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 }}


[[Comma list]]: {{monzo| -53 10 16 }} = 9010162353515625 / 9007199254740992
{{Mapping|legend=1| 2 1 6 | 0 8 -5 }}


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


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


{{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 ==
Astro temperament tempers out the astro comma, [91 -12 -31⟩ (tribisa-thiwegu). 7-limit extensions of the astro temperament include [[Horwell temperaments#Kastro|kastro]].
Astro temperament tempers out the [[astro comma]], {{monzo| 91 -12 -31 }}. 7-limit extensions of the astro temperament include [[Horwell temperaments #Kastro|kastro]].


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[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 5 1 | 0 -31 12 }}
 
: mapping generators: ~2, ~{{monzo| -38 5 13 }}


[[POTE generator]]: ~296630859375/274877906944 = 132.194668
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194857
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194668


{{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): 1.47


== 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 list]]: {{monzo| -70 72 -19 }}


[[Comma]]: {{monzo|-70 72 -19}}
{{Mapping|legend=1| 1 11 38 | 0 -19 -72 }}


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


[[POTE generator]]: ~1638400000/1162261467 = 594.634565
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634456
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634565


{{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): 2.08


== 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 17 14 | 0 -33 -25 }}
 
: mapping generators: ~2, ~864/625


[[POTE generator]]: ~864/625 = 560.546826
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~864/625 = 560.546952
* [[POTE]]: ~2 = 1200.000000, ~864/625 = 560.546826


{{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 (53 &amp; 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}. For its 7-limit extension, see [[Ragismic microtemperaments #Monzismic]].
Monzismic ({{nowrap| 53 & 612 }}) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}. For its 7-limit extension, see [[Ragismic microtemperaments #Monzismic]].


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


[[Comma list]]: {{monzo| 54 -37 2 }} = 450359962737049600 / 450283905890997363
[[Comma list]]: {{monzo| 54 -37 2 }}
 
{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}


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


[[POTE generator]]: ~387420489/335544320 = 249.018406
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981427
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981594


{{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 15 16 | 0 -51 -52 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~6/5 = 315.647899
* [[POTE]]: ~2 = 1200.000000, ~6/5 = 315.647919
 
{{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 }}
 
Mapping: {{mapping| 1 15 16 -16 | 0 -102 -104 143 }}


[[POTE generator]]: ~6/5 = 315.647919
Optimal tunings:  
* CTE: ~2 = 1200.000000, ~36756909/33554432 = 157.823952
* CWE: ~2 = 1200.000000, ~36756909/33554432 = 157.823954


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


[[Badness]]: 0.041819
Badness (Sintel): 0.313


== 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 13 -36 | 0 -14 47 }}
 
: mapping generators: ~2, 7381125/4194304


[[POTE generator]]: ~8388608/7381125 = 221.567899
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567861
* [[POTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567899


{{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 111: Line 155:
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: {{mapping| 1 13 -36 -263 | 0 -14 47 326 }}


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


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


Badness: 0.023704
Badness (Sintel): 0.600


=== 11-limit ===
=== 11-limit ===
Line 124: Line 170:
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: {{mapping| 1 13 -36 -263 -573 | 0 -14 47 326 707 }}


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


Optimal GPV sequence: {{Val list| 612, 2513, 3125, 3737, 8086 }}
{{Optimal ET sequence|legend=1| 612, 2513, 3125, 3737, 8086, 19909d }}


Badness: 0.036641
Badness (Sintel): 1.21


== 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 11 19 | 0 -35 -62 }}
 
: mapping generators: ~2, ~{{monzo| -6 23 -13 }}


[[POTE generator]]: ~94143178827/78125000000 = 322.801358
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801349
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801358


{{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 45 -18 | 0 -47 22 }}
 
: mapping generators: ~2, ~{{monzo| -45 7 15 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.530979
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.531001
 
{{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): 1.09
 
; 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
 
== Quectismic ==
{{See also| Quectisma }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 554 -351 1 }}


[[POTE generator]]: ~1953125000000000/1853020188851841 = 91.531001
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}


{{Optimal GPV sequence|legend=1| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467 }}
: mapping generators: ~2, ~3


[[Badness]]: 0.046315
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.955310
* [[CWE]]: ~2 = 1200.000000, ~3/2 = 701.955310
 
{{Optimal ET sequence|legend=1| 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, {{monzo| 82 -19 -3 -16 }}
 
Mapping: {{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|legend=1| 306, 665, 1636, 2301, 2966 }}
 
Badness (Sintel): 15.3
 
=== Conanoquectismic ===
Subgroup: 2.3.5.7
 
Comma list: {{monzo| 33 -26 12 -7 }}, {{monzo| -32 13 17 -10 }}
 
Mapping: {{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|legend=1| 306d, 359cd, 665 }}
 
Badness (Sintel): 55.0
 
== Counterquectismic ==
{{See also| Counterquectisma }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -500 314 1 }}
 
{{Mapping|legend=1| 1 0 500 | 0 1 -314 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~3/2 = 701.954415
* [[CWE]]: ~2 = 1200.000000, ~3/2 = 701.954416
 
{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}
 
[[Badness]] (Sintel): 12.0


== 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 28 73 | 0 -37 -99 }}
 
: mapping generators: ~2, ~6561/4000


[[POTE generator]]: ~8000/6561 = 343.296098
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~8000/6561 = 343.296094
* [[POTE]]: ~2 = 1200.000000, ~8000/6561 = 343.296098


{{Optimal GPV sequence|legend=1| 388, 783, 1171, 1954, 3125, 4296, 5467, 7421, 9763, 14059, 18355 }}
{{Optimal ET sequence|legend=1| 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 43 15 | 0 -49 -15 }}


[[POTE generator]]: ~{{monzo| 24 4 -13 }} = 185.754179
: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}


{{Optimal GPV sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 24 4 -13 }} = 185.754186
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 24 4 -13 }} = 185.754179


[[Badness]]: 0.005656
{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}
 
[[Badness]] (Sintel): 0.133


== Atomic ==
== Atomic ==
{{see also | Kirnberger's atom }}
{{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
[[Subgroup]]: 2.3.5
Line 197: Line 338:
[[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


[[POTE generator]]: ~32805/32768 = 1.955162
[[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 GPV sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}
{{Optimal ET sequence|legend=1| 12, …, 576, 588, 600, 612, 2460, 3072, 3684, 4296, 12276, 16572, 20868, 25164, 46032 }}


[[Badness]]: 0.003803
[[Badness]] (Sintel): 0.0892


=== 7-limit ===
=== 7-limit ===
Line 210: Line 355:
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}
Comma list: 250047/250000, {{monzo| -55 30 2 1 }}


Mapping: [{{val| 12 0 161 338 }}, {{val| 0 1 -7 -16 }}]
Mapping: {{mapping| 12 0 161 338 | 0 1 -7 -16 }}


POTE generator: ~32805/32768 = 1.949608
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 GPV sequence: {{Val list| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}
{{Optimal ET sequence|legend=0| 12, …, 600, 612, 1236, 1848, 4308, 10464, 14772, 25236c, 40008ccd }}


Badness: 0.045818
Badness (Sintel): 1.16


=== 11-limit ===
=== 11-limit ===
Line 223: Line 370:
Comma list: 9801/9800, 151263/151250, 184549376/184528125
Comma list: 9801/9800, 151263/151250, 184549376/184528125


Mapping: [{{val| 12 0 161 338 517 }}, {{val| 0 1 -7 -16 -25 }}]
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


POTE generator: ~32805/32768 = 1.947988
[[Comma list]]: {{monzo| -32 -64 71 }}


Optimal GPV sequence: {{Val list| 12, …, 600e, 612, 1236, 1848 }}
{{Mapping|legend=1| 1 35 32 | 0 -71 -64 }}


Badness: 0.016029
[[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 =
= Rank-3 temperaments =
== Sasaquinbizo-atriyo (171 &amp; 1106 &amp; 3566) ==
== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
Subgroup: 2.3.5.7
[[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 }}


Comma list: 282475249000/282429536481
[[Badness]] (Sintel): 1.24


Mapping: [{{val| 1 0 9 -3 }}, {{val| 0 1 8 0 }}, {{val| 0 0 10 -3 }}]
== Saquadtrizo-asepgu (171 & 2019 & 3566) ==
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 171, 935, 1106, 1277, 2289, 2460, 3395, 3566, 3737}}
[[Comma list]]: {{monzo| 0 -11 -7 12 }} (13841287201/13839609375)


Badness: 0.000281
{{Mapping|legend=1| 1 0 0 0 | 0 1 7 5 | 0 0 -12 -7 }}


== Saquadtrizo-asepgu (171 &amp; 2019 &amp; 3566) ==
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9584, ~16807/10125 = 877.2825
Subgroup: 2.3.5.7


Comma list: 13841287201/13839609375
{{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 }}


Mapping: [{{val| 1 0 0 0 }}, {{val| 0 1 7 5 }}, {{val| 0 0 12 7 }}]
[[Badness]] (Sintel): 0.595


Vals: {{Val list| 171, 1547, 1677, 1718, 1848, 2019, 3395, 3566, 3737}}
== Laquinzo-aquadquadgu (171 & 441 & 4973) ==
[[Subgroup]]: 2.3.5.7


Badness: 0.000135
[[Comma list]]: {{monzo| -7 19 -16 5 }} (19534128475869/19531250000000)


== Laquinzo-aquadquadgu (171 &amp; 441 &amp; 4973) ==
{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}
Subgroup: 2.3.5.7


Comma list: {{monzo| -7 19 -16 5 }} = 19534128475869/19531250000000
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9501, ~250/189 = 484.2956


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


Vals: {{Val list| 171, 270, 441, 612, 783, 4802, 4973, 5144}}
[[Badness]] (Sintel): 1.31


Badness: 0.000297
== Laleruyo (171 & 1547 & 3125) ==
[[Subgroup]]: 2.3.5.7


== Laleruyo (171 &amp; 1547 &amp; 3125) ==
[[Comma list]]: {{monzo| -1 4 11 -11 }} (3955078125/3954653486)
Subgroup: 2.3.5.7


Comma list: 3955078125/3954653486
{{Mapping|legend=1| 1 3 0 1 | 0 -11 0 -4 | 0 0 1 1 }}


Mapping: [{{val| 1 3 0 1 }}, {{val| 0 11 0 4 }}, {{val| 0 0 1 1 }}]
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~375/343 = 154.3678, ~5/4 = 386.3070


Vals: {{Val list| 171, 1407, 1547, 1578, 1718, 2783, 2954, 3125, 3296}}
{{Optimal ET sequence|legend=1| 171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987 }}


Badness: 0.000114
[[Badness]] (Sintel): 0.501


== Starscape or Latritriru-ayo ==
== Starscape or Latritriru-ayo ==
Line 281: Line 510:
[[Comma list]]: 645700815/645657712
[[Comma list]]: 645700815/645657712


[[Mapping]]: [{{val| 1 0 4 0 }}, {{val| 0 1 1 2 }}, {{val| 0 0 9 1 }}]
{{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 }}


{{Val list|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


[[Badness]]: 0.000545
== Sepbizo-asegu (171 & 3566 & 4973) ==
[[Subgroup]]: 2.3.5.7


== Sepbizo-asegu (171 &amp; 3566 &amp; 4973) ==
[[Comma list]]: {{monzo| -3 2 -17 14 }} (6104007655641/6103515625000)
Subgroup: 2.3.5.7


Comma list: 6104007655641/6103515625000
{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 -14 -17 }}


Mapping: [{{val| 1 0 13 16 }}, {{val| 0 1 10 12 }}, {{val| 0 0 14 17 }}]
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9548, ~31250/16807 = 1073.8021


Vals: {{Val list| 171, 1236, 1407, 3566, 3737, 3908, 4802, 4973, 8539, 8710}}
{{Optimal ET sequence|legend=1| 171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653 }}


Badness: 0.000223
[[Badness]] (Sintel): 0.982


== Euzenius or Sabiruquinyo ==
== Euzenius or Sabiruquinyo ==
{{See also| Euzenius comma }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 78125000/78121827
[[Comma list]]: 78125000/78121827


[[Mapping]]: [{{val| 1 0 0 -5 }}, {{val| 0 2 0 -13 }}, {{val| 0 0 1 5 }}]
{{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 }}


{{Val list|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


[[Badness]]: 0.000205
== Nommismic or Quinla-zoyoyo ==
{{See also| Nommisma }}


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


[[Comma list]]: {{monzo| -55 30 2 1 }}
[[Comma list]]: {{monzo| -55 30 2 1 }}


[[Mapping]]: [{{val| 1 0 0 55 }}, {{val| 0 1 0 -30 }}, {{val| 0 0 1 -2 }}]
{{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


{{Val list|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795 }}
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
[[Subgroup]]: 2.3.5.7


[[Badness]]: 0.000366
[[Comma list]]: {{monzo| 1 -15 -18 23 }}


== Satwethezo-atritribigu (171 &amp; 1547 &amp; 4973) ==
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}
Subgroup: 2.3.5.7
 
[[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, }} …


Comma list: {{monzo| 1 -15 -18 23 }}
[[Badness]] (Sintel): 0.505


Mapping: [{{val| 1 0 9 7 }}, {{val| 0 1 3 3 }}, {{val| 0 0 23 18 }}]
== Technologismic or Trisa-quinbiru-agu ==
[[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| 51 -13 -1 -10 }}


Badness: 0.000114
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}


== Trisa-quinbiru-agu (441 &amp; 1848 &amp; 3125) ==
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.9553, ~63/32 = 1172.7374
Subgroup: 2.3.5.7


Comma list: {{monzo| 51 -13 -1 -10 }}
{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …


Mapping: [{{val| 1 0 1 5 }}, {{val| 0 1 7 -2 }}, {{val| 0 0 10 -1 }}]
[[Badness]] (Sintel): 0.116


Vals: {{Val list| 441, 1848, 3125, 3566, 4973, 6691, 8098, 8539, 9816, 11664, 14789, 15230, 18355, 21480, 30019, 33144, 39835, 51499}}
== Quinbisa-quinbizo-alegu (1848 & 11664 & 21480) ==
[[Subgroup]]: 2.3.5.7


Badness: 0.000263
[[Comma list]]: {{monzo| 110 -71 -11 10 }}


== Quinbisa-quinbizo-alegu (21480 &amp; 33144 &amp; 103169) ==
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}
Subgroup: 2.3.5.7


Comma list: {{monzo| 110 -71 -11 10 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 701.955, ~32805/28672 = 233.128


Mapping: [{{val| 1 0 0 -11 }}, {{val| 0 1 -1 6 }}, {{val| 0 0 10 11 }}]
{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …


Vals: {{Val list| 11664, 21480, 33144, 36881, 48545, 54624, 70025, 81689, 91505, 103169}}
[[Badness]] (Sintel): 0.538


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