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This is a collection of microtemperaments 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:  
Temperaments discussed elsewhere are:  
* [[Superpyth]], {64/63, 245/243} → [[Archytas clan #Superpyth|Archytas clan]]
* [[Superpyth]] (+64/63 or 245/243) → [[Archytas clan #Superpyth|Archytas clan]]
* ''[[Cloudtone]]'', {81/80, 16875/16384} → [[Meantone family #Cloudtone|Meantone family]]
* ''[[Gariwizmic]]'' (+33554432/33480783) → [[Garischismic clan #Gariwizmic|Garischismic clan]]
* [[Oolong]], {126/125, 117649/116640} → [[Starling temperaments #Oolong|Starling temperaments]]
* ''[[Cloudtone]]'' (+81/80) → [[Meantone family #Cloudtone|Meantone family]]
* [[Wizard]], {225/224, 118098/117649} → [[Marvel temperaments #Wizard|Marvel temperaments]]
* [[Tsaharuk]] (+32805/32768) → [[Schismatic family #Tsaharuk|Schismatic family]]
* ''[[Hemiseven]]'', {1029/1024, 19683/19600} → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* [[Ennealimmal]], {2401/2400, 4375/4374} → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Sengagen]]'', {3136/3125, 420175/419904} → [[Hemimean clan #Sengagen|Hemimean clan]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125 or 10976/10935) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Septiquarter]]'', {5120/5103, 420175/419904} → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Witcher]]'' (+1600000/1594323) → [[Amity family #Witcher|Amity family]]
* ''[[Sqrtphi]]'', {15625/15552, 16875/16807} → [[Kleismic family #Sqrtphi|Kleismic family]]
* [[Oolong]] (+126/125) → [[Starling temperaments #Oolong|Starling temperaments]]
* [[Tsaharuk]], {32805/32768, 420175/419904} → [[Schismatic family #Tsaharuk|Schismatic family]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Fifthplus]]'', {65625/65536, 420175/419904} → [[Horwell temperaments #Fifthplus|Horwell temperaments]]
* ''[[Fifthplus]]'' (+65625/65536) → [[Horwell temperaments #Fifthplus|Horwell temperaments]]
* ''[[Octowerck]]'', {321489/320000, 420175/419904} → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* ''[[Octowerck]]'' (+321489/320000) → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* ''[[Quinwell]]'', {420175/419904, 2109375/2097152} → [[Semicomma family #Quinwell|Semicomma family]]
* ''[[Sengagen]]'' (+3136/3125) → [[Hemimean clan #Sengagen|Hemimean clan]]
* ''[[Decavish]]'', {420175/419904, 2202927104/2197265625} → [[Vishnuzmic family #Decavish|Vishnuzmic family]]
* [[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, tertiathirds, and witch.  
Considered below are qak, tokko, and tertiathirds, in the order of increasing [[badness]].  


== Qak ==
== Qak ==
Subgroup: 2.3.5.7
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


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


[[Mapping]]: [{{val| 1 27 11 40 }}, {{val| 0 -41 -14 -60 }}]
{{Mapping|legend=1| 1 -14 -3 -20 | 0 41 14 60 }}
: mapping generators: ~2, ~125/96


Mapping generators: ~2, ~192/125
[[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 }}


{{Multival|legend=1| 41 14 60 -73 -20 100 }}
{{Optimal ET sequence|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}


[[POTE generator]]: ~192/125 = 743.856
[[Badness]] (Sintel): 0.741


{{Val list|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}
== Tokko ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Tokko (5-limit)]].''


[[Badness]]: 0.029267
Tokko tempers out the [[mitonisma]] in addition to the wizma, and may be described as the {{nowrap| 5 & 171 }} temperament.  


== Tokko ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val| 1 12 56 -2 }}, {{val| 0 -13 -67 6 }}]
{{Mapping|legend=1| 1 -1 -11 4 | 0 13 67 -6 }}
 
: mapping generators: ~2, ~147/128
Mapping generators: ~2, ~256/147
 
{{Multival|legend=1| 13 67 -6 76 -46 -202 }}


[[POTE generator]]: ~256/147 = 961.401
[[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 }}


{{Val list|legend=1| 5, 161c, 166, 171, 1544d, 1715d, 1886d, 2057d, 2228d, 2399d, 2570d, 2741d, 2912dd, 3083cdd }}
{{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd }}


[[Badness]]: 0.044417
[[Badness]] (Sintel): 1.12


== Tertiathirds ==
== Tertiathirds ==
The ''tertiathirds'' temperament (121&270) 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.
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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 48 5 76 }}, {{val| 0 -52 -3 -82 }}]
{{Mapping|legend=1| 1 -4 2 -6 | 0 52 3 82 }}
 
: mapping generators: ~2, ~3375/3136
Mapping generators: ~2, ~6272/3375
 
{{Multival|legend=1| 52 3 82 -116 -16 182 }}


[[POTE generator]]: ~6272/3375 = 1071.1121
[[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 }}


{{Val list|legend=1| 121, 149, 270, 2309c, 2579c, 2849c, 3119bc, 3389bc, 3659bcc, 3929bcc, 4199bcc }}
{{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c }}


[[Badness]]: 0.093005
[[Badness]] (Sintel): 2.35


=== 11-limit ===
=== 11-limit ===
Line 78: Line 90:
Comma list: 3025/3024, 5632/5625, 117649/117612
Comma list: 3025/3024, 5632/5625, 117649/117612


Mapping: [{{val| 1 48 5 76 107 }}, {{val| 0 -52 -3 -82 -116 }}]
Mapping: {{mapping| 1 -4 2 -6 -9 | 0 52 3 82 116 }}


POTE generator: ~245/132 = 1071.1102
Optimal tunings:  
* WE: ~2 = 1199.9251{{c}}, ~264/245 = 128.8818{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~264/245 = 128.8890{{c}}


Optimal GPV sequence: {{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 ===
=== 13-limit ===
Line 91: Line 105:
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224


Mapping: [{{val| 1 48 5 76 107 76 }}, {{val| 0 -52 -3 -82 -116 -81 }}]
Mapping: {{mapping| 1 -4 2 -6 -9 -5 | 0 52 3 82 116 81 }}


POTE generator: ~13/7 = 1071.1098
Optimal tunings:  
* WE: ~2 = 1199.9510{{c}}, ~14/13 = 128.8849{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}


Optimal GPV sequence: {{Val list| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.019494
Badness (Sintel): 0.806


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224
Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224
 
Mapping: [{{val| 1 48 5 76 107 76 63 }}, {{val| 0 -52 -3 -82 -116 -81 -66 }}]
 
POTE generator: ~13/7 = 1071.1088
 
Optimal GPV sequence: {{Val list| 121, 149, 270 }}
 
Badness: 0.019107
 
== Witch ==
{{see also|Amity family}}
 
The witch temperament (99&166) tempers out wizma (420175/419904) and the amity comma (1600000/1594323) as well as the ''hemfiness'' comma (4096000/4084101, saquinru-atriyo).
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 420175/419904, 1600000/1594323
 
[[Mapping]]: [{{val| 1 18 45 12 }}, {{val| 0 -25 -65 -14 }}]
 
Mapping generators: ~2, ~63/40
 
{{Multival|legend=1| 25 65 14 45 -48 -150 }}
 
[[POTE generator]]: ~63/40 = 787.902
 
{{Val list|legend=1| 99, 265, 364, 463, 562, 1025 }}
 
[[Badness]]: 0.080608
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4356, 1240029/1239040
 
Mapping: [{{val| 1 18 45 12 77 }}, {{val| 0 -25 -65 -14 -112 }}]
 
POTE generator: ~63/40 = 787.921
 
Optimal GPV sequence: {{Val list| 99e, 166, 265 }}
 
Badness: 0.082607
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4375/4356, 10648/10647


Mapping: [{{val| 1 18 45 12 77 93 }}, {{val| 0 -25 -65 -14 -112 -136 }}]
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 | 0 52 3 82 116 81 66 }}


POTE generator: ~52/33 = 787.930
Optimal tunings:  
* WE: ~2 = 1199.8754{{c}}, ~14/13 = 128.8778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8898{{c}}


Optimal GPV sequence: {{Val list| 99ef, 166, 265, 431c, 696cc }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.057980
Badness (Sintel): 0.973


=== Semiwitch ===
=== 19-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 9801/9800, 41503/41472, 585640/583443
 
Mapping: [{{val| 2 11 25 10 16 }}, {{val| 0 -25 -65 -14 -29 }}]
 
Mapping generators: ~99/70, ~49/44
 
POTE generator: ~49/44 = 187.904
 
Optimal GPV sequence: {{Val list| 166, 198, 364, 562, 926, 1488bd, 2414bdd }}
 
Badness: 0.088189
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280
 
Mapping: [{{val| 2 11 25 10 16 24 }}, {{val| 0 -25 -65 -14 -29 -53 }}]
 
POTE generator: ~39/35 = 187.904
 
Optimal GPV sequence: {{Val list| 166, 198, 364, 562, 926 }}
 
Badness: 0.038009
 
=== Hitch ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 117649/117612, 1600000/1594323
 
Mapping: [{{val| 1 18 45 12 -10 }}, {{val| 0 -50 -130 -28 41 }}]
 
Mapping generators: ~2, ~1120/891
 
POTE generator: ~1120/891 = 393.950
 
Optimal GPV sequence: {{Val list| 198, 463, 661, 1124bde }}
 
Badness: 0.093482
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323
 
Mapping: [{{val| 1 18 45 12 -10 71 }}, {{val| 0 -50 -130 -28 41 -205 }}]
 
POTE generator: ~1120/891 = 393.948
 
Optimal GPV sequence: {{Val list| 198, 463, 661 }}
 
Badness: 0.060716
 
==== Hitcher ====
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561
Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125


Mapping: [{{val| 1 18 45 12 -10 6 }}, {{val| 0 -50 -130 -28 41 -7 }}]
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 5 | 0 52 3 82 116 81 66 -7 }}


POTE generator: ~49/39 = 393.949
Optimal tunings:  
* WE: ~2 = 1199.8964{{c}}, ~14/13 = 128.8793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}


Optimal GPV sequence: {{Val list| 198, 463f, 661ff }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.059591
Badness (Sintel): 0.950


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