Wizmic microtemperaments: Difference between revisions

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{{Technical data page}}
A wizmic temperament is one which tempers out the wizma, | -6 -8 2 5 > = 420175/419904.
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).


=Qak=
Temperaments discussed elsewhere are:
Commas: 420175/419904, 703125/702464
* [[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]]


POTE generator: ~125/96 = 456.144
Considered below are qak, tokko, and tertiathirds, in the order of increasing [[badness]].  


Map: [<1 27 11 40|, <0 -41 -14 -60|]
== Qak ==
Qak tempers out the [[meter]] in addition to the wizma, and may be described as the {{nowrap| 50 & 171 }} temperament.


Wedgie: <<41 14 60 -73 -20 100||
[[Subgroup]]: 2.3.5.7


EDOs: 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd
[[Comma list]]: 420175/419904, 703125/702464


Badness: 0.0293
{{Mapping|legend=1| 1 -14 -3 -20 | 0 41 14 60 }}
: mapping generators: ~2, ~125/96


=Tokko=
[[Optimal tuning]]s:
Commas: 420175/419904, 5250987/5242880
* [[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 generator: ~147/128 = 238.599
{{Optimal ET sequence|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}


Map: [<1 12 56 -2|, <0 -13 -67 6|]
[[Badness]] (Sintel): 0.741


Wedgie: <<13 67 -6 76 -46 -202||
== Tokko ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Tokko (5-limit)]].''


EDOs: 5, 166, 171, 1544d, 1715d, 1886d, 2057d
Tokko tempers out the [[mitonisma]] in addition to the wizma, and may be described as the {{nowrap| 5 & 171 }} temperament.


Badness: 0.0444
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 420175/419904, 5250987/5242880
 
{{Mapping|legend=1| 1 -1 -11 4 | 0 13 67 -6 }}
: mapping generators: ~2, ~147/128
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd }}
 
[[Badness]] (Sintel): 1.12
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 420175/419904, 29360128/29296875
 
{{Mapping|legend=1| 1 -4 2 -6 | 0 52 3 82 }}
: mapping generators: ~2, ~3375/3136
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c }}
 
[[Badness]] (Sintel): 2.35
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 5632/5625, 117649/117612
 
Mapping: {{mapping| 1 -4 2 -6 -9 | 0 52 3 82 116 }}
 
Optimal tunings:
* WE: ~2 = 1199.9251{{c}}, ~264/245 = 128.8818{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~264/245 = 128.8890{{c}}
 
{{Optimal ET sequence|legend=0| 121, 149, 270 }}
 
Badness (Sintel): 1.10
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224
 
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}}
 
{{Optimal ET sequence|legend=0| 121, 149, 270 }}
 
Badness (Sintel): 0.973
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125
 
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 5 | 0 52 3 82 116 81 66 -7 }}
 
Optimal tunings:
* WE: ~2 = 1199.8964{{c}}, ~14/13 = 128.8793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}
 
{{Optimal ET sequence|legend=0| 121, 149, 270 }}
 
Badness (Sintel): 0.950
 
[[Category:Temperament collections]]
[[Category:Wizmic microtemperaments| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 14:59, 3 June 2026

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.

This is a collection of rank-2 microtemperaments which temper out the wizma (monzo: [-6 -8 2 5⟩, ratio: 420175/419904).

Temperaments discussed elsewhere are:

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 50 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 703125/702464

Mapping: [⟨1 -14 -3 -20], ⟨0 41 14 60]]

mapping generators: ~2, ~125/96

Optimal tunings:

  • WE: ~2 = 1200.0566 ¢, ~125/96 = 456.1655 ¢
error map: ⟨+0.057 +0.039 -0.166 -0.027]
  • CWE: ~2 = 1200.0000 ¢, ~125/96 = 456.1450 ¢
error map: ⟨0.000 -0.009 -0.283 -0.125]

Optimal ET sequence: 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd

Badness (Sintel): 0.741

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 5 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 5250987/5242880

Mapping: [⟨1 -1 -11 4], ⟨0 13 67 -6]]

mapping generators: ~2, ~147/128

Optimal tunings:

  • WE: ~2 = 1200.0850 ¢, ~147/128 = 238.6157 ¢
error map: ⟨+0.085 -0.036 +0.004 -0.180]
  • CWE: ~2 = 1200.0000 ¢, ~147/128 = 238.5997 ¢
error map: ⟨0.000 -0.159 -0.133 -0.424]

Optimal ET sequence: 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd

Badness (Sintel): 1.12

Tertiathirds

Tertiathirds tempers out the quasiorwellisma, 29360128/29296875 in the 7-limit, and may be described as the 121 & 149 temperament. It splits the interval of 5/4 into three 14/13 generators.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 29360128/29296875

Mapping: [⟨1 -4 2 -6], ⟨0 52 3 82]]

mapping generators: ~2, ~3375/3136

Optimal tunings:

  • WE: ~2 = 1199.9305 ¢, ~3375/3136 = 128.8804 ¢
error map: ⟨-0.070 +0.104 +0.188 -0.216]
  • CWE: ~2 = 1200.0000 ¢, ~3375/3136 = 128.8871 ¢
error map: ⟨0.000 +0.173 +0.348 -0.085]

Optimal ET sequence: 121, 149, 270, 2309c, 2579c

Badness (Sintel): 2.35

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 117649/117612

Mapping: [⟨1 -4 2 -6 -9], ⟨0 52 3 82 116]]

Optimal tunings:

  • WE: ~2 = 1199.9251 ¢, ~264/245 = 128.8818 ¢
  • CWE: ~2 = 1200.0000 ¢, ~264/245 = 128.8890 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 1.10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 -4 2 -6 -9 -5], ⟨0 52 3 82 116 81]]

Optimal tunings:

  • WE: ~2 = 1199.9510 ¢, ~14/13 = 128.8849 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 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: [⟨1 -4 2 -6 -9 -5 -3], ⟨0 52 3 82 116 81 66]]

Optimal tunings:

  • WE: ~2 = 1199.8754 ¢, ~14/13 = 128.8778 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8898 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.973

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125

Mapping: [⟨1 -4 2 -6 -9 -5 -3 5], ⟨0 52 3 82 116 81 66 -7]]

Optimal tunings:

  • WE: ~2 = 1199.8964 ¢, ~14/13 = 128.8793 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.950

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