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 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 }}


EDOs: [[118edo|118]], [[494edo|494]], [[612edo|612]], [[730edo|730]], [[848edo|848]], [[1342edo|1342]], [[1954edo|1954]]
: mapping generators: ~94921875/67108864, ~1125/1024


Badness: 0.0141
[[Optimal tuning]]s:  
* [[CTE]]: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.743279
* [[POTE]]: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.742626


==Astro==
{{Optimal ET sequence|legend=1| 22, 74, 96, 118, 376, 494, 612, 1342, 3296, 4638, 7934 }}
Comma: |91 -12 -31&gt;


POTE generator: ~296630859375/274877906944 = 132.1947
[[Badness]] (Sintel): 0.331


Map: [&lt;1 5 1|, &lt;0 31 -12|]
== Astro ==
Astro temperament tempers out the [[astro comma]], {{monzo| 91 -12 -31 }}. 7-limit extensions of the astro temperament include [[Horwell temperaments #Kastro|kastro]].


EDOs: [[118edo|118]], [[935edo|935]], [[1053edo|1053]], [[1171edo|1171]], [[1289edo|1289]], [[1407edo|1407]], [[2224edo|2224]], [[2460edo|2460]]
[[Subgroup]]: 2.3.5


Badness: 0.0628
[[Comma list]]: {{monzo| 91 -12 -31 }}


==Whoosh==
{{Mapping|legend=1| 1 5 1 | 0 -31 12 }}
Comma: |37 25 -33&gt;


POTE generator: ~864/625 = 560.5468
: mapping generators: ~2, ~{{monzo| -38 5 13 }}


Map: [&lt;1 -16 -11|, &lt;0 33 25|]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194857
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194668


EDOs: [[289edo|289]], [[441edo|441]], [[730edo|730]], [[1171edo|1171]], [[1612edo|1612]], [[1901edo|1901]], [[2631edo|2631]]
{{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: 0.0257
[[Badness]] (Sintel): 1.47


==Monzismic==
== Gaster ==
Comma: |54 -37 2&gt; = 450359962737049600 / 450283905890997363
{{Main| Gaster temperament }}
{{See also| Mirkwai clan #Gaster }}


POTE generator: ~387420489/335544320 = 249.0184
[[Subgroup]]: 2.3.5


Map: [&lt;1 0 -27|, &lt;0 2 37|]
[[Comma list]]: {{monzo| -70 72 -19 }}


EDOs: [[53edo|53]], [[506edo|506]], [[559edo|559]], [[612edo|612]], [[1783edo|1783]], [[2954edo|2954]], [[6520edo|6520]]
{{Mapping|legend=1| 1 11 38 | 0 -19 -72 }}


Badness: 0.0104
: mapping generators: ~2, ~{{monzo| 19 -19 5 }}


==Egads==
[[Optimal tuning]]s:
Comma: |-36 -52 51&gt;
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634456
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634565


POTE generator: ~6/5 = 315.6479
{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}


Map: [&lt;1 15 16|, &lt;0 -51 -52|]
[[Badness]] (Sintel): 2.08


EDOs: [[441edo|441]], [[901edo|901]], [[1342edo|1342]], [[1783edo|1783]], [[2224edo|2224]], [[3125edo|3125]], [[4467edo|4467]], [[4908edo|4908]]
== Whoosh ==
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]].


Badness: 0.0418
[[Subgroup]]: 2.3.5


==Fortune==
[[Comma list]]: {{monzo| 37 25 -33 }}
Comma: |-107 47 14&gt;


POTE generator: ~8388608/7381125 = 221.5679
{{Mapping|legend=1| 1 17 14 | 0 -33 -25 }}


Map: [&lt;1 -1 11|, &lt;0 14 -47|]
: mapping generators: ~2, ~864/625


EDOs: [[612edo|612]], [[1289edo|1289]], [[1901edo|1901]], [[2513edo|2513]], [[3125edo|3125]], [[3737edo|3737]], [[4414edo|4414]], [[5638edo|5638]]
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~864/625 = 560.546952
* [[POTE]]: ~2 = 1200.000000, ~864/625 = 560.546826


Badness: 0.0279
{{Optimal ET sequence|legend=1| 15, …, 122, 137, 152, 289, 441, 730, 1171, 1901, 3072, 4243, 7315, 25017, 32332, 39647c }}


===7-limit===
[[Badness]] (Sintel): 0.602
Commas: 78125000/78121827, |-52 17 12 -1&gt;


POTE generator: ~8388608/7381125 = 221.5680
== Monzismic ==
Monzismic ({{nowrap| 53 & 612 }}) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}. For its 7-limit extension, see [[Ragismic microtemperaments #Monzismic]].


Map: [&lt;1 -1 11 63|, &lt;0 14 -47 -326|]
[[Subgroup]]: 2.3.5


EDOs: 612, 2513, 3125, 3737, 4349, 5638, 6862, 9987
[[Comma list]]: {{monzo| 54 -37 2 }}


Badness: 0.0237
{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}


===11-limit===
: mapping generators: ~2, ~{{monzo| 27 -18 1 }}
Commas: 151263/151250, 21437500/21434787, 369140625/369098752


POTE generator: ~8388608/7381125 = 221.5681
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981427
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981594


Map: [&lt;1 -1 11 63 134|, &lt;0 14 -47 -326 -707|]
{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}


EDOs: 612, 2513, 3125, 3737, 4349, 4961, 6862, 8086
[[Badness]] (Sintel): 0.244


Badness: 0.0366
== Egads ==
[[Subgroup]]: 2.3.5


==Senior==
[[Comma list]]: {{monzo| -36 -52 51 }}
Comma: |-17 62 -35&gt;


POTE generator: ~94143178827/78125000000 = 322.8014
{{Mapping|legend=1| 1 15 16 | 0 -51 -52 }}


Map: [&lt;1 11 19|, &lt;0 -35 -62|]
: mapping generators: ~2, ~6/5


EDOs: [[171edo|171]], [[1171edo|1171]], [[1342edo|1342]], [[2513edo|2513]], [[3684edo|3684]], [[3855edo|3855]], [[4855edo|4855]], [[6197edo|6197]]
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~6/5 = 315.647899
* [[POTE]]: ~2 = 1200.000000, ~6/5 = 315.647919


Badness: 0.0212
{{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 }}


==Gross==
[[Badness]] (Sintel): 0.981
Comma: |144 -22 -47&gt;


POTE generator: ~1953125000000000/1853020188851841 = 91.5310
=== Hemiegads ===
Subgroup: 2.3.5.7


Map: [&lt;1 -2 4|, &lt;0 47 -22|]
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}


EDOs: [[118edo|118]], [[1783edo|1783]], [[1901edo|1901]], [[3684edo|3684]], [[5467edo|5467]], [[5585edo|5585]], [[7250edo|7250]]
Mapping: {{mapping| 1 15 16 -16 | 0 -102 -104 143 }}


Badness: 0.0463
Optimal tunings:  
* CTE: ~2 = 1200.000000, ~36756909/33554432 = 157.823952
* CWE: ~2 = 1200.000000, ~36756909/33554432 = 157.823954


==Raider==
{{Optimal ET sequence|legend=0| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}
Comma: |71 -99 37&gt;


POTE generator: ~8000/6561 = 343.2961
Badness (Sintel): 0.313


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 13 -36 | 0 -14 47 }}
Comma: |-90 -15 49&gt;


POTE generator: ~1358954496/1220703125 = 185.7542
: mapping generators: ~2, 7381125/4194304


Map: [&lt;1 -6 0|, &lt;0 49 15|]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567861
* [[POTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567899


EDOs: [[730edo|730]], [[1783edo|1783]], [[2513edo|2513]], [[4296edo|4296]], [[6079edo|6079]], [[6809edo|6809]], [[10375edo|10375]], [[11105edo|11105]]
{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}


Badness: 0.0057
[[Badness]] (Sintel): 0.654


==Atomic==
=== 7-limit ===
Comma: |161 -84 -12&gt;
Subgroup: 2.3.5.7


POTE generator: ~32805/32768 = 1.95516
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}


Map: [&lt;12 0 161|, &lt;0 1 -7|]
Mapping: {{mapping| 1 13 -36 -263 | 0 -14 47 326 }}


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


Badness: 0.0038
{{Optimal ET sequence|legend=0| 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}


===7-limit===
Badness (Sintel): 0.600
Commas: 250047/250000, |-55 30 2 1&gt;


POTE generator: ~32805/32768 = 1.94961
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;12 0 161 338|, &lt;0 1 -7 -16|]
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752


EDOs: 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 4308
Mapping: {{mapping| 1 13 -36 -263 -573 | 0 -14 47 326 707 }}


Badness: 0.0458
Optimal tunings:  
* CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568116
* POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568139


===11-limit===
{{Optimal ET sequence|legend=1| 612, 2513, 3125, 3737, 8086, 19909d }}
Commas: 9801/9800, 151263/151250, 184549376/184528125


POTE generator: ~32805/32768 = 1.94799
Badness (Sintel): 1.21


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


EDOs: 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 4308
[[Subgroup]]: 2.3.5


Badness: 0.0160
[[Comma list]]: {{monzo| -17 62 -35 }}


{{Mapping|legend=1| 1 11 19 | 0 -35 -62 }}


=Rank three temperaments=
: mapping generators: ~2, ~{{monzo| -6 23 -13 }}
==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|]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801349
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801358


EDOs: 171, 935, 1106, 1277, 2289, 2460, 3395, 3566, 3737
{{Optimal ET sequence|legend=1| 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}


Badness: 0.000281
[[Badness]] (Sintel): 0.498


==Saquadtrizo-asepgu (171 &amp; 2019 &amp; 3566)==
== Gross ==
Comma: 13841287201/13839609375
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.


Map: [&lt;1 0 0 0|, &lt;0 1 7 5|, &lt;0 0 12 7|]
[[Subgroup]]: 2.3.5


EDOs: 171, 1547, 1677, 1718, 1848, 2019, 3395, 3566, 3737
[[Comma list]]: {{monzo| 144 -22 -47 }}


Badness: 0.000135
{{Mapping|legend=1| 1 45 -18 | 0 -47 22 }}


==Laquinzo-aquadquadgu (171 &amp; 441 &amp; 4973)==
: mapping generators: ~2, ~{{monzo| -45 7 15 }}
Comma: |-7 19 -16 5&gt; = 19534128475869/19531250000000


Map: [&lt;1 0 3 11|, &lt;0 1 4 9|, &lt;0 0 5 16|]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.530979
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.531001


EDOs: 171, 270, 441, 612, 783, 4802, 4973, 5144
{{Optimal ET sequence|legend=1| 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c }}


Badness: 0.000297
[[Badness]] (Sintel): 1.09


==Laleruyo (171 &amp; 1547 &amp; 3125)==
; Music
Comma: 3955078125/3954653486
* ''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


Map: [&lt;1 3 0 1|, &lt;0 11 0 4|, &lt;0 0 1 1|]
== Quectismic ==
{{See also| Quectisma }}


EDOs: 171, 1407, 1547, 1578, 1718, 2783, 2954, 3125, 3296
[[Subgroup]]: 2.3.5


Badness: 0.000114
[[Comma list]]: {{monzo| 554 -351 1 }}


==Starscape==
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}
Comma: 645700815/645657712


Map: [&lt;1 0 4 0|, &lt;0 1 1 2|, &lt;0 0 9 1|]
: mapping generators: ~2, ~3


EDOs: 171, 1848, 2019, 2783, 2954, 3125, 3296, 4973, 5144
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.955310
* [[CWE]]: ~2 = 1200.000000, ~3/2 = 701.955310


Badness: 0.000545
{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}


==Sepbizo-asegu (171 &amp; 3566 &amp; 4973)==
[[Badness]] (Sintel): 8.74
Comma: 6104007655641/6103515625000


Map: [&lt;1 0 13 16|, &lt;0 1 10 12|, &lt;0 0 14 17|]
=== Nanoquectismic ===
Subgroup: 2.3.5.7


EDOs: 171, 1236, 1407, 3566, 3737, 3908, 4802, 4973, 8539, 8710
Comma list: 43046721/43025920, {{monzo| 82 -19 -3 -16 }}


Badness: 0.000223
Mapping: {{mapping| 1 0 -554 109 | 0 1 351 -67 }}


==Euzenius==
Optimal tunings:
Comma: 78125000/78121827
* CTE: ~2 = 1200.000000, ~3/2 = 701.955371
* CWE: ~2 = 1200.000000, ~3/2 = 701.955457


Map: [&lt;1 0 0 -5|, &lt;0 2 0 -13|, &lt;0 0 1 5|]
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}


EDOs: 171, 441, 612, 2954, 3125, 3566, 3737, 6520, 6691
Badness (Sintel): 15.3


Badness: 0.000205
=== Conanoquectismic ===
Subgroup: 2.3.5.7


==Quinla-zoyoyo (612 &amp; 1848 &amp; 3125)==
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
Comma: |-55 30 2 1&gt;


Map: [&lt;1 0 0 55|, &lt;0 1 0 -30|, &lt;0 0 1 -2|]
Mapping: {{mapping| 1 0 -554 -945 | 0 1 351 598 }}


EDOs: 612, 665, 1236, 1277, 1848, 1889, 2460, 2513, 3125, 3737, 4973, 5585, 6862, 8098, 8710, 11835, 14960
Optimal tunings:  
* CTE: ~2 = 1200.000000, ~3/2 = 701.954810
* CWE: ~2 = 1200.000000, ~3/2 = 701.954771


Badness: 0.000366
{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}


==Sa-twentythezo-atritribigu (171 &amp; 1547 &amp; 4973)==
Badness (Sintel): 55.0
Comma: |1 -15 -18 23&gt;


Map: [&lt;1 0 9 7|, &lt;0 1 3 3|, &lt;0 0 23 18|]
== Counterquectismic ==
{{See also| Counterquectisma }}


EDOs: 171, 1547, 4802, 4973, 5144, 6520, 6691, 6862, 11493, 11664, 11835, 18184, 18355, 18526, 25046, 30019, 30190, 48545
[[Subgroup]]: 2.3.5


Badness: 0.000114
[[Comma list]]: {{monzo| -500 314 1 }}


==Trisa-quinbiru-agu (441 &amp; 1848 &amp; 3125)==
{{Mapping|legend=1| 1 0 500 | 0 1 -314 }}
Comma: |51 -13 -1 -10&gt;


Map: [&lt;1 0 1 5|, &lt;0 1 7 -2|, &lt;0 0 10 -1|]
: mapping generators: ~2, ~3


EDOs: 441, 1848, 3125, 3566, 4973, 6691, 8098, 8539, 9816, 11664, 14789, 15230, 18355, 21480, 30019, 33144, 39835, 51499
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.954415
* [[CWE]]: ~2 = 1200.000000, ~3/2 = 701.954416


Badness: 0.000263
{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}


==Quinbisa-quinbizo-alegu (21480 &amp; 33144 &amp; 103169)==
[[Badness]] (Sintel): 12.0
Comma: |110 -71 -11 10&gt;


Map: [&lt;1 0 0 -11|, &lt;0 1 -1 6|, &lt;0 0 10 11|]
== Raider ==
{{See also| Syntonic–chromatic equivalence continuum }}


EDOs: 11664, 21480, 33144, 36881, 48545, 54624, 70025, 81689, 91505, 103169
[[Subgroup]]: 2.3.5


[[category:todo:add definition]]
[[Comma list]]: {{monzo| 71 -99 37 }}
[[category:todo:intro]]
 
{{Mapping|legend=1| 1 28 73 | 0 -37 -99 }}
 
: mapping generators: ~2, ~6561/4000
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~8000/6561 = 343.296094
* [[POTE]]: ~2 = 1200.000000, ~8000/6561 = 343.296098
 
{{Optimal ET sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355, 22651, 26947, 31243, 35539, 66782, 102321, 137860 }}
 
[[Badness]] (Sintel): 0.302
 
== Pirate ==
{{See also| Hemfiness temperaments #Piracy }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -90 -15 49 }}
 
{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}
 
: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 24 4 -13 }} = 185.754186
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 24 4 -13 }} = 185.754179
 
{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
 
[[Badness]] (Sintel): 0.133
 
== Atomic ==
{{See also | Kirnberger's atom }}
 
Atomic observes the [[schisma]], and the 3/2 is one schisma sharp of its [[12edo]] value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the [[Pythagorean comma]] is twelve schismas, and hence [[81/80]], the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.
 
Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 161 -84 -12 }}
 
{{Mapping|legend=1| 12 0 161 | 0 1 -7 }}
 
: mapping generators: ~{{monzo| -67 35 5 }}, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000, ~3/2 = 701.955176 (~32805/32768 = 1.955176)
* [[POTE]]: ~{{monzo| -67 35 5 }} = 100.000000, ~3/2 = 701.955162 (~32805/32768 = 1.955162)
 
{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
 
[[Badness]] (Sintel): 0.0892
 
=== 7-limit ===
Subgroup: 2.3.5.7
 
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
 
Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}
 
Optimal tunings:
* CTE: ~30375/28672 = 100.000000, ~3/2 = 701.949917 (or ~32805/32768 = 1.949917)
* POTE: ~30375/28672 = 100.000000, ~3/2 = 701.949608 (or ~32805/32768 = 1.949608)
 
{{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
 
Badness (Sintel): 1.16
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 151263/151250, 184549376/184528125
 
Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}
 
Optimal tunings:
* CTE: ~30375/28672 = 100.000000, ~3/2 = 701.948456 (or ~32805/32768 = 1.948456)
* POTE: ~30375/28672 = 100.000000, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)
 
{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}
 
Badness (Sintel): 0.530
 
== Revopentic (2.3.7) ==
{{See also| Revopentisma }}
 
Subgroup: 2.3.7
 
Comma list: 11399736556781568/11398895185373143
 
{{Mapping|legend=2| 1 12 5 | 0 -19 -4 }}
 
: mapping generators: ~2, ~24576/16807
 
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000000, ~24576/16807 = 657.791865
* [[CWE]]: ~2 = 1200.000000, ~24576/16807 = 657.791796
 
{{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 12 -155 5 | 0 -19 287 -4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000000, ~24576/16807 = 657.792032
* CWE: ~2 = 1200.000000, ~24576/16807 = 657.792032
 
{{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 }}
 
: sval mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.957820
* [[POTE]]: ~2 = 1200.000000, ~3/2 = 701.957784
 
{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}
 
[[Badness]] (Sintel): 0.474
 
== Icarus ==
This temperament is named after a very distant star. Curiously, the names of two even more distant stars coincide with named temperaments, those being [[godzilla]] and [[mothra]].
 
[[Subgroup]]: 3.5.7
 
[[Comma list]]: {{monzo| -32 -64 71 }}
 
{{Mapping|legend=1| 1 35 32 | 0 -71 -64 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~3 = 1901.955, ~{{monzo| 5 9 -10 }} = 898.340
* [[CWE]]: ~3 = 1901.955, ~{{monzo| 5 9 -10 }} = 898.340
 
[[Optimal ET sequence]]: [[163edt|b163cd]], [[199edt|b199c]], [[235edt|b235]], [[271edt|b271]], [[4643edt|b4643]], [[4914edt|b4914]], …, [[8708edt|b8708]], [[8979edt|b8979]], [[18229edt|b18229]], [[27208edt|b27208]], [[99853edt|b99853]], [[127061edt|b127061]], [[154269edt|b154269]], [[181477edt|b181477]], [[208685edt|b208685]], [[390162edt|b390162]]
 
[[Badness]] (Sintel): 0.00493
 
= Rank-3 temperaments =
== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 3 -24 3 10 }} (282475249000/282429536481)
 
{{Mapping|legend=1| 1 0 9 -3 | 0 1 8 0 | 0 0 10 -3 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9624, ~6561/3430 = 1122.9387
 
{{Optimal ET sequence|legend=1| 171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069 }}
 
[[Badness]] (Sintel): 1.24
 
== Saquadtrizo-asepgu (171 & 2019 & 3566) ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 0 -11 -7 12 }} (13841287201/13839609375)
 
{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9584, ~16807/10125 = 877.2825
 
{{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, ~3/2 = 701.9501, ~250/189 = 484.2956
 
{{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, ~375/343 = 154.3678, ~5/4 = 386.3070
 
{{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 or Latritriru-ayo ==
[[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, ~3/2 = 701.9514, ~9/7 = 435.0709
 
{{Optimal ET sequence|legend=1| 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634 }}
 
[[Badness]] (Sintel): 0.240
 
== Sepbizo-asegu (171 & 3566 & 4973) ==
[[Subgroup]]: 2.3.5.7
 
[[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, ~3/2 = 701.9548, ~31250/16807 = 1073.8021
 
{{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 or Sabiruquinyo ==
{{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, ~6250/5103 = 350.9787, ~5/4 = 386.3099
 
{{Optimal ET sequence|legend=1| 171, 441, 612, 1730, 1901, 2072, 2171, 2342, 2513, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 16336, 16507, 23027, 26152, 32843, 42659, 58995, 68811, 101654 }}
 
[[Badness]] (Sintel): 0.0902
 
== Nommismic or Quinla-zoyoyo ==
{{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, ~3/2 = 701.9516, ~5/4 = 386.3132
 
{{Optimal ET sequence|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, }} …
 
[[Badness]] (Sintel): 1.62
 
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: {{monzo| 1 -15 -18 23 }}
 
{{Mapping|legend=1| 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|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 or Trisa-quinbiru-agu ==
[[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, ~3/2 = 701.9553, ~63/32 = 1172.7374
 
{{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 & 11664 & 21480) ==
[[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 = 1200.0000, ~3/2 = 701.955, ~32805/28672 = 233.128
 
{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …
 
[[Badness]] (Sintel): 0.538
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]

Latest revision as of 15:37, 14 July 2025

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:

  • CTE: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.743279
  • POTE: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.742626

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 5 1], ⟨0 -31 12]]

mapping generators: ~2, ~[-38 5 13⟩

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~[-38 5 13⟩ = 132.194857
  • POTE: ~2 = 1200.000000, ~[-38 5 13⟩ = 132.194668

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): 1.47

Gaster

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

Subgroup: 2.3.5

Comma list: [-70 72 -19⟩

Mapping: [⟨1 11 38], ⟨0 -19 -72]]

mapping generators: ~2, ~[19 -19 5⟩

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~[19 -19 5⟩ = 594.634456
  • POTE: ~2 = 1200.000000, ~[19 -19 5⟩ = 594.634565

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

Badness (Sintel): 2.08

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 17 14], ⟨0 -33 -25]]

mapping generators: ~2, ~864/625

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~864/625 = 560.546952
  • POTE: ~2 = 1200.000000, ~864/625 = 560.546826

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:

  • CTE: ~2 = 1200.000000, ~[27 -18 1⟩ = 950.981427
  • POTE: ~2 = 1200.000000, ~[27 -18 1⟩ = 950.981594

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 15 16], ⟨0 -51 -52]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~6/5 = 315.647899
  • POTE: ~2 = 1200.000000, ~6/5 = 315.647919

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 15 16 -16], ⟨0 -102 -104 143]]

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~36756909/33554432 = 157.823952
  • CWE: ~2 = 1200.000000, ~36756909/33554432 = 157.823954

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

Badness (Sintel): 0.313

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 13 -36], ⟨0 -14 47]]

mapping generators: ~2, 7381125/4194304

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.567861
  • POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.567899

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 13 -36 -263], ⟨0 -14 47 326]]

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568013
  • POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568031

Optimal ET sequence: 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 13 -36 -263 -573], ⟨0 -14 47 326 707]]

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568116
  • POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568139

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

Badness (Sintel): 1.21

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 11 19], ⟨0 -35 -62]]

mapping generators: ~2, ~[-6 23 -13⟩

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~[-6 23 -13⟩ = 322.801349
  • POTE: ~2 = 1200.000000, ~[-6 23 -13⟩ = 322.801358

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 45 -18], ⟨0 -47 22]]

mapping generators: ~2, ~[-45 7 15⟩

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~[46 -7 -15⟩ = 91.530979
  • POTE: ~2 = 1200.000000, ~[46 -7 -15⟩ = 91.531001

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): 1.09

Music

Quectismic

See also: Quectisma

Subgroup: 2.3.5

Comma list: [554 -351 1⟩

Mapping: [⟨1 0 -554], ⟨0 1 351]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~3/2 = 701.955310
  • CWE: ~2 = 1200.000000, ~3/2 = 701.955310

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

Badness (Sintel): 8.74

Nanoquectismic

Subgroup: 2.3.5.7

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~3/2 = 701.955371
  • CWE: ~2 = 1200.000000, ~3/2 = 701.955457

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

Badness (Sintel): 15.3

Conanoquectismic

Subgroup: 2.3.5.7

Comma list: [33 -26 12 -7⟩, [-32 13 17 -10⟩

Mapping: [⟨1 0 -554 -945], ⟨0 1 351 598]]

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~3/2 = 701.954810
  • CWE: ~2 = 1200.000000, ~3/2 = 701.954771

Optimal ET sequence: 306d, 359cd, 665

Badness (Sintel): 55.0

Counterquectismic

See also: Counterquectisma

Subgroup: 2.3.5

Comma list: [-500 314 1⟩

Mapping: [⟨1 0 500], ⟨0 1 -314]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~3/2 = 701.954415
  • CWE: ~2 = 1200.000000, ~3/2 = 701.954416

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

Badness (Sintel): 12.0

Raider

See also: Syntonic–chromatic equivalence continuum

Subgroup: 2.3.5

Comma list: [71 -99 37⟩

Mapping: [⟨1 28 73], ⟨0 -37 -99]]

mapping generators: ~2, ~6561/4000

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~8000/6561 = 343.296094
  • POTE: ~2 = 1200.000000, ~8000/6561 = 343.296098

Optimal ET sequence: 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 43 15], ⟨0 -49 -15]]

Mapping generators: ~2, ~[-23 -4 13⟩

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~[24 4 -13⟩ = 185.754186
  • POTE: ~2 = 1200.000000, ~[24 4 -13⟩ = 185.754179

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

Badness (Sintel): 0.133

Atomic

See also: Kirnberger's atom

Atomic observes the schisma, and the 3/2 is one schisma sharp of its 12edo value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the Pythagorean comma is twelve schismas, and hence 81/80, the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.

Atomic extensions discussed elsewhere include minutes and hafnium.

Subgroup: 2.3.5

Comma list: [161 -84 -12⟩

Mapping: [⟨12 0 161], ⟨0 1 -7]]

mapping generators: ~[-67 35 5⟩, ~3

Optimal tunings:

  • CTE: ~[-67 35 5⟩ = 100.000000, ~3/2 = 701.955176 (~32805/32768 = 1.955176)
  • POTE: ~[-67 35 5⟩ = 100.000000, ~3/2 = 701.955162 (~32805/32768 = 1.955162)

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

Badness (Sintel): 0.0892

7-limit

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-55 30 2 1⟩

Mapping: [⟨12 0 161 338], ⟨0 1 -7 -16]]

Optimal tunings:

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

Optimal ET sequence: 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd

Badness (Sintel): 1.16

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250, 184549376/184528125

Mapping: [⟨12 0 161 338 517], ⟨0 1 -7 -16 -25]]

Optimal tunings:

  • CTE: ~30375/28672 = 100.000000, ~3/2 = 701.948456 (or ~32805/32768 = 1.948456)
  • POTE: ~30375/28672 = 100.000000, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)

Optimal ET sequence: 12, …, 600e, 612, 1236, 1848

Badness (Sintel): 0.530

Revopentic (2.3.7)

See also: Revopentisma

Subgroup: 2.3.7

Comma list: 11399736556781568/11398895185373143

Sval mapping: [⟨1 12 5], ⟨0 -19 -4]]

mapping generators: ~2, ~24576/16807

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~24576/16807 = 657.791865
  • CWE: ~2 = 1200.000000, ~24576/16807 = 657.791796

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 12 -155 5], ⟨0 -19 287 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~24576/16807 = 657.792032
  • CWE: ~2 = 1200.000000, ~24576/16807 = 657.792032

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⟩

Sval mapping: [⟨1 0 109], ⟨0 1 -67]]

sval mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1200.000000, ~3/2 = 701.957820
  • POTE: ~2 = 1200.000000, ~3/2 = 701.957784

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

Badness (Sintel): 0.474

Icarus

This temperament is named after a very distant star. Curiously, the names of two even more distant stars coincide with named temperaments, those being godzilla and mothra.

Subgroup: 3.5.7

Comma list: [-32 -64 71⟩

Mapping: [⟨1 35 32], ⟨0 -71 -64]]

Optimal tunings:

  • CTE: ~3 = 1901.955, ~[5 9 -10⟩ = 898.340
  • CWE: ~3 = 1901.955, ~[5 9 -10⟩ = 898.340

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

Badness (Sintel): 0.00493

Rank-3 temperaments

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

Saquadtrizo-asepgu (171 & 2019 & 3566)

Subgroup: 2.3.5.7

Comma list: [0 -11 -7 12⟩ (13841287201/13839609375)

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

Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 701.9584, ~16807/10125 = 877.2825

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 or Latritriru-ayo

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 or Sabiruquinyo

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 or Quinla-zoyoyo

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 or Trisa-quinbiru-agu

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 & 11664 & 21480)

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 = 1200.0000, ~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

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