = 594.634565
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.002817{{c}}, ~{{monzo| -18 19 -5 }} = 605.366856{{c}}
: [[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 GPV sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243 }}
{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}


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


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


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 37 25 -33 }}
[[Comma list]]: {{monzo| 37 25 -33 }}


[[Mapping]]: [{{val| 1 -16 -11 }}, {{val| 0 33 25 }}]
{{Mapping|legend=1| 1 -16 -11 | 0 33 25 }}
: mapping generators: ~2, ~625/432


[[POTE generator]]: ~864/625 = 560.546826
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.997806{{c}}, ~625/432 = 639.452005{{c}}
: [[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 GPV 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 temperament (53&amp;559) tempers out the [[monzisma]], {{Monzo|54 -37 2}}. 7-limit extensions of the monzismic temperament include [[Ragismic microtemperaments #Monzism|monzism]].
Monzismic ({{nowrap| 53 & 612 }}) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}. For its 7-limit extension, see [[Ragismic microtemperaments #Monzismic]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


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


[[POTE generator]]: ~387420489/335544320 = 249.018406
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.997524{{c}}, ~{{monzo| 27 -18 1 }} = 950.979631{{c}}
: [[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 GPV 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 ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


[[Mapping]]: [{{val| 1 15 16 }}, {{val| 0 -51 -52 }}]
{{Mapping|legend=1| 1 -36 -36 | 0 51 52 }}
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.000553{{c}}, ~5/3 = 884.352488{{c}}
: [[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 }}
 
[[Badness]] (Sintel): 0.981
 
=== Hemiegads ===
Subgroup: 2.3.5.7
 
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}


[[POTE generator]]: ~6/5 = 315.647919
{{Mapping|legend=0| 1 -87 -88 127 | 0 102 104 -143 }}
: mapping generators: ~2, ~67108864/36756909


{{Optimal GPV sequence|legend=1| 19, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}
Optimal tunings:
* WE: ~2 = 1200.000308{{c}}, ~67108864/36756909 = 1042.176313{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~67108864/36756909 = 1042.176046{{c}}


[[Badness]]: 0.041819
{{Optimal ET sequence|legend=0| 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]], {{monzo| -4 20 -9 0 0 -4 }}
 
{{Mapping|legend=0| 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 temperament tempers out the fortune comma, {{Monzo|-107 47 14}} (tritrila-sepbiyo). It can be described as 612&amp;2513 temperament, which tempers out 78125000/78121827 (euzenius comma) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.
Fortune tempers out the [[fortune comma]], {{monzo| -107 47 14 }}. It can be described as {{nowrap| 612 & 2513 }} temperament, which tempers out 78125000/78121827 ([[euzenius comma]]) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -107 47 14 }}
[[Comma list]]: {{monzo| -107 47 14 }}


[[Mapping]]: [{{val| 1 -1 11 }}, {{val| 0 14 -47 }}]
{{Mapping|legend=1| 1 -1 11 | 0 14 -47 }}
: mapping generators: ~2, ~8388608/7381125


[[POTE generator]]: ~8388608/7381125 = 221.567899
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.000553{{c}}, ~8388608/7381125 = 221.568202{{c}}
: [[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 GPV sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151 }}
{{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 109: Line 178:
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}


Mapping: [{{val| 1 -1 11 63 }}, {{val| 0 14 -47 -326 }}]
{{Mapping|legend=0| 1 -1 11 63 | 0 14 -47 -326 }}


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


Optimal GPV sequence: {{Val list| 612, 1901, 2513, 3125, 6862, 9987 }}
{{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 122: Line 193:
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752


Mapping: [{{val| 1 -1 11 63 134 }}, {{val| 0 14 -47 -326 -707 }}]
{{Mapping|legend=0| 1 -1 11 63 134 | 0 14 -47 -326 -707 }}


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


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


Badness: 0.036641
Badness (Sintel): 1.211


== Senior ==
== Senior ==
Senior temperament (171&amp;1171) tempers out the senior comma, {{Monzo|-17 62 -35}} (quadla-sepquingu). 7-limit extensions of the senior temperament include [[Ragismic microtemperaments #Seniority|seniority]].
Senior ({{nowrap| 171 & 1171 }}) tempers out the [[senior comma]], {{monzo| -17 62 -35 }}. 7-limit extensions of the senior temperament include [[Ragismic microtemperaments #Seniority|seniority]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| -17 62 -35 }}
[[Comma list]]: {{monzo| -17 62 -35 }}


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


[[POTE generator]]: ~94143178827/78125000000 = 322.801358
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.000236{{c}}, ~{{monzo| 7 -23 13 }} = 877.198814{{c}}
: [[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 GPV sequence|legend=1| 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
{{Optimal ET sequence|legend=1| 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}


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


== Gross ==
== Gross ==
Gross temperament (118&amp;1783) tempers out the gross comma, {{Monzo|144 -22 -47}}, by which a stack of 22 [[tritave]]s (third harmonic) and 47 [[5/1|pentave]]s (fifth harmonic) falls short of 144 (gross) octaves.
Gross ({{nowrap| 118 & 1783 }}) tempers out the gross comma, {{monzo| 144 -22 -47 }}, by which a stack of 22 [[3/1|3rd harmonics]] and 47 [[5/1|5th harmonics]] falls short of 144 (gross) octaves.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 144 -22 -47 }}
[[Comma list]]: {{monzo| 144 -22 -47 }}


[[Mapping]]: [{{val| 1 -2 4 }}, {{val| 0 47 -22 }}]
{{Mapping|legend=1| 1 -2 4 | 0 47 -22 }}
: mapping generators: ~2, ~{{monzo| 46 -7 -15 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.998956{{c}}, ~{{monzo| 46 -7 -15 }} = 91.530922{{c}}
: [[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| 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
* ''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
 
== Septarium ==
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
 
[[Comma list]]: {{monzo| 73 77 -84 }}
 
{{Mapping|legend=1| 7 1 7 | 0 12 11 }}
: mapping generators: ~{{monzo| 21 22 -24 }}, ~{{monzo| -20 -21 23 }}


[[POTE generator]]: ~1953125000000000/1853020188851841 = 91.531001
[[Optimal tuning]]s:  
* [[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 GPV sequence|legend=1| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467 }}
{{Optimal ET sequence|legend=1| 133, 308, 441, 1631, 2072, 2513, 5467, 7980 }}


[[Badness]]: 0.046315
[[Badness]] (Sintel): 1.395
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
Comma list: 78125000/78121827, 140737488355328/140710042265625
 
Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}
 
Optimal tunings:
* 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 ==
== Raider ==
{{See also| Syntonic-chromatic equivalence continuum }}
{{See also| Syntonic–chromatic equivalence continuum }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


[[Mapping]]: [{{val| 1 -9 -26 }}, {{val| 0 37 99 }}]
{{Mapping|legend=1| 1 -9 -26 | 0 37 99 }}
: mapping generators: ~2, ~8000/6561


[[POTE generator]]: ~8000/6561 = 343.296098
[[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 GPV sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355 }}
{{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 ==
Subgroup: 2.3.5
{{See also| Hemfiness temperaments #Piracy }}
 
[[Subgroup]]: 2.3.5


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


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


[[POTE generator]]: ~1358954496/1220703125 = 185.754179
[[Optimal tuning]]s:  
* [[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 GPV sequence|legend=1| 730, 1783, 2513, 4296, 6079, 6809, 10375, 11105 }}
{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}


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


== Atomic ==
== Atomic ==
{{see also | Kirnberger's atom }}
: ''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.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


[[Mapping]]: [{{val| 12 0 161 }}, {{val| 0 1 -7 }}]
{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}
: mapping generators: ~{{monzo| -67 35 5 }}, ~3
 
[[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 }}
 
Subgroup: 2.3.7


[[POTE generator]]: ~32805/32768 = 1.955162
Comma list: {{monzo| 47 4 -19 }}


{{Optimal GPV sequence|legend=1| 12, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
{{Mapping|legend=3| 1 -7 1 | 0 19 4 }}
: mapping generators: ~2, ~16807/12288


[[Badness]]: 0.003803
[[Optimal tuning]]s:
* [[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 }}
 
{{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): 0.0704
 
=== Revopent ===
Revopent can be described as the {{nowrap| 1848 & 3125 }} temperament.


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


Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}
 
{{Mapping|legend=0| 1 -7 132 1 | 0 19 -287 4 }}
 
Optimal tunings:
* WE: ~2 = 1200.000110{{c}}, ~16807/12288 = 542.208018{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~16807/12288 = 542.207968{{c}}
 
{{Optimal ET sequence|legend=0| 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]]: {{monzo| 109 -67 -1 }}
 
{{Mapping|legend=3| 1 0 109 | 0 1 -67 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[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 }}
 
[[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]]: {{monzo| -32 -64 71 }}
 
{{Mapping|legend=3| 1 -36 -32 | 0 71 64 }}
: mapping generators: ~3, ~{{monzo| -4 -9 10 }}
 
[[Optimal tuning]]s:
* [[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]]
 
[[Badness]] (Sintel): TBD
 
= Rank-3 temperaments =
== Akjaysmic ==
{{See also| Akjaysma }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 47 -7 -7 -7 }}


Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]
{{Mapping|legend=1| 7 0 0 47 | 0 1 0 -1 | 0 0 1 -1 }}
: mapping generators: ~1157625/1048576, ~3, ~5


POTE generator: ~32805/32768 = 1.949608
[[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 GPV sequence: {{Val list| 12, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
{{Optimal ET sequence|legend=1| 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}


Badness: 0.045818
[[Badness]] (Sintel): 2.22


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


Comma list: 9801/9800, 151263/151250, 184549376/184528125
Comma list: 184549376/184528125, 199297406/199290375
 
{{Mapping|legend=0| 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 }}
 
[[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 }}
 
[[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 }}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 13841287201/13839609375
 
{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
: mapping generators: ~2, ~3, ~16807/10125
 
[[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 }}
 
[[Badness]] (Sintel): 0.595
 
== Laquinzo-aquadquadgu (171 & 441 & 4973) ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]
[[Comma list]]: {{monzo| -7 19 -16 5 }} (19534128475869/19531250000000)


POTE generator: ~32805/32768 = 1.947988
{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}


Optimal GPV sequence: {{Val list| 12, 600e, 612, 1236, 1848 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9501{{c}}, ~250/189 = 484.2956{{c}}


Badness: 0.016029
{{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 }}


= Rank-3 temperaments =
[[Badness]] (Sintel): 1.31
== Sasaquinbizo-atriyo (171 &amp; 1106 &amp; 3566) ==
 
Subgroup: 2.3.5.7
== Laleruyo (171 & 1547 & 3125) ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -1 4 11 -11 }} (3955078125/3954653486)
 
{{Mapping|legend=1| 1 3 0 1 | 0 -11 0 -4 | 0 0 1 1 }}
 
[[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 }}
 
[[Badness]] (Sintel): 0.501
 
== Starscape ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 645700815/645657712
 
{{Mapping|legend=1| 1 0 4 0 | 0 1 1 2 | 0 0 -9 -1 }}
 
[[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 }}
 
[[Badness]] (Sintel): 0.240
 
== Sepbizo-asegu (171 & 3566 & 4973) ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -3 2 -17 14 }} (6104007655641/6103515625000)
 
{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 -14 -17 }}
 
[[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 }}
 
[[Badness]] (Sintel): 0.982
 
== Euzenius ==
{{See also| Euzenius comma }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 78125000/78121827
 
{{Mapping|legend=1| 1 1 0 -5 | 0 2 0 -13 | 0 0 1 5 }}
 
[[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 }}
 
[[Badness]] (Sintel): 0.0902


Comma list: 282475249000/282429536481
== Nommismic ==
{{See also| Nommisma }}


Mapping: [{{val| 1 0 9 -3 }}, {{val| 0 1 8 0 }}, {{val| 0 0 10 -3 }}]
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 171, 935, 1106, 1277, 2289, 2460, 3395, 3566, 3737}}
[[Comma list]]: {{monzo| -55 30 2 1 }}


Badness: 0.000281
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}


== Saquadtrizo-asepgu (171 &amp; 2019 &amp; 3566) ==
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9516{{c}}, ~5/4 = 386.3132{{c}}
Subgroup: 2.3.5.7


Comma list: 13841287201/13839609375
{{Optimal ET sequence|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, }} …


Mapping: [{{val| 1 0 0 0 }}, {{val| 0 1 7 5 }}, {{val| 0 0 12 7 }}]
[[Badness]] (Sintel): 1.62


Vals: {{Val list| 171, 1547, 1677, 1718, 1848, 2019, 3395, 3566, 3737}}
== Microorganic ==
[[Subgroup]]: 2.3.7.11


Badness: 0.000135
An insanely accurate [[gary]] or no-5 [[superkleismic]] detemperament. It does not have good strong extensions to the 11-limit and beyond. It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]].


== Laquinzo-aquadquadgu (171 &amp; 441 &amp; 4973) ==
[[Comma list]]: [[67110351/67108864]]
Subgroup: 2.3.5.7


Comma list: {{monzo| -7 19 -16 5 }} = 19534128475869/19531250000000
{{Mapping|legend=1|1 0 7 7|0 1 1 -2| 0 0 3 -5}}


Mapping: [{{val| 1 0 3 11 }}, {{val| 0 1 4 9 }}, {{val| 0 0 5 16 }}]
[[Optimal tuning|Optimal tunings]]:


Vals: {{Val list| 171, 270, 441, 612, 783, 4802, 4973, 5144}}
* [[WE]]: ~2 = 1200.001014{{c}}, ~3/2 = 1901.953889{{c}}, ~539/512 = 88.956151{{c}}


Badness: 0.000297
: [[error map]]: ⟨+0.001014  -0.000098  -0.001537  -0.001401]


== Laleruyo (171 &amp; 1547 &amp; 3125) ==
* [[CWE]]: ~2 = 1200.000000 ¢, ~3/2 =  701.953627¢, ~539/512 =  88.955923 ¢
Subgroup: 2.3.5.7


Comma list: 3955078125/3954653486
: error map: ⟨0.000 +0.001374 -0.004512  -0.004808]


Mapping: [{{val| 1 3 0 1 }}, {{val| 0 11 0 4 }}, {{val| 0 0 1 1 }}]
{{Optimal ET sequence|legend=1|11, 12e, 26, 41, 94, 135, 566, 701, 836, 877, 1012, 1713, 1848, 4573, 4708, 6556, 11264, 17820}} …


Vals: {{Val list| 171, 1407, 1547, 1578, 1718, 2783, 2954, 3125, 3296}}
[[Badness]] (Sintel): 0.032


Badness: 0.000114
Notable temperaments not appearing in the optimal ET sequence are {{Edos|15, 217, 311}}.


== Starscape or Latritriru-ayo ==
=== Microbiology ===
Subgroup: 2.3.5.7
A weak extension of microorganic adding the [[chalmersia]] and the [[insanisma]], which surfaces as a detemperament of [[vulture]].


Comma list: 645700815/645657712
[[Subgroup]]: 2.3.5.7.11.13


Mapping: [{{val| 1 0 4 0 }}, {{val| 0 1 1 2 }}, {{val| 0 0 9 1 }}]
[[Comma list]]: [[67110351/67108864]], [[123201/123200]], [[5767168/5767125]]


Vals: {{Val list| 171, 1848, 2019, 2783, 2954, 3125, 3296, 4973, 5144}}
{{Mapping|legend=1|1 0 4 -11 27 15|0 2 3 -1 1 -3| 0 0 5 18 -30 11}}
[[Optimal tuning|Optimal tunings]]:


Badness: 0.000545
* [[WE]]: ~2 =  1200.000535{{c}}, ~359424/207515 =  950.976364{{c}}, ~2695/1536 =  973.322574{{c}}


== Sepbizo-asegu (171 &amp; 3566 &amp; 4973) ==
: [[error map]]: ⟨+0.000535 −0.002272 +0.004652 −0.001831 −0.004338 +0.002963]
Subgroup: 2.3.5.7


Comma list: 6104007655641/6103515625000
* [[CWE]]: ~2 = 1200.000000 ¢, ~3/2 =  950.975971 ¢, ~539/512 =  973.322136  ¢


Mapping: [{{val| 1 0 13 16 }}, {{val| 0 1 10 12 }}, {{val| 0 0 14 17 }}]
: error map: ⟨0.000000 −0.003059 +0.003520 −0.003432 −0.006047 +0.000931]


Vals: {{Val list| 171, 1236, 1407, 3566, 3737, 3908, 4802, 4973, 8539, 8710}}
{{Optimal ET sequence|legend=1|53, 217, 270, 1551f, 1604, 1821, 1874, 2091, 2144, 2361, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808}} …


Badness: 0.000223
Badness (Sintel): 0.349


== Euzenius or Sabiruquinyo ==
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Comma list: 78125000/78121827
[[Comma list]]: {{monzo| 1 -15 -18 23 }}


Mapping: [{{val| 1 0 0 -5 }}, {{val| 0 2 0 -13 }}, {{val| 0 0 1 5 }}]
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}


Vals: {{Val list| 171, 441, 612, 2954, 3125, 3566, 3737, 6520, 6691}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9551{{c}}, ~56953125/40353607 = 596.5022{{c}}


Badness: 0.000205
{{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, }} …


== Quinla-zoyoyo (612 &amp; 1848 &amp; 3125) ==
[[Badness]] (Sintel): 0.505
Subgroup: 2.3.5.7


Comma list: {{monzo| -55 30 2 1 }}
== Technologismic ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 0 55 }}, {{val| 0 1 0 -30 }}, {{val| 0 0 1 -2 }}]
[[Comma list]]: {{monzo| 51 -13 -1 -10 }}


Vals: {{Val list| 612, 665, 1236, 1277, 1848, 1889, 2460, 2513, 3125, 3737, 4973, 5585, 6862, 8098, 8710, 11835, 14960}}
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}


Badness: 0.000366
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9553{{c}}, ~63/32 = 1172.7374{{c}}


== Sa-twentythezo-atritribigu (171 &amp; 1547 &amp; 4973) ==
{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …
Subgroup: 2.3.5.7


Comma list: {{monzo| 1 -15 -18 23 }}
[[Badness]] (Sintel): 0.116


Mapping: [{{val| 1 0 9 7 }}, {{val| 0 1 3 3 }}, {{val| 0 0 23 18 }}]
== Quinbisa-quinbizo-alegu (1848 & 3737 & 6079) ==
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 171, 1547, 4802, 4973, 5144, 6520, 6691, 6862, 11493, 11664, 11835, 18184, 18355, 18526, 25046, 30019, 30190, 48545}}
[[Comma list]]: {{monzo| 110 -71 -11 10 }}


Badness: 0.000114
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}


== Trisa-quinbiru-agu (441 &amp; 1848 &amp; 3125) ==
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~32805/28672 = 233.128{{c}}
Subgroup: 2.3.5.7


Comma list: {{monzo| 51 -13 -1 -10 }}
{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …


Mapping: [{{val| 1 0 1 5 }}, {{val| 0 1 7 -2 }}, {{val| 0 0 10 -1 }}]
[[Badness]] (Sintel): 0.538


Vals: {{Val list| 441, 1848, 3125, 3566, 4973, 6691, 8098, 8539, 9816, 11664, 14789, 15230, 18355, 21480, 30019, 33144, 39835, 51499}}
== Septrongismic ==
{{See also| Septrongisma }}
[[Subgroup]]: 2.3.5.7


Badness: 0.000263
[[Comma list]]: {{monzo| -7 30 -9 -7 }} = 205891132094649/205885750000000


== Quinbisa-quinbizo-alegu (21480 &amp; 33144 &amp; 103169) ==
{{Mapping|legend=1| 1 0 0 -1 | 0 1 1 3 | 0 0 7 -9 }}
Subgroup: 2.3.5.7


Comma list: {{monzo| 110 -71 -11 10 }}
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.954{{c}}, ~1715000/1594323 = 126.337{{c}}


Mapping: [{{val| 1 0 0 -11 }}, {{val| 0 1 -1 6 }}, {{val| 0 0 10 11 }}]
{{Optimal ET sequence|legend=1| 171, 2783, 2954, 3125, 8539, 11664, 14789, }} …


Vals: {{Val list| 11664, 21480, 33144, 36881, 48545, 54624, 70025, 81689, 91505, 103169}}
[[Badness]] (Sintel): 0.320


[[Category:Temperament collection]]
[[Category:Temperament collections]]
{{Todo| add definition | intro }}

Latest revision as of 16:13, 20 October 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

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

Microorganic

Subgroup: 2.3.7.11

An insanely accurate gary or no-5 superkleismic detemperament. It does not have good strong extensions to the 11-limit and beyond. It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered olympia~argyria.

Comma list: 67110351/67108864

Mapping: [⟨1 0 7 7], ⟨0 1 1 -2], ⟨0 0 3 -5]]

Optimal tunings:

  • WE: ~2 = 1200.001014 ¢, ~3/2 = 1901.953889 ¢, ~539/512 = 88.956151 ¢
error map: ⟨+0.001014 -0.000098 -0.001537 -0.001401]
  • CWE: ~2 = 1200.000000 ¢, ~3/2 = 701.953627¢, ~539/512 = 88.955923 ¢
error map: ⟨0.000 +0.001374 -0.004512 -0.004808]

Optimal ET sequence: 11, 12e, 26, 41, 94, 135, 566, 701, 836, 877, 1012, 1713, 1848, 4573, 4708, 6556, 11264, 17820 …

Badness (Sintel): 0.032

Notable temperaments not appearing in the optimal ET sequence are 15, 217, 311.

Microbiology

A weak extension of microorganic adding the chalmersia and the insanisma, which surfaces as a detemperament of vulture.

Subgroup: 2.3.5.7.11.13

Comma list: 67110351/67108864, 123201/123200, 5767168/5767125

Mapping: [⟨1 0 4 -11 27 15], ⟨0 2 3 -1 1 -3], ⟨0 0 5 18 -30 11]] Optimal tunings:

  • WE: ~2 = 1200.000535 ¢, ~359424/207515 = 950.976364 ¢, ~2695/1536 = 973.322574 ¢
error map: ⟨+0.000535 −0.002272 +0.004652 −0.001831 −0.004338 +0.002963]
  • CWE: ~2 = 1200.000000 ¢, ~3/2 = 950.975971 ¢, ~539/512 = 973.322136 ¢
error map: ⟨0.000000 −0.003059 +0.003520 −0.003432 −0.006047 +0.000931]

Optimal ET sequence: 53, 217, 270, 1551f, 1604, 1821, 1874, 2091, 2144, 2361, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808 …

Badness (Sintel): 0.349

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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