Wizmic microtemperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Sorting. Fix grammar. - redundant categories
- witcher (more properly addressed in amity family)
 
(One intermediate revision by the same user not shown)
Line 4: Line 4:
Temperaments 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]]
* ''[[Gariwizmic]]'' (+33554432/33480783) → [[Garischismic clan #Gariwizmic|Garischismic clan]]
* ''[[Cloudtone]]'' (+81/80) → [[Meantone family #Cloudtone|Meantone family]]
* ''[[Cloudtone]]'' (+81/80) → [[Meantone family #Cloudtone|Meantone family]]
* [[Tsaharuk]] (+32805/32768) → [[Schismatic family #Tsaharuk|Schismatic family]]
* [[Tsaharuk]] (+32805/32768) → [[Schismatic family #Tsaharuk|Schismatic family]]
Line 10: Line 11:
* ''[[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]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel 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]]
* [[Oolong]] (+126/125) → [[Starling temperaments #Oolong|Starling temperaments]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
Line 19: Line 21:
* ''[[Decavish]]'' (+2202927104/2197265625) → [[Vishnuzmic family #Decavish|Vishnuzmic family]]
* ''[[Decavish]]'' (+2202927104/2197265625) → [[Vishnuzmic family #Decavish|Vishnuzmic family]]


Considered below are qak, tokko, tertiathirds, witcher, and gariwizmic.  
Considered below are qak, tokko, and tertiathirds, in the order of increasing [[badness]].  


== Qak ==
== Qak ==
Line 142: Line 144:


Badness (Sintel): 0.950
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 {{nowrap| 99 & 166 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 420175/419904, 1600000/1594323
{{Mapping|legend=1| 1 -7 -20 -2 | 0 25 65 14 }}
: mapping generators: ~2, ~80/63
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8698{{c}}, ~80/63 = 412.0531{{c}}
: [[error map]]: {{val| -0.130 +0.283 -0.260 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/63 = 412.0956{{c}}
: error map: {{val| 0.000 +0.435 -0.100 +0.512 }}
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -7 -20 -2 -35 | 0 25 65 14 112 }}
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~80/63 = 412.0758{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/63 = 412.0788{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -7 -20 -2 -35 -43 | 0 25 65 14 112 136 }}
Optimal tunings:
* WE: ~2 = 1200.0660{{c}}, ~33/26 = 412.0928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/26 = 412.0709{{c}}
{{Optimal ET sequence|legend=0| 99ef, 166, 265 }}
Badness (Sintel): 2.40
=== Semiwitcher ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 41503/41472, 585640/583443
Mapping: {{mapping| 2 -14 -40 -4 -13 | 0 25 65 14 29 }}
: mapping generators: ~99/70, ~80/63
Optimal tunings:
* WE: ~99/70 = 599.9717{{c}}, ~80/63 = 412.0766{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~80/63 = 412.0951{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 11 25 10 16 24 | 0 25 65 14 29 53 }}
Optimal tunings:
* WE: ~99/70 = 599.9666{{c}}, ~33/26 = 412.0726{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/26 = 412.0944{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -32 -85 -16 31 | 0 50 130 28 -41 }}
: mapping generators: ~2, ~891/560
Optimal tunings:
* WE: ~2 = 1199.8455{{c}}, ~891/560 = 805.9457{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0488{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -32 -85 -16 31 -134 | 0 50 130 28 -41 205 }}
Optimal tunings:
* WE: ~2 = 1199.8415{{c}}, ~891/560 = 805.9451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0508{{c}}
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -32 -85 -16 31 -1 | 0 50 130 28 -41 7 }}
Optimal tunings:
* WE: ~2 = 1199.7690{{c}}, ~78/49 = 805.8958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/49 = 806.0500{{c}}
{{Optimal ET sequence|legend=0| 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 {{nowrap| 94 & 176 }} temperament. It assumes a [[semioctave]] period and a [[3/2|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|legend=1| 2 0 -119 50 | 0 1 39 -14 }}
: mapping generators: ~46305/32768, ~3
[[Optimal tuning]]s:
* [[WE]]: ~46305/32768 = 599.9657{{c}}, ~3/2 = 702.1765{{c}}
: [[error map]]: {{val|-0.069 +0.153 -0.021 -0.053 }}
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~3/2 = 702.2161{{c}}
: error map: {{val| 0.000 +0.261 +0.114 +0.149 }}
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 0 -119 50 -66 | 0 1 39 -14 23 }}
Optimal tunings:
* WE: ~99/70 = 599.9790{{c}}, ~3/2 = 702.1938{{c}}
: error map: {{val| -0.042 +0.197 +0.106 -0.001 -0.440 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2179{{c}}
: error map: {{val| 0.000 +0.263 +0.185 +0.123 -0.306 }}
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 -119 50 -66 93 | 0 1 39 -14 23 -27 }}
Optimal tunings:
* WE: ~99/70 = 599.9958{{c}}, ~3/2 = 702.2096{{c}}
: error map: {{val| -0.008 +0.246 +0.035 +0.146 -0.412 -0.353 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2145{{c}}
: error map: {{val| 0.000 +0.260 +0.054 +0.170 -0.383 -0.321 }}
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 -119 50 -66 93 -131 | 0 1 39 -14 23 -27 44 }}
Optimal tunings:
* WE: ~99/70 = 599.9969{{c}}, ~3/2 = 702.2114{{c}}
: error map: {{val| -0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2150{{c}}
: error map: {{val| 0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055 }}
{{Optimal ET sequence|legend=0| 94, 176, 270, 634, 904, 1174 }}
Badness (Sintel): 0.655


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

Retrieved from "https://en.xen.wiki/index.php?title=Wizmic_microtemperaments&oldid=231515"