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Wizmic microtemperaments: Difference between revisions

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A wizmic temperament is one which tempers out the wizma, {{monzo|-6 -8 2 5}} = 420175/419904.
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[microtemperament]]s which [[tempering out|temper out]] the [[wizma]] ({{monzo|legend=1| -6 -8 2 5 }}, [[ratio]]: 420175/419904).


Discussed elsewhere are [[Vishnuzmic family #Decavish|decavish]], [[ennealimmal]], [[Horwell temperaments #Fifthplus|fifthplus]], [[Gamelismic clan #Hemiseven|hemiseven]], [[Porwell temperaments #Hendecatonic|hendecatonic]], [[Varunismic temperaments #Octowerck|octowerck]], [[Hemimean clan #Sengagen|sengagen]], [[Hemifamity temperaments #Septiquarter|septiquarter]], [[tsaharuk]], and [[wizard]].
Temperaments discussed elsewhere are:
* [[Superpyth]] (+64/63 or 245/243) → [[Archytas clan #Superpyth|Archytas clan]]
* ''[[Gariwizmic]]'' (+33554432/33480783) → [[Garischismic clan #Gariwizmic|Garischismic clan]]
* ''[[Cloudtone]]'' (+81/80) → [[Meantone family #Cloudtone|Meantone family]]
* [[Tsaharuk]] (+32805/32768) → [[Schismatic family #Tsaharuk|Schismatic family]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125 or 10976/10935) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Witcher]]'' (+1600000/1594323) → [[Amity family #Witcher|Amity family]]
* [[Oolong]] (+126/125) → [[Starling temperaments #Oolong|Starling temperaments]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Fifthplus]]'' (+65625/65536) → [[Horwell temperaments #Fifthplus|Horwell temperaments]]
* ''[[Octowerck]]'' (+321489/320000) → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* ''[[Sengagen]]'' (+3136/3125) → [[Hemimean clan #Sengagen|Hemimean clan]]
* [[Sqrtphi]] (+15625/15552 or 16875/16807) → [[Kleismic family #Sqrtphi|Kleismic family]]
* ''[[Quinwell]]'' (+2109375/2097152) → [[Semicomma family #Quinwell|Semicomma family]]
* ''[[Decavish]]'' (+2202927104/2197265625) → [[Vishnuzmic family #Decavish|Vishnuzmic family]]
 
Considered below are qak, tokko, and tertiathirds, in the order of increasing [[badness]].
 
== Qak ==
Qak tempers out the [[meter]] in addition to the wizma, and may be described as the {{nowrap| 50 & 171 }} temperament.
 
[[Subgroup]]: 2.3.5.7


== Qak  ==
[[Comma list]]: 420175/419904, 703125/702464
[[Comma list]]: 420175/419904, 703125/702464


[[Mapping]]: [{{val|1 -14 -3 -20}}, {{val|0 41 14 60}}]
{{Mapping|legend=1| 1 -14 -3 -20 | 0 41 14 60 }}
: mapping generators: ~2, ~125/96


[[Wedgie]]: {{multival|41 14 60 -73 -20 100}}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0566{{c}}, ~125/96 = 456.1655{{c}}
: [[error map]]: {{val| +0.057 +0.039 -0.166 -0.027 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~125/96 = 456.1450{{c}}
: error map: {{val| 0.000 -0.009 -0.283 -0.125 }}


[[POTE tuning|POTE generator]]: ~125/96 = 456.144
{{Optimal ET sequence|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}


{{Val list|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}
[[Badness]] (Sintel): 0.741


[[Badness]]: 0.029267
== Tokko ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Tokko (5-limit)]].''
 
Tokko tempers out the [[mitonisma]] in addition to the wizma, and may be described as the {{nowrap| 5 & 171 }} temperament.
 
[[Subgroup]]: 2.3.5.7


== Tokko  ==
[[Comma list]]: 420175/419904, 5250987/5242880
[[Comma list]]: 420175/419904, 5250987/5242880


[[Mapping]]: [{{val|1 -1 -11 4}}, {{val|0 13 67 -6}}]
{{Mapping|legend=1| 1 -1 -11 4 | 0 13 67 -6 }}
: mapping generators: ~2, ~147/128


[[Wedgie]]: {{multival|13 67 -6 76 -46 -202}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0850{{c}}, ~147/128 = 238.6157{{c}}
: [[error map]]: {{val| +0.085 -0.036 +0.004 -0.180 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~147/128 = 238.5997{{c}}
: error map: {{val| 0.000 -0.159 -0.133 -0.424 }}


[[POTE tuning|POTE generator]]: ~147/128 = 238.599
{{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd }}


{{Val list|legend=1| 5, 166, 171, 1544d, 1715d, 1886d, 2057d, 2228d, 2399d, 2570d, 2741d, 2912dd, 3083cdd }}
[[Badness]] (Sintel): 1.12


[[Badness]]: 0.044417
== Tertiathirds ==
Tertiathirds tempers out the quasiorwellisma, [[29360128/29296875]] in the 7-limit, and may be described as the {{nowrap| 121 & 149 }} temperament. It splits the interval of [[5/4]] into three [[14/13]] generators.


== Tertiathirds  ==
[[Subgroup]]: 2.3.5.7
The ''tertiathirds'' or ''trienthirds'' temperament (121&270, named by [[User:Xenllium|Xenllium]]) tempers out the quasiorwellisma, 29360128/29296875 in the 7-limit, as well as the wizma. This temperament splits the interval of 5/4 into three [[14/13]] generators.


[[Comma list]]: 420175/419904, 29360128/29296875
[[Comma list]]: 420175/419904, 29360128/29296875


[[Mapping]]: [{{val|1 -4 2 -6}}, {{val|0 52 3 82}}]
{{Mapping|legend=1| 1 -4 2 -6 | 0 52 3 82 }}
: mapping generators: ~2, ~3375/3136


[[Wedgie]]: {{multival|52 3 82 -116 -16 182}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9305{{c}}, ~3375/3136 = 128.8804{{c}}
: [[error map]]: {{val| -0.070 +0.104 +0.188 -0.216 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3375/3136 = 128.8871{{c}}
: error map: {{val| 0.000 +0.173 +0.348 -0.085 }}


[[POTE tuning|POTE generator]]: ~3375/3136 = 128.8879
{{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c }}


{{Val list|legend=1| 121, 149, 270, 2309c, 2579c, 2849c, 3119bc, 3389bc, 3659bcc, 3929bcc, 4199bcc }}
[[Badness]] (Sintel): 2.35


[[Badness]]: 0.093005
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 3025/3024, 5632/5625, 117649/117612
Comma list: 3025/3024, 5632/5625, 117649/117612


Mapping: [{{val|1 -4 2 -6 -9}}, {{val|0 52 3 82 116}}]
Mapping: {{mapping| 1 -4 2 -6 -9 | 0 52 3 82 116 }}


POTE generator: ~264/245 = 128.8898
Optimal tunings:  
* WE: ~2 = 1199.9251{{c}}, ~264/245 = 128.8818{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~264/245 = 128.8890{{c}}


Vals: {{Val list| 121, 149, 270, 2551bc, 2821bc, 3091bc, 3361bc, 3631bbcc, 3901bbccd, 4171bbccd, 4441bbccd, 4711bbccd }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.033322
Badness (Sintel): 1.10
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224


Mapping: [{{val|1 -4 2 -6 -9 -5}}, {{val|0 52 3 82 116 81}}]
Mapping: {{mapping| 1 -4 2 -6 -9 -5 | 0 52 3 82 116 81 }}
 
Optimal tunings:
* WE: ~2 = 1199.9510{{c}}, ~14/13 = 128.8849{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}
 
{{Optimal ET sequence|legend=0| 121, 149, 270 }}
 
Badness (Sintel): 0.806
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224
 
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 | 0 52 3 82 116 81 66 }}
 
Optimal tunings:
* WE: ~2 = 1199.8754{{c}}, ~14/13 = 128.8778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8898{{c}}


POTE generator: ~14/13 = 128.8902
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Vals: {{Val list| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
Badness (Sintel): 0.973


Badness: 0.019494
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


=== 17-limit  ===
Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224


Mapping: [{{val|1 -4 2 -6 -9 -5 -3}}, {{val|0 52 3 82 116 81 66}}]
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 5 | 0 52 3 82 116 81 66 -7 }}


POTE generator: ~14/13 = 128.8912
Optimal tunings:  
* WE: ~2 = 1199.8964{{c}}, ~14/13 = 128.8793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}


Vals: {{Val list| 121, 149, 270 }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.019107
Badness (Sintel): 0.950


[[Category:Temperament]]
[[Category:Temperament collections]]
[[Category:Wizmic microtemperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Wizmic]]
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