Wizmic microtemperaments: Difference between revisions

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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]]
* [[Superpyth]] (+64/63 or 245/243) → [[Archytas clan #Superpyth|Archytas clan]]
* ''[[Cloudtone]]'', {81/80, 16875/16384} → [[Meantone family #Cloudtone]]
* ''[[Gariwizmic]]'' (+33554432/33480783) → [[Garischismic clan #Gariwizmic|Garischismic clan]]
* [[Oolong]], {126/125, 117649/116640} → [[Starling temperaments #Oolong]]
* ''[[Cloudtone]]'' (+81/80) → [[Meantone family #Cloudtone|Meantone family]]
* [[Wizard]], {225/224, 118098/117649} → [[Marvel temperaments #Wizard]]
* [[Tsaharuk]] (+32805/32768) → [[Schismatic family #Tsaharuk|Schismatic family]]
* ''[[Hemiseven]]'', {1029/1024, 19683/19600} → [[Gamelismic clan #Hemiseven]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* [[Ennealimmal]], {2401/2400, 4375/4374} → [[Ragismic microtemperaments #Ennealimmal]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Sengagen]]'', {3136/3125, 420175/419904} → [[Hemimean clan #Sengagen]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125 or 10976/10935) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Septiquarter]]'', {5120/5103, 420175/419904} → [[Hemifamity temperaments #Septiquarter]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic]]
* ''[[Witcher]]'' (+1600000/1594323) → [[Amity family #Witcher|Amity family]]
* ''[[Sqrtphi]]'', {15625/15552, 16875/16807} → [[Kleismic family #Sqrtphi]]
* [[Oolong]] (+126/125) → [[Starling temperaments #Oolong|Starling temperaments]]
* [[Tsaharuk]], {32805/32768, 420175/419904} → [[Schismatic family #Tsaharuk]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Fifthplus]]'', {65625/65536, 420175/419904} → [[Horwell temperaments #Fifthplus]]
* ''[[Fifthplus]]'' (+65625/65536) → [[Horwell temperaments #Fifthplus|Horwell temperaments]]
* ''[[Octowerck]]'', {321489/320000, 420175/419904} → [[Varunismic temperaments #Octowerck]]
* ''[[Octowerck]]'' (+321489/320000) → [[Varunismic temperaments #Octowerck|Varunismic temperaments]]
* ''[[Decavish]]'', {420175/419904, 2202927104/2197265625} → [[Vishnuzmic family #Decavish]]
* ''[[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 ==
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 -14 -3 -20 }}, {{val| 0 41 14 60 }}]
{{Mapping|legend=1| 1 -14 -3 -20 | 0 41 14 60 }}
: mapping generators: ~2, ~125/96
 
[[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]]: ~125/96 = 456.144
[[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 -1 -11 4 }}, {{val| 0 13 67 -6 }}]
{{Mapping|legend=1| 1 -1 -11 4 | 0 13 67 -6 }}
 
: mapping generators: ~2, ~147/128
{{Multival|legend=1| 13 67 -6 76 -46 -202 }}


[[POTE generator]]: ~147/128 = 238.599
[[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 -4 2 -6 }}, {{val| 0 52 3 82 }}]
{{Mapping|legend=1| 1 -4 2 -6 | 0 52 3 82 }}
 
: mapping generators: ~2, ~3375/3136
{{Multival|legend=1| 52 3 82 -116 -16 182 }}


[[POTE generator]]: ~3375/3136 = 128.8879
[[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 69: Line 90:
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 ===
=== 13-limit ===
Line 82: 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 -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 }}


POTE generator: ~14/13 = 128.8902
Optimal tunings:  
* WE: ~2 = 1199.9510{{c}}, ~14/13 = 128.8849{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}


Vals: {{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 -4 2 -6 -9 -5 -3 }}, {{val| 0 52 3 82 116 81 66 }}]
 
POTE generator: ~14/13 = 128.8912
 
Vals: {{Val list| 121, 149, 270 }}


Badness: 0.019107
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 | 0 52 3 82 116 81 66 }}


== Witch ==
Optimal tunings:
{{see also|Amity family}}
* WE: ~2 = 1199.8754{{c}}, ~14/13 = 128.8778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8898{{c}}


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).
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Subgroup: 2.3.5.7
Badness (Sintel): 0.973


[[Comma list]]: 420175/419904, 1600000/1594323
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
[[Mapping]]: [{{val| 1 -7 -20 -2 }}, {{val| 0 25 65 14 }}
 
{{Multival|legend=1| 25 65 14 45 -48 -150 }}
 
[[POTE generator]]: ~80/63 = 412.098
 
{{Val list|legend=1| 99, 265, 364, 463, 562, 1025 }}
 
[[Badness]]: 0.080608
 
=== Semiwitch ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 41503/41472, 585640/583443
 
Mapping: [{{val|2 11 25 10 16}}, {{val|0 -25 -65 -14 -29}}]
 
POTE generator: ~49/44 = 187.904
 
Vals: {{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
 
Vals: {{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}}]
 
POTE generator: ~1120/891 = 393.950
 
Vals: {{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
 
Vals: {{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}}


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


Badness: 0.059591
Badness (Sintel): 0.950


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Wizmic microtemperaments| ]] <!-- main article -->
[[Category:Wizmic microtemperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Microtemperament]]

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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