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Very high accuracy temperaments: Difference between revisions

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- septimal atomic (addressed in landscape microtemps). Rework on intro to the 5-limit temp.
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Agh Linnaeus isn't a microbiologist choose better name
 
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Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}
Comma list: 78125000/78121827, {{monzo| -48 0 11 8 }}


Mapping: {{mapping| 1 -87 -88 127 | 0 102 104 -143 }}
{{Mapping|legend=0| 1 -87 -88 127 | 0 102 104 -143 }}
: mapping generators: ~2, ~67108864/36756909
: mapping generators: ~2, ~67108864/36756909


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Comma list: [[140625/140608]], {{monzo| -4 20 -9 0 0 -4 }}
Comma list: [[140625/140608]], {{monzo| -4 20 -9 0 0 -4 }}


Mapping: {{mapping| 1 -36 -36 -98 | 0 51 52 138 }}
{{Mapping|legend=0| 1 -36 -36 -98 | 0 51 52 138 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}
Comma list: 78125000/78121827, {{monzo| -52 17 12 -1 }}


Mapping: {{mapping| 1 -1 11 63 | 0 14 -47 -326 }}
{{Mapping|legend=0| 1 -1 11 63 | 0 14 -47 -326 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 151263/151250, 21437500/21434787, 369140625/369098752
Comma list: 151263/151250, 21437500/21434787, 369140625/369098752


Mapping: {{mapping| 1 -1 11 63 134 | 0 14 -47 -326 -707 }}
{{Mapping|legend=0| 1 -1 11 63 134 | 0 14 -47 -326 -707 }}


Optimal tunings:  
Optimal tunings:  
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[[Badness]] (Sintel): 0.133
[[Badness]] (Sintel): 0.133
== 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:
* [[WE]]: ~2 = 1200.000184840{{c}}, ~3/2 = 701.954524421{{c}}
: [[error map]]: {{val| +0.000185 -0.000292 -0.000002 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.954415845{{c}}
: error map: {{val| 0.000000 -0.000585 -0.000289 }}
{{Optimal ET sequence|legend=1| 306c, 359, 665, 2354, 3019, 3684, 11717, 15401, 19085, 34486 }}
[[Badness]] (Sintel): 11.971
== Quectismic ==
{{See also| Quectisma }}
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 554 -351 1 }}
{{Mapping|legend=1| 1 0 -554 | 0 1 351 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.999902567{{c}}, ~3/2 = 701.955253373{{c}}
: [[error map]]: {{val| -0.000097 +0.000155 -0.000001 }}
* [[CWE]]: ~2 = 1200.000000000{{c}}, ~3/2 = 701.955310152{{c}}
: error map: {{val| 0.000000 +0.000309 +0.000149 }}
{{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): 8.736
=== 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:
* WE: ~2 = 1199.979407{{c}}, ~3/2 = 701.943437{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.955457{{c}}
{{Optimal ET sequence|legend=1| 306, 665, 1636, 2301, 2966 }}
Badness (Sintel): 15.285
=== 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:
* WE: ~2 = 1200.010750{{c}}, ~3/2 = 701.961041{{c}}
* CWE: ~2 = 1200.000000{{c}}, ~3/2 = 701.954771{{c}}
{{Optimal ET sequence|legend=1| 306d, 359cd, 665 }}
Badness (Sintel): 54.974


== Atomic ==
== Atomic ==
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Comma list: {{monzo| 47 4 -19 }}
Comma list: {{monzo| 47 4 -19 }}


{{Mapping|legend=2| 1 -7 1 | 0 19 4 }}
{{Mapping|legend=3| 1 -7 1 | 0 19 4 }}
: mapping generators: ~2, ~16807/12288
: mapping generators: ~2, ~16807/12288


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Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}
Comma list: 645700815/645657712, {{monzo| 51 -13 -1 -10 }}


Mapping: {{mapping| 1 -7 132 1 | 0 19 -287 4 }}
{{Mapping|legend=0| 1 -7 132 1 | 0 19 -287 4 }}


Optimal tunings:  
Optimal tunings:  
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[[Comma list]]: {{monzo| 109 -67 -1 }}
[[Comma list]]: {{monzo| 109 -67 -1 }}


{{Mapping|legend=2| 1 0 109 | 0 1 -67 }}
{{Mapping|legend=3| 1 0 109 | 0 1 -67 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


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[[Comma list]]: {{monzo| -32 -64 71 }}
[[Comma list]]: {{monzo| -32 -64 71 }}


{{Mapping|legend=2| 1 -36 -32 | 0 71 64 }}
{{Mapping|legend=3| 1 -36 -32 | 0 71 64 }}
: mapping generators: ~3, ~{{monzo| -4 -9 10 }}
: mapping generators: ~3, ~{{monzo| -4 -9 10 }}


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= Rank-3 temperaments =
= Rank-3 temperaments =
== Akjaysmic ==
{{See also| Akjaysma }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: {{monzo| 47 -7 -7 -7 }}
{{Mapping|legend=1| 7 0 0 47 | 0 1 0 -1 | 0 0 1 -1 }}
: mapping generators: ~1157625/1048576, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}
: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}
* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}
: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}
{{Optimal ET sequence|legend=1| 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}
[[Badness]] (Sintel): 2.22
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 184549376/184528125, 199297406/199290375
{{Mapping|legend=0| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}
: mapping generators: ~29160/26411, ~3, ~5
Optimal tunings:
* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}
* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}
{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}
Badness (Sintel): 1.32
== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
== Sasaquinbizo-atriyo (171 & 1106 & 3566) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Badness]] (Sintel): 1.62
[[Badness]] (Sintel): 1.62
== Microorganic ==
[[Subgroup]]: 2.3.7.11
An insanely accurate [[gary]] or no-5 [[superkleismic]] detemperament. It does not have good strong extensions to the 11-limit and beyond. It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]].
[[Comma list]]: [[67110351/67108864]]
{{Mapping|legend=1|1 0 7 7|0 1 1 -2| 0 0 3 -5}}
[[Optimal tuning|Optimal tunings]]:
* [[WE]]: ~2 = 1200.001014{{c}}, ~3/2 = 1901.953889{{c}}, ~539/512 = 88.956151{{c}}
: [[error map]]: ⟨+0.001014  -0.000098  -0.001537  -0.001401]
* [[CWE]]: ~2 = 1200.000000 ¢, ~3/2 =  701.953627¢, ~539/512 =  88.955923 ¢
: error map: ⟨0.000 +0.001374 -0.004512  -0.004808]
{{Optimal ET sequence|legend=1|11, 12e, 26, 41, 94, 135, 566, 701, 836, 877, 1012, 1713, 1848, 4573, 4708, 6556, 11264, 17820}} …
[[Badness]] (Sintel): 0.032
Notable temperaments not appearing in the optimal ET sequence are {{Edos|15, 217, 311}}.
=== Microbiology ===
A weak extension of microorganic adding the [[chalmersia]] and the [[insanisma]], which surfaces as a detemperament of [[vulture]].
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: [[67110351/67108864]], [[123201/123200]], [[5767168/5767125]]
{{Mapping|legend=1|1 0 4 -11 27 15|0 2 3 -1 1 -3| 0 0 5 18 -30 11}}
[[Optimal tuning|Optimal tunings]]:
* [[WE]]: ~2 =  1200.000535{{c}}, ~359424/207515 =  950.976364{{c}}, ~2695/1536 =  973.322574{{c}}
: [[error map]]: ⟨+0.000535 −0.002272 +0.004652 −0.001831 −0.004338 +0.002963]
* [[CWE]]: ~2 = 1200.000000 ¢, ~3/2 =  950.975971 ¢, ~539/512 =  973.322136  ¢
: error map: ⟨0.000000 −0.003059 +0.003520 −0.003432 −0.006047 +0.000931]
{{Optimal ET sequence|legend=1|53, 217, 270, 1551f, 1604, 1821, 1874, 2091, 2144, 2361, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808}} …
Badness (Sintel): 0.349


== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
== Satwethezo-atritribigu (171 & 1547 & 4973)  ==
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