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


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


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


{{Mapping|legend=1| 2 1 6 | 0 8 -5 }}
{{Mapping|legend=1| 2 1 6 | 0 8 -5 }}
Line 14: Line 15:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~94921875/67108864 = 1\2, ~1125/1024 = 162.7433
* [[CTE]]: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.743279
* [[POTE]]: ~94921875/67108864 = 1\2, ~1125/1024 = 162.742626
* [[POTE]]: ~94921875/67108864 = 600.000000, ~1125/1024 = 162.742626


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


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


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 33: Line 34:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| -38 5 13 }} = 132.1949
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194857
* [[POTE]]: ~2 = 1\1, ~{{monzo| -38 5 13 }} = 132.194668
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -38 5 13 }} = 132.194668


{{Optimal ET sequence|legend=1| 9, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}
{{Optimal ET sequence|legend=1| 9, …, 109, 118, 699, 817, 935, 1053, 1171, 13934, 15105, 16276, 17447, 18618, 19789, 20960, 22131, 23302, 24473, 25644, 26815, 27986c }}


[[Badness]]: 0.062778
[[Badness]] (Sintel): 1.47


== Gaster ==
== Gaster ==
Line 53: Line 54:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 19 -19 5 }} = 594.6345
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634456
* [[POTE]]: ~2 = 1\1, ~{{monzo| 19 -19 5 }} = 594.634565
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 19 -19 5 }} = 594.634565


{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}
{{Optimal ET sequence|legend=1| 111, 224, 335, 559, 1342, 1901, 3243, 8387, 11630 }}


[[Badness]]: 0.088530
[[Badness]] (Sintel): 2.08


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 72: Line 73:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~864/625 = 560.5470
* [[CTE]]: ~2 = 1200.000000, ~864/625 = 560.546952
* [[POTE]]: ~2 = 1\1, ~864/625 = 560.546826
* [[POTE]]: ~2 = 1200.000000, ~864/625 = 560.546826


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


[[Badness]]: 0.025653
[[Badness]] (Sintel): 0.602


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


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


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


{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}
{{Mapping|legend=1| 1 0 -27 | 0 2 37 }}
Line 91: Line 92:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 27 -18 1 }} = 950.9814
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981427
* [[POTE]]: ~2 = 1\1, ~{{monzo| 27 -18 1 }} = 950.981594
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 27 -18 1 }} = 950.981594


{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}
{{Optimal ET sequence|legend=1| 53, 347, 400, 453, 506, 559, 612, 1171, 1783, 4737, 6520 }}


[[Badness]]: 0.010391
[[Badness]] (Sintel): 0.244


== Egads ==
== Egads ==
Line 108: Line 109:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 315.6479
* [[CTE]]: ~2 = 1200.000000, ~6/5 = 315.647899
* [[POTE]]: ~2 = 1\1, ~6/5 = 315.647919
* [[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 }}
{{Optimal ET sequence|legend=1| 19, …, 365, 384, 403, 422, 441, 901, 1342, 1783, 3125, 4908, 45955, 50863, 55771, 60679, 65587, 70495, 75403, 80311, 85219, 90127 }}


[[Badness]]: 0.041819
[[Badness]] (Sintel): 0.981


=== Hemiegads ===
=== Hemiegads ===
Line 123: Line 124:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~36756909/33554432 = 157.8240
* CTE: ~2 = 1200.000000, ~36756909/33554432 = 157.823952
* CWE: ~2 = 1200.000000, ~36756909/33554432 = 157.823954


{{Optimal ET sequence|legend=1| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}
{{Optimal ET sequence|legend=0| 38, 403, 441, 1802, 2243, 2684, 3125, 6691, 9816, 16507, 75844, 125365c }}


Badness: 0.0124
Badness (Sintel): 0.313


== Fortune ==
== Fortune ==
Fortune tempers out the fortune comma, {{monzo| -107 47 14 }}. It can be described as 612 &amp; 2513 temperament, which tempers out 78125000/78121827 (euzenius comma) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.
Fortune tempers out the [[fortune comma]], {{monzo| -107 47 14 }}. It can be described as {{nowrap| 612 & 2513 }} temperament, which tempers out 78125000/78121827 ([[euzenius comma]]) in the 7-limit; 151263/151250, 21437500/21434787, and 369140625/369098752 in the 11-limit.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 141: Line 143:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~8388608/7381125 = 221.5679
* [[CTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567861
* [[POTE]]: ~2 = 1\1, ~8388608/7381125 = 221.567899
* [[POTE]]: ~2 = 1200.000000, ~8388608/7381125 = 221.567899


{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}
{{Optimal ET sequence|legend=1| 65, 352, 417, 482, 547, 612, 1901, 2513, 5638, 8151, 18815, 45781, 64596, 110377, 174973bc }}


[[Badness]]: 0.027876
[[Badness]] (Sintel): 0.654


=== 7-limit ===
=== 7-limit ===
Line 156: Line 158:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~8388608/7381125 = 221.5680
* CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568013
* POTE: ~2 = 1\1, ~8388608/7381125 = 221.568031
* POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568031


{{Optimal ET sequence|legend=1| 612, 1901, 2513, 3125, 6862, 9987, 13112, 36211, 49323 }}
{{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 171: Line 173:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~8388608/7381125 = 221.5681
* CTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568116
* POTE: ~2 = 1\1, ~8388608/7381125 = 221.568139
* POTE: ~2 = 1200.000000, ~8388608/7381125 = 221.568139


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


Badness: 0.036641
Badness (Sintel): 1.21


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 190: Line 192:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| -6 23 -13 }} = 322.8013
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801349
* [[POTE]]: ~2 = 1\1, ~{{monzo| -6 23 -13 }} = 322.801358
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| -6 23 -13 }} = 322.801358


{{Optimal ET sequence|legend=1| 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}
{{Optimal ET sequence|legend=1| 26, …, 145, 171, 658, 829, 1000, 1171, 2513, 6197, 39695, 45892, 52089, 58286, 64483, 70680 }}


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


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 209: Line 211:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 46 -7 -15 }} = 91.5310
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.530979
* [[POTE]]: ~2 = 1\1, ~{{monzo| 46 -7 -15 }} = 91.531001
* [[POTE]]: ~2 = 1200.000000, ~{{monzo| 46 -7 -15 }} = 91.531001


{{Optimal ET sequence|legend=1| 13, 105, 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c }}
{{Optimal ET sequence|legend=1| 118, 957, 1075, 1193, 1311, 1429, 1547, 1665, 1783, 3684, 5467, 36486, 41953, 47420, 52887, 63821, 69288, 74755, 144043c, 218798c }}


[[Badness]]: 0.046315
[[Badness]] (Sintel): 1.09


; Music
; Music
Line 231: Line 233:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9553
* [[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 }}
{{Optimal ET sequence|legend=1| 306, 359c, 665, 2301, 2966, 3631, 4296, 46591, 50887, 55183, 59479, 63775, 68071, 72367, 76663, 80959, 85255, 89551 }}


[[Badness]]: 0.372
[[Badness]] (Sintel): 8.74


=== Nanoquectismic ===
=== Nanoquectismic ===
Line 245: Line 248:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.955
* 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 }}
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}


Badness: 0.604
Badness (Sintel): 15.3


=== Conanoquectismic ===
=== Conanoquectismic ===
Line 259: Line 263:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 701.9548
* CTE: ~2 = 1200.000000, ~3/2 = 701.954810
* CWE: ~2 = 1200.000000, ~3/2 = 701.954771


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


Badness: 2.17
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
Line 277: Line 301:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~8000/6561 = 343.2961
* [[CTE]]: ~2 = 1200.000000, ~8000/6561 = 343.296094
* [[POTE]]: ~2 = 1\1, ~8000/6561 = 343.296098
* [[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 }}
{{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 ==
{{See also| Hemfiness temperaments #Piracy }}
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 291: Line 317:
{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}
{{Mapping|legend=1| 1 43 15 | 0 -49 -15 }}


: mapping generators: ~2, ~{{monzo| -23 -4 13 }}
: Mapping generators: ~2, ~{{monzo| -23 -4 13 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~{{monzo| 24 4 -13 }} = 185.7542
* [[CTE]]: ~2 = 1200.000000, ~{{monzo| 24 4 -13 }} = 185.754186
* [[POTE]]: ~2 = 1\1, ~{{monzo| 24 4 -13 }} = 185.754179
* [[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 … }}
{{Optimal ET sequence|legend=1| 84, 239, 323, 730, 1783, 2513, 4296, 15401, 19697, 23993, 52282, 76275, 128557, 204832 … }}


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


== Atomic ==
== Atomic ==
{{See also | Kirnberger's atom }}
{{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]].
Atomic extensions discussed elsewhere include [[minutes]] and [[72nd-octave temperaments#Hafnium|hafnium]].
Atomic does not temper out the [[schisma]], so 3/2 is one schisma sharp of its [[12edo]] value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the [[Pythagorean comma]] is twelve schismas, and hence [[81/80]], the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 317: Line 343:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~{{monzo| -67 35 5 }} = 1\12, ~3/2 = 701.9552 (~32805/32768 = 1.9552)
* [[CTE]]: ~{{monzo| -67 35 5 }} = 100.000000, ~3/2 = 701.955176 (~32805/32768 = 1.955176)
* [[POTE]]: ~{{monzo| -67 35 5 }} = 1\12, ~3/2 = 701.955162 (~32805/32768 = 1.955162)
* [[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 }}
{{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 332: Line 358:


Optimal tunings:  
Optimal tunings:  
* CTE: ~30375/28672 = 1\12, ~3/2 = 701.9499 (or ~32805/32768 = 1.9499)
* CTE: ~30375/28672 = 100.000000, ~3/2 = 701.949917 (or ~32805/32768 = 1.949917)
* POTE: ~30375/28672 = 1\12, ~3/2 = 701.949608 (or ~32805/32768 = 1.949608)
* POTE: ~30375/28672 = 100.000000, ~3/2 = 701.949608 (or ~32805/32768 = 1.949608)


{{Optimal ET sequence|legend=1| 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 347: Line 373:


Optimal tunings:  
Optimal tunings:  
* CTE: ~30375/28672 = 1\12, ~3/2 = 701.9485 (or ~32805/32768 = 1.9485)
* CTE: ~30375/28672 = 100.000000, ~3/2 = 701.948456 (or ~32805/32768 = 1.948456)
* POTE: ~30375/28672 = 1\12, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)
* POTE: ~30375/28672 = 100.000000, ~3/2 = 701.947988 (or ~32805/32768 = 1.947988)


{{Optimal ET sequence|legend=1| 12, …, 600e, 612, 1236, 1848 }}
{{Optimal ET sequence|legend=0| 12, …, 600e, 612, 1236, 1848 }}


Badness: 0.016029
Badness (Sintel): 0.530


== Revopentic (2.3.7) ==
== Revopentic (2.3.7) ==
Line 363: Line 389:
{{Mapping|legend=2| 1 12 5 | 0 -19 -4 }}
{{Mapping|legend=2| 1 12 5 | 0 -19 -4 }}


: mapping generators: ~2 = 1\1, 24576/16807
: mapping generators: ~2, ~24576/16807


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~24576/16807 = 657.792
* [[CTE]]: ~2 = 1200.000000, ~24576/16807 = 657.791865
* [[CWE]]: ~2 = 1200.000000, ~24576/16807 = 657.791796


[[Support]]ing [[ET]]s: {{EDOs| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794 }}
{{Optimal ET sequence|legend=1| 31, 73, 104, 135, 436, 571, 706, 1277, 8233, 9510, 10787, 12064, 13341, 14618, 15895, 33067, 48962, 113819d, 162781d, … }}


Badness (Sintel's TE logflat): 0.070
[[Badness]] (Sintel): 0.0704


=== Revopent ===
=== Revopent ===
Revopent can be described as the 1848 & 3125 temperament.
Revopent can be described as the {{nowrap| 1848 & 3125 }} temperament.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 382: Line 409:


Optimal tunings:  
Optimal tunings:  
* CTE: ~24576/16807 = 657.792
* CTE: ~2 = 1200.000000, ~24576/16807 = 657.792032
* CWE: ~2 = 1200.000000, ~24576/16807 = 657.792032


{{Optimal ET sequence|legend=1| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}
{{Optimal ET sequence|legend=0| 135, 436c, 571, 1277, 1848, 3125, 8098, 11223, 25571, 36794, 121605d }}


Badness: 0.263
Badness (Sintel): 0.665


== Nanic ==
== Nanic ==
Line 395: Line 423:
[[Comma list]]: {{monzo| 109 -67 -1 }}
[[Comma list]]: {{monzo| 109 -67 -1 }}


{{Mapping|legend=1| 1 0 109 | 0 1 -67 }}
{{Mapping|legend=2| 1 0 109 | 0 1 -67 }}


: sval mapping generators: ~2, ~3
: sval mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1\1, ~3/2 = 701.9578
* [[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 }}
{{Optimal ET sequence|legend=1| 53, 200, 253, 306, 665, 971, 1277, 6691, 7968, 9245, 10522, 11799, 13076, 40505, 53581, 66657, 79733 }}


[[Badness]]: 0.0138
[[Badness]] (Sintel): 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 =
= 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
[[Comma list]]: {{monzo| 3 -24 3 10 }} (282475249000/282429536481)


{{Mapping|legend=1| 1 0 9 -3 | 0 1 8 0 | 0 0 10 -3 }}
{{Mapping|legend=1| 1 0 9 -3 | 0 1 8 0 | 0 0 10 -3 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9624, ~6561/3430 = 1122.9387
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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 }}
{{Optimal ET sequence|legend=1| 171, 764, 935, 1106, 1277, 2118, 2289, 2460, 3224, 3395, 3566, 26068, 26239, 29634, 29805, 33200, 33371, 36937, 40503, 44069 }}


[[Badness]]: 0.281 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.24


== Saquadtrizo-asepgu (171 &amp; 2019 &amp; 3566) ==
== Saquadtrizo-asepgu (171 & 2019 & 3566) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| 0 -11 -7 12 }} = 13841287201/13839609375
[[Comma list]]: {{monzo| 0 -11 -7 12 }} (13841287201/13839609375)


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


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9584, ~16807/10125 = 877.2825
[[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 }}
{{Optimal ET sequence|legend=1| 171, 814, 985, 1034, 1205, 1376, 1547, 1718, 1848, 2019, 3224, 3395, 3566, 8980, 9151, 12717, 47131, 50697, 59848d, 63414d, 76131d }}


[[Badness]]: 0.135 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.595


== Laquinzo-aquadquadgu (171 &amp; 441 &amp; 4973) ==
== Laquinzo-aquadquadgu (171 & 441 & 4973) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}
{{Mapping|legend=1| 1 0 3 11 | 0 1 4 9 | 0 0 5 16 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9501, ~250/189 = 484.2956
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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 }}
{{Optimal ET sequence|legend=1| 171, 441, 612, 2696, 2795, 2867, 2966, 3137, 3308, 3407, 3479, 3578, 3749, 3920, 4091, 4190, 4361, 4532, 4703, 4802, 4973, 5144, 10117, 10558, 15531, 15702, 52079, 62637, 67781, 78339 }}


[[Badness]]: 0.297 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.31


== Laleruyo (171 &amp; 1547 &amp; 3125) ==
== Laleruyo (171 & 1547 & 3125) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| -1 4 11 -11 }} = 3955078125/3954653486
[[Comma list]]: {{monzo| -1 4 11 -11 }} (3955078125/3954653486)


{{Mapping|legend=1| 1 3 0 1 | 0 -11 0 -4 | 0 0 1 1 }}
{{Mapping|legend=1| 1 3 0 1 | 0 -11 0 -4 | 0 0 1 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~375/343 = 154.3678, ~5/4 = 386.3070
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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 }}
{{Optimal ET sequence|legend=1| 171, 863, 894, 1034, 1065, 1205, 1236, 1376, 1407, 1547, 1578, 1718, 2612, 2783, 2954, 3125, 12329, 15454, 18579, 21704, 40283, 61987 }}


[[Badness]]: 0.114 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.501


== Starscape or Latritriru-ayo ==
== Starscape or Latritriru-ayo ==
Line 466: Line 512:
{{Mapping|legend=1| 1 0 4 0 | 0 1 1 2 | 0 0 -9 -1 }}
{{Mapping|legend=1| 1 0 4 0 | 0 1 1 2 | 0 0 -9 -1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9514, ~9/7 = 435.0709
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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 }}
{{Optimal ET sequence|legend=1| 171, 742, 764, 935, 1106, 1277, 1677, 1848, 2019, 2783, 2954, 3125, 8098, 11223, 25571, 33840, 36794, 45063, 48188, 59411, 70634 }}


[[Badness]]: 0.545 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.240


== Sepbizo-asegu (171 &amp; 3566 &amp; 4973) ==
== Sepbizo-asegu (171 & 3566 & 4973) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| -3 2 -17 14 }} = 6104007655641/6103515625000
[[Comma list]]: {{monzo| -3 2 -17 14 }} (6104007655641/6103515625000)


{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 -14 -17 }}
{{Mapping|legend=1| 1 0 13 16 | 0 1 10 12 | 0 0 -14 -17 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9548, ~31250/16807 = 1073.8021
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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 }}
{{Optimal ET sequence|legend=1| 171, 894, 1065, 1236, 1407, 1578, 3053, 3224, 3395, 3566, 5144, 8368, 8539, 8710, 17249, 20815, 29525, 38064, 67589, 105653 }}


[[Badness]]: 0.223 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.982


== Euzenius or Sabiruquinyo ==
== Euzenius or Sabiruquinyo ==
Line 494: Line 540:
{{Mapping|legend=1| 1 1 0 -5 | 0 2 0 -13 | 0 0 1 5 }}
{{Mapping|legend=1| 1 1 0 -5 | 0 2 0 -13 | 0 0 1 5 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6250/5103 = 350.9787, ~5/4 = 386.3099
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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 }}
{{Optimal ET sequence|legend=1| 171, 441, 612, 1730, 1901, 2072, 2171, 2342, 2513, 2684, 2783, 2954, 3125, 6520, 6691, 9816, 16336, 16507, 23027, 26152, 32843, 42659, 58995, 68811, 101654 }}


[[Badness]]: 0.205 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.0902


== Nommismic or Quinla-zoyoyo ==
== Nommismic or Quinla-zoyoyo ==
Line 509: Line 555:
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 0 55 | 0 1 0 -30 | 0 0 1 -2 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9516, ~5/4 = 386.3132
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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, }} …
{{Optimal ET sequence|legend=1| 612, 1236, 1277, 1848, 2513, 3125, 6862, 8098, 8710, 11835, 14960, 26795, 195663b, }} …


[[Badness]]: 0.366 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.62


== Satwethezo-atritribigu (171 &amp; 1547 &amp; 4973)  ==
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 522: Line 568:
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}
{{Mapping|legend=1| 1 0 9 7 | 0 1 3 3 | 0 0 -23 -18 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9551, ~56953125/40353607 = 596.5022
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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, }} …
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 3255, 3426, 3597, 4460, 4631, 4802, 4973, 5144, 6520, 6691, 11664, 18355, 48374, 48545, 66900, 145464, }} …


[[Badness]]: 0.114 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.505


== Technologismic or Trisa-quinbiru-agu (441 &amp; 1848 &amp; 3125)  ==
== Technologismic or Trisa-quinbiru-agu ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 535: Line 581:
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}
{{Mapping|legend=1| 1 0 1 5 | 0 1 7 -2 | 0 0 -10 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9553, ~63/32 = 1172.7374
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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, }} …
{{Optimal ET sequence|legend=1| 441, 966, 1277, 1407, 1718, 1848, 2684, 3125, 6691, 9816, 11664, 14789, 18355, 33144, 51499, 124478, 175977, }} …


[[Badness]]: 0.263 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.116


== Quinbisa-quinbizo-alegu (1848 &amp; 11664 &amp; 21480) ==
== Quinbisa-quinbizo-alegu (1848 & 11664 & 21480) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 548: Line 594:
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}
{{Mapping|legend=1| 1 0 0 -11 | 0 1 9 17 | 0 0 -10 -11 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.955, ~32805/28672 = 233.128
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~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, }} …
{{Optimal ET sequence|legend=1| 1848, 3737, 5585, 6079, 7927, 9816, 11664, 21480, 33144, 54624, 70025, 81689, 103169, }} …


[[Badness]]: 0.122 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.538


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
Retrieved from "https://en.xen.wiki/w/Very_high_accuracy_temperaments"