Wizmic microtemperaments: Difference between revisions

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


Discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Superpyth]] (+64/63 or 245/243) → [[Archytas clan #Superpyth|Archytas clan]]
* [[Superpyth]] (+64/63 or 245/243) → [[Archytas clan #Superpyth|Archytas clan]]
* ''[[Cloudtone]]'' (+81/80) → [[Meantone family #Cloudtone|Meantone family]]
* ''[[Cloudtone]]'' (+81/80) → [[Meantone family #Cloudtone|Meantone family]]
* [[Oolong]] (+126/125) → [[Starling temperaments #Oolong|Starling temperaments]]
* [[Tsaharuk]] (+32805/32768) → [[Schismatic family #Tsaharuk|Schismatic family]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Sengagen]]'' (+3136/3125) → [[Hemimean clan #Sengagen|Hemimean clan]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125 or 10976/10935) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125 or 10976/10935) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Sqrtphi]]'' (+15625/15552 or 16875/16807) → [[Kleismic family #Sqrtphi|Kleismic family]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* [[Tsaharuk]] (+32805/32768) → [[Schismatic family #Tsaharuk|Schismatic family]]
* [[Oolong]] (+126/125) → [[Starling temperaments #Oolong|Starling temperaments]]
* ''[[Fifthplus]]'' (+5625/65536) → [[Horwell temperaments #Fifthplus|Horwell 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]]
* ''[[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]]
* ''[[Quinwell]]'' (+2109375/2097152) → [[Semicomma family #Quinwell|Semicomma family]]
* ''[[Decavish]]'' (+2202927104/2197265625) → [[Vishnuzmic family #Decavish|Vishnuzmic family]]
* ''[[Decavish]]'' (+2202927104/2197265625) → [[Vishnuzmic family #Decavish|Vishnuzmic family]]
Line 345: Line 345:
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Wizmic microtemperaments| ]] <!-- main article -->
[[Category:Wizmic microtemperaments| ]] <!-- main article -->
[[Category:Wizmic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Microtemperaments]]

Revision as of 19:14, 11 May 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, tertiathirds, witcher, and gariwizmic.

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

Witcher

See also: Amity family

Witcher tempers out the wizma (420175/419904) and the amity comma (1600000/1594323). It may be described as the 99 & 166 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 1600000/1594323

Mapping: [⟨1 -7 -20 -2], ⟨0 25 65 14]]

mapping generators: ~2, ~80/63

Optimal tunings:

  • WE: ~2 = 1199.8698 ¢, ~80/63 = 412.0531 ¢
error map: ⟨-0.130 +0.283 -0.260 +0.178]
  • CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0956 ¢
error map: ⟨0.000 +0.435 -0.100 +0.512]

Optimal ET sequence: 32c, 67c, 99, 265, 364, 463, 562, 1025

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4356, 1240029/1239040

Mapping: [⟨1 -7 -20 -2 -35], ⟨0 25 65 14 112]]

Optimal tunings:

  • WE: ~2 = 1199.9909 ¢, ~80/63 = 412.0758 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0788 ¢

Optimal ET sequence: 99e, 166, 265

Badness (Sintel): 2.73

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4375/4356, 10648/10647

Mapping: [⟨1 -7 -20 -2 -35 -43], ⟨0 25 65 14 112 136]]

Optimal tunings:

  • WE: ~2 = 1200.0660 ¢, ~33/26 = 412.0928 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/26 = 412.0709 ¢

Optimal ET sequence: 99ef, 166, 265

Badness (Sintel): 2.40

Semiwitcher

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 41503/41472, 585640/583443

Mapping: [⟨2 -14 -40 -4 -13], ⟨0 25 65 14 29]]

mapping generators: ~99/70, ~80/63

Optimal tunings:

  • WE: ~99/70 = 599.9717 ¢, ~80/63 = 412.0766 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~80/63 = 412.0951 ¢

Optimal ET sequence: 32c, 166, 198, 364, 562, 926

Badness (Sintel): 2.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280

Mapping: [⟨2 11 25 10 16 24], ⟨0 25 65 14 29 53]]

Optimal tunings:

  • WE: ~99/70 = 599.9666 ¢, ~33/26 = 412.0726 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/26 = 412.0944 ¢

Optimal ET sequence: 32cf, 166, 198, 364, 562, 926

Badness (Sintel): 1.57

Hitch

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 117649/117612, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31], ⟨0 50 130 28 -41]]

mapping generators: ~2, ~891/560

Optimal tunings:

  • WE: ~2 = 1199.8455 ¢, ~891/560 = 805.9457 ¢
  • CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0488 ¢

Optimal ET sequence: 67c, 198, 463, 661, 1124bde

Badness (Sintel): 3.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31 -134], ⟨0 50 130 28 -41 205]]

Optimal tunings:

  • WE: ~2 = 1199.8415 ¢, ~891/560 = 805.9451 ¢
  • CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0508 ¢

Optimal ET sequence: 198, 463, 661

Badness (Sintel): 2.51

Hitcher

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561

Mapping: [⟨1 -32 -85 -16 31 -1], ⟨0 50 130 28 -41 7]]

Optimal tunings:

  • WE: ~2 = 1199.7690 ¢, ~78/49 = 805.8958 ¢
  • CWE: ~2 = 1200.0000 ¢, ~78/49 = 806.0500 ¢

Optimal ET sequence: 67c, 198, 463f, 661ff

Badness (Sintel): 2.46

Gariwizmic

Gariwizmic tempers out the wizma (420175/419904) and the garischisma (33554432/33480783), and may be described as the 94 & 176 temperament. It assumes a semioctave period and a perfect fifth generator that is slightly sharp of just. It finds 5/4 39 fifths away, shifted by a semioctave. It extends extremely well to the 2.3.5.7.11.13.19 subgroup.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 33554432/33480783

Mapping: [⟨2 0 -119 50], ⟨0 1 39 -14]]

mapping generators: ~46305/32768, ~3

Optimal tunings:

  • WE: ~46305/32768 = 599.9657 ¢, ~3/2 = 702.1765 ¢
error map: ⟨-0.069 +0.153 -0.021 -0.053]
  • CWE: ~46305/32768 = 600.0000 ¢, ~3/2 = 702.2161 ¢
error map: ⟨0.000 +0.261 +0.114 +0.149]

Optimal ET sequence: 94, 176, 270, 904, 1174, 1444, 1714, 3158b, 4872bbcd

Badness (Sintel): 2.22

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 19712/19683, 41503/41472

Mapping: [⟨2 0 -119 50 -66], ⟨0 1 39 -14 23]]

Optimal tunings:

  • WE: ~99/70 = 599.9790 ¢, ~3/2 = 702.1938 ¢
error map: ⟨-0.042 +0.197 +0.106 -0.001 -0.440]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2179 ¢
error map: ⟨0.000 +0.263 +0.185 +0.123 -0.306]

Optimal ET sequence: 94, 176, 270, 1174, 1444, 1714, 1984e

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683

Mapping: [⟨2 0 -119 50 -66 93], ⟨0 1 39 -14 23 -27]]

Optimal tunings:

  • WE: ~99/70 = 599.9958 ¢, ~3/2 = 702.2096 ¢
error map: ⟨-0.008 +0.246 +0.035 +0.146 -0.412 -0.353]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2145 ¢
error map: ⟨0.000 +0.260 +0.054 +0.170 -0.383 -0.321]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.822

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079

Mapping: [⟨2 0 -119 50 -66 93 -131], ⟨0 1 39 -14 23 -27 44]]

Optimal tunings:

  • WE: ~99/70 = 599.9969 ¢, ~3/2 = 702.2114 ¢
error map: ⟨-0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2150 ¢
error map: ⟨0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.655

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