Very high accuracy temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
TallKite (talk | contribs)
autopatrolled, Sysops
4,197 edits
→sa-23zo-atritribigu (171 & 1547 & 4973): expanded the color name to the spoken form
Hkm (talk | contribs)
autopatrolled
1,058 edits
No edit summary
 
(109 intermediate revisions by 19 users not shown)
Line 1: Line 1:
Below are listed some very high accuracy temperaments ([[Tenney-Euclidean temperament measures|TE error]] < 0.005 cents/octave).
{{Technical data page}}
Below are listed some very high accuracy temperaments (Breed's [[Tenney-Euclidean temperament measures|TE error]] < 0.005 cents/octave).


=Rank two temperaments=
= Rank-2 temperaments =
== 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==
[[Subgroup]]: 2.3.5
Comma: |-53 10 16&gt; = 9010162353515625 / 9007199254740992


POTE generator: ~1125/1024 = 162.743
[[Comma list]]: {{monzo| -53 10 16 }}


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


EDOs: [[118edo|118]], [[494edo|494]], [[612edo|612]], [[730edo|730]], [[848edo|848]], [[1342edo|1342]], [[1954edo|1954]]
[[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 }}


Badness: 0.0141
{{Optimal ET sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}


==Astro==
[[Badness]] (Sintel): 0.331
Comma: |91 -12 -31&gt;


POTE generator: ~296630859375/274877906944 = 132.1947
== Astro ==
Astro temperament tempers out the [[astro comma]], {{monzo| 91 -12 -31 }}. 7-limit extensions of the astro temperament include [[Horwell temperaments #Kastro|kastro]].


Map: [&lt;1 5 1|, &lt;0 31 -12|]
[[Subgroup]]: 2.3.5


EDOs: [[118edo|118]], [[935edo|935]], [[1053edo|1053]], [[1171edo|1171]], [[1289edo|1289]], [[1407edo|1407]], [[2224edo|2224]], [[2460edo|2460]]
[[Comma list]]: {{monzo| 91 -12 -31 }}


Badness: 0.0628
{{Mapping|legend=1| 1 -26 13 | 0 31 -12 }}
: mapping generators: ~2, ~{{monzo| 39 -5 -13 }}


==Whoosh==
[[Optimal tuning]]s:
Comma: |37 25 -33&gt;
* [[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 }}


POTE generator: ~864/625 = 560.5468
{{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 }}


Map: [&lt;1 -16 -11|, &lt;0 33 25|]
[[Badness]] (Sintel): 1.473


EDOs: [[289edo|289]], [[441edo|441]], [[730edo|730]], [[1171edo|1171]], [[1612edo|1612]], [[1901edo|1901]], [[2631edo|2631]]
== Gaster ==
{{Main| Gaster temperament }}
{{See also| Mirkwai clan #Gaster }}


Badness: 0.0257
[[Subgroup]]: 2.3.5


==Monzismic==
[[Comma list]]: {{monzo| -70 72 -19 }}
Comma: |54 -37 2&gt; = 450359962737049600 / 450283905890997363


POTE generator: ~387420489/335544320 = 249.0184
{{Mapping|legend=1| 1 -8 -34 | 0 19 72 }}
: mapping generators: ~2, ~{{monzo| -18 19 -5 }}


Map: [&lt;1 0 -27|, &lt;0 2 37|]
[[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 }}


EDOs: [[53edo|53]], [[506edo|506]], [[559edo|559]], [[612edo|612]], [[1783edo|1783]], [[2954edo|2954]], [[6520edo|6520]]
{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}


Badness: 0.0104
[[Badness]] (Sintel): 2.077


==Egads==
== Whoosh ==
Comma: |-36 -52 51&gt;
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]].


POTE generator: ~6/5 = 315.6479
[[Subgroup]]: 2.3.5


Map: [&lt;1 15 16|, &lt;0 -51 -52|]
[[Comma list]]: {{monzo| 37 25 -33 }}


EDOs: [[441edo|441]], [[901edo|901]], [[1342edo|1342]], [[1783edo|1783]], [[2224edo|2224]], [[3125edo|3125]], [[4467edo|4467]], [[4908edo|4908]]
{{Mapping|legend=1| 1 -16 -11 | 0 33 25 }}
: mapping generators: ~2, ~625/432


Badness: 0.0418
[[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 }}


==Fortune==
{{Optimal ET sequence|legend=1| 15, …, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}
Comma: |-107 47 14&gt;


POTE generator: ~8388608/7381125 = 221.5679
[[Badness]] (Sintel): 0.602


Map: [&lt;1 -1 11|, &lt;0 14 -47|]
== Monzismic ==
Monzismic ({{nowrap| 53 & 612 }}) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}. For its 7-limit extension, see [[Ragismic microtemperaments #Monzismic]].


EDOs: [[612edo|612]], [[1289edo|1289]], [[1901edo|1901]], [[2513edo|2513]], [[3125edo|3125]], [[3737edo|3737]], [[4414edo|4414]], [[5638edo|5638]]
[[Subgroup]]: 2.3.5


Badness: 0.0279
[[Comma list]]: {{monzo| 54 -37 2 }}


===7-limit===
{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}
Commas: 78125000/78121827, |-52 17 12 -1&gt;
: mapping generators: ~2, ~{{monzo| 27 -18 1 }}


POTE generator: ~8388608/7381125 = 221.5680
[[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 }}


Map: [&lt;1 -1 11 63|, &lt;0 14 -47 -326|]
{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}


EDOs: 612, 2513, 3125, 3737, 4349, 5638, 6862, 9987
[[Badness]] (Sintel): 0.244


Badness: 0.0237
== Egads ==
[[Subgroup]]: 2.3.5


===11-limit===
[[Comma list]]: {{monzo| -36 -52 51 }}
Commas: 151263/151250, 21437500/21434787, 369140625/369098752


POTE generator: ~8388608/7381125 = 221.5681
{{Mapping|legend=1| 1 -36 -36 | 0 51 52 }}
: mapping generators: ~2, ~5/3


Map: [&lt;1 -1 11 63 134|, &lt;0 14 -47 -326 -707|]
[[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 }}


EDOs: 612, 2513, 3125, 3737, 4349, 4961, 6862, 8086
{{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: 0.0366
[[Badness]] (Sintel): 0.981


==Senior==
=== Hemiegads ===
Comma: |-17 62 -35&gt;
Subgroup: 2.3.5.7


POTE generator: ~94143178827/78125000000 = 322.8014
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}


Map: [&lt;1 11 19|, &lt;0 -35 -62|]
Mapping: {{mapping| 1 -87 -88 127 | 0 102 104 -143 }}
: mapping generators: ~2, ~67108864/36756909


EDOs: [[171edo|171]], [[1171edo|1171]], [[1342edo|1342]], [[2513edo|2513]], [[3684edo|3684]], [[3855edo|3855]], [[4855edo|4855]], [[6197edo|6197]]
Optimal tunings:  
* WE: ~2 = 1200.000308{{c}}, ~67108864/36756909 = 1042.176313{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~67108864/36756909 = 1042.176046{{c}}


Badness: 0.0212
{{Optimal ET sequence|legend=0| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}


==Gross==
Badness (Sintel): 0.313
Comma: |144 -22 -47&gt;


POTE generator: ~1953125000000000/1853020188851841 = 91.5310
=== Catabolic ===
Subgroup: 2.3.5.13


Map: [&lt;1 -2 4|, &lt;0 47 -22|]
Comma list: [[140625/140608]], {{monzo| -4 20 -9 0 0 -4 }}


EDOs: [[118edo|118]], [[1783edo|1783]], [[1901edo|1901]], [[3684edo|3684]], [[5467edo|5467]], [[5585edo|5585]], [[7250edo|7250]]
Mapping: {{mapping| 1 -36 -36 -98 | 0 51 52 138 }}


Badness: 0.0463
Optimal tunings:  
* WE: ~2 = 1200.000308{{c}}, ~5/3 = 884.354064{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~5/3 = 884.351835{{c}}


==Raider==
{{Optimal ET sequence|legend=0| 19, …, 441, 460, 901, 1342 }}
Comma: |71 -99 37&gt;


POTE generator: ~8000/6561 = 343.2961
Badness (Sintel): 0.222


Map: [&lt;1 -9 -26|, &lt;0 37 99|]
== Fortune ==
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.


EDOs: [[1171edo|1171]], [[1954edo|1954]], [[3125edo|3125]], [[4296edo|4296]], [[5467edo|5467]], [[7421edo|7421]], [[9763edo|9763]], [[14059edo|14059]], [[18355edo|18355]]
[[Subgroup]]: 2.3.5


Badness: 0.0129
[[Comma list]]: {{monzo| -107 47 14 }}


==Pirate==
{{Mapping|legend=1| 1 -1 11 | 0 14 -47 }}
Comma: |-90 -15 49&gt;
: mapping generators: ~2, ~8388608/7381125


POTE generator: ~1358954496/1220703125 = 185.7542
[[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 }}


Map: [&lt;1 -6 0|, &lt;0 49 15|]
{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}


EDOs: [[730edo|730]], [[1783edo|1783]], [[2513edo|2513]], [[4296edo|4296]], [[6079edo|6079]], [[6809edo|6809]], [[10375edo|10375]], [[11105edo|11105]]
[[Badness]] (Sintel): 0.654


Badness: 0.0057
=== 7-limit ===
Subgroup: 2.3.5.7


==Atomic==
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}
Comma: |161 -84 -12&gt;


POTE generator: ~32805/32768 = 1.95516
Mapping: {{mapping| 1 -1 11 63 | 0 14 -47 -326 }}


Map: [&lt;12 0 161|, &lt;0 1 -7|]
Optimal tunings:  
* WE: ~2 = 1200.001970{{c}}, ~8388608/7381125 = 221.568395{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568026{{c}}


EDOs: [[12edo|12]], [[612edo|612]], [[624edo|624]], [[1236edo|1236]], [[1848edo|1848]], [[2460edo|2460]], [[3072edo|3072]], [[3084edo|3084]], [[3684edo|3684]], [[4296edo|4296]], [[4308edo|4308]], [[4908edo|4908]], [[7980edo|7980]], [[12276edo|12276]], [[16572edo|16572]], [[20868edo|20868]], [[25164edo|25164]], [[29460edo|29460]], [[33756edo|33756]], [[46032edo|46032]]
{{Optimal ET sequence|legend=0| 65d, …, 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}


Badness: 0.0038
Badness (Sintel): 0.600


===7-limit===
=== 11-limit ===
Commas: 250047/250000, |-55 30 2 1&gt;
Subgroup: 2.3.5.7.11


POTE generator: ~32805/32768 = 1.94961
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752


Map: [&lt;12 0 161 338|, &lt;0 1 -7 -16|]
Mapping: {{mapping| 1 -1 11 63 134 | 0 14 -47 -326 -707 }}


EDOs: 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 4308
Optimal tunings:  
* WE: ~2 = 1200.003854{{c}}, ~8388608/7381125 = 221.568850{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~8388608/7381125 = 221.568133{{c}}


Badness: 0.0458
{{Optimal ET sequence|legend=0| 65dee, …, 612, 2513, 3125, 3737, 8086, 19909d }}


===11-limit===
Badness (Sintel): 1.211
Commas: 9801/9800, 151263/151250, 184549376/184528125


POTE generator: ~32805/32768 = 1.94799
== Senior ==
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]].


Map: [&lt;12 0 161 338 517|, &lt;0 1 -7 -16 -25|]
[[Subgroup]]: 2.3.5


EDOs: 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 4308
[[Comma list]]: {{monzo| -17 62 -35 }}


Badness: 0.0160
{{Mapping|legend=1| 1 -24 -43 | 0 35 62 }}
: mapping generators: ~2, ~{{monzo| 7 -23 13 }}


[[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 }}


=Rank three temperaments=
{{Optimal ET sequence|legend=1| 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
==sasaquinbizo-atriyo (171 &amp; 1106 &amp; 3566)==
Comma: 282475249000/282429536481


Map: [&lt;1 0 9 -3|, &lt;0 1 8 0|, &lt;0 0 10 -3|]
[[Badness]] (Sintel): 0.498


EDOs: 171, 935, 1106, 1277, 2289, 2460, 3395, 3566, 3737
== Gross ==
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.


Badness: 0.000281
[[Subgroup]]: 2.3.5


==saquadtrizo-asepgu (171 &amp; 2019 &amp; 3566)==
[[Comma list]]: {{monzo| 144 -22 -47 }}
Comma: 13841287201/13839609375


Map: [&lt;1 0 0 0|, &lt;0 1 7 5|, &lt;0 0 12 7|]
{{Mapping|legend=1| 1 -2 4 | 0 47 -22 }}
: mapping generators: ~2, ~{{monzo| 46 -7 -15 }}


EDOs: 171, 1547, 1677, 1718, 1848, 2019, 3395, 3566, 3737
[[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 }}


Badness: 0.000135
{{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 }}


==laquinzo-aquadquadgu (171 &amp; 441 &amp; 4973)==
[[Badness]] (Sintel): 1.086
Comma: |-7 19 -16 5&gt; = 19534128475869/19531250000000


Map: [&lt;1 0 3 11|, &lt;0 1 4 9|, &lt;0 0 5 16|]
; 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


EDOs: 171, 270, 441, 612, 783, 4802, 4973, 5144
== 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.


Badness: 0.000297
[[Subgroup]]: 2.3.5


==laleruyo (171 &amp; 1547 &amp; 3125)==
[[Comma list]]: {{monzo| 73 77 -84 }}
Comma: 3955078125/3954653486


Map: [&lt;1 3 0 1|, &lt;0 11 0 4|, &lt;0 0 1 1|]
{{Mapping|legend=1| 7 1 7 | 0 12 11 }}
: mapping generators: ~{{monzo| 21 22 -24 }}, ~{{monzo| -20 -21 23 }}


EDOs: 171, 1407, 1547, 1578, 1718, 2783, 2954, 3125, 3296
[[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}}


Badness: 0.000114
{{Optimal ET sequence|legend=1| 133, 308, 441, 1631, 2072, 2513, 5467, 7980 }}


==Starscape==
[[Badness]] (Sintel): 1.395
Comma: 645700815/645657712


Map: [&lt;1 0 4 0|, &lt;0 1 1 2|, &lt;0 0 9 1|]
=== 7-limit ===
Subgroup: 2.3.5.7


EDOs: 171, 1848, 2019, 2783, 2954, 3125, 3296, 4973, 5144
Comma list: 78125000/78121827, 140737488355328/140710042265625


Badness: 0.000545
Mapping: {{Mapping| 7 1 7 39 | 0 12 11 -23 }}


==sepbizo-segu (171 &amp; 3566 &amp; 4973)==
Optimal tunings:
Comma: 6104007655641/6103515625000
* WE: ~1157625/1048576 = 171.428{{c}}, ~2097152/1929375 = 144.211{{c}}
* CWE: ~1157625/1048576 = 171.429{{c}}, ~2097152/1929375 = 144.212{{c}}


Map: [&lt;1 0 13 16|, &lt;0 1 10 12|, &lt;0 0 14 17|]
{{Optimal ET sequence|legend=0| 133, 308, 441, 1631, 2072, 2513, 2954 }}


EDOs: 171, 1236, 1407, 3566, 3737, 3908, 4802, 4973, 8539, 8710
Badness (Sintel): 0.777


Badness: 0.000223
== Raider ==
{{See also| Syntonic–chromatic equivalence continuum }}


==Euzenius==
[[Subgroup]]: 2.3.5
Comma: 78125000/78121827


Map: [&lt;1 0 0 -5|, &lt;0 2 0 -13|, &lt;0 0 1 5|]
[[Comma list]]: {{monzo| 71 -99 37 }}


EDOs: 171, 441, 612, 2954, 3125, 3566, 3737, 6520, 6691
{{Mapping|legend=1| 1 -9 -26 | 0 37 99 }}
: mapping generators: ~2, ~8000/6561


Badness: 0.000205
[[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 }}


==quinla-zoyoyo (612 &amp; 1848 &amp; 3125)==
{{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 }}
Comma: |-55 30 2 1&gt;


Map: [&lt;1 0 0 55|, &lt;0 1 0 -30|, &lt;0 0 1 -2|]
[[Badness]] (Sintel): 0.302


EDOs: 612, 665, 1236, 1277, 1848, 1889, 2460, 2513, 3125, 3737, 4973, 5585, 6862, 8098, 8710, 11835, 14960
== Pirate ==
{{See also| Hemfiness temperaments #Piracy }}


Badness: 0.000366
[[Subgroup]]: 2.3.5


==sa-twenty-thezo-atritribigu (171 &amp; 1547 &amp; 4973)==
[[Comma list]]: {{monzo| -90 -15 49 }}
Comma: |1 -15 -18 23&gt;


Map: [&lt;1 0 9 7|, &lt;0 1 3 3|, &lt;0 0 23 18|]
{{Mapping|legend=1| 1 -6 0 | 0 49 15 }}
: mapping generators: ~2, ~{{monzo| 24 4 -13 }}


EDOs: 171, 1547, 4802, 4973, 5144, 6520, 6691, 6862, 11493, 11664, 11835, 18184, 18355, 18526, 25046, 30019, 30190, 48545
[[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 }}


Badness: 0.000114
{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}


==trisa-quinbiru-agu (441 &amp; 1848 &amp; 3125)==
[[Badness]] (Sintel): 0.133
Comma: |51 -13 -1 -10&gt;


Map: [&lt;1 0 1 5|, &lt;0 1 7 -2|, &lt;0 0 10 -1|]
== Atomic ==
: ''For extensions, see [[Landscape microtemperaments #Atomic]].''


EDOs: 441, 1848, 3125, 3566, 4973, 6691, 8098, 8539, 9816, 11664, 14789, 15230, 18355, 21480, 30019, 33144, 39835, 51499
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.


Badness: 0.000263
[[Subgroup]]: 2.3.5


[[category:todo:add definition]]
[[Comma list]]: {{monzo| 161 -84 -12 }}
[[category:todo:intro]]
 
{{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
 
Comma list: {{monzo| 47 4 -19 }}
 
{{Mapping|legend=2| 1 -7 1 | 0 19 4 }}
: mapping generators: ~2, ~16807/12288
 
[[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.
 
Subgroup: 2.3.5.7
 
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}
 
Mapping: {{mapping| 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=2| 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=2| 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|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 }}
 
[[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
 
[[Comma list]]: {{monzo| -7 19 -16 5 }} (19534128475869/19531250000000)
 
{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}
 
[[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 }}
 
[[Badness]] (Sintel): 1.31
 
== 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
 
== Nommismic ==
{{See also| Nommisma }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| -55 30 2 1 }}
 
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}
 
[[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, }} …
 
[[Badness]] (Sintel): 1.62
 
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 1 -15 -18 23 }}
 
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}
 
[[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, }} …
 
[[Badness]] (Sintel): 0.505
 
== Technologismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 51 -13 -1 -10 }}
 
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}
 
[[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, }} …
 
[[Badness]] (Sintel): 0.116
 
== Quinbisa-quinbizo-alegu (1848 & 3737 & 6079) ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 110 -71 -11 10 }}
 
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}
 
[[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, }} …
 
[[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]]

Latest revision as of 05:23, 20 June 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

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

Retrieved from "https://en.xen.wiki/index.php?title=Very_high_accuracy_temperaments&oldid=232456"