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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|-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).


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]]
* [[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) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[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]]'' (+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]]
* ''[[Witcher]]'' (+1600000/1594323) → [[Amity family #Witcher|Amity family]]
* ''[[Fifthplus]]'' (+5625/65536) → [[Horwell temperaments #Fifthplus|Horwell temperaments]]
* [[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]]
* ''[[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]]


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


== Qak ==
== Qak ==
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
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 27 11 40 | 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]] ([[POTE]]): ~2 = 1\1, ~125/96 = 456.144
[[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 }}


{{Optimal ET sequence|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}
{{Optimal ET sequence|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}


[[Badness]]: 0.029267
[[Badness]] (Sintel): 0.741


== Tokko ==
== Tokko ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #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 {{nowrap| 5 & 171 }} temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 43: Line 52:
[[Comma list]]: 420175/419904, 5250987/5242880
[[Comma list]]: 420175/419904, 5250987/5242880


{{Mapping|legend=1| 1 12 56 -2 | 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
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 238.599
{{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd }}


{{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, 1886d, 2057d, 2228d, 2399d, 2570d, 2741d, 2912dd, 3083cdd }}
[[Badness]] (Sintel): 1.12
 
[[Badness]]: 0.044417


== 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
Line 60: Line 72:
[[Comma list]]: 420175/419904, 29360128/29296875
[[Comma list]]: 420175/419904, 29360128/29296875


{{Mapping|legend=1| 1 48 5 76 | 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
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~3375/3136 = 128.888
{{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c }}


{{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c, 2849c, 3119bc, 3389bc, 3659bcc, 3929bcc, 4199bcc }}
[[Badness]] (Sintel): 2.35
 
[[Badness]]: 0.093005


=== 11-limit ===
=== 11-limit ===
Line 75: Line 90:
Comma list: 3025/3024, 5632/5625, 117649/117612
Comma list: 3025/3024, 5632/5625, 117649/117612


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


Optimal tuning (POTE): ~2 = 1\1, ~264/245 = 128.890
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=1| 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 88: Line 105:
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224


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


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.890
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=1| 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 ===
Line 101: Line 120:
Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224
Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224


Mapping: {{mapping| 1 48 5 76 107 76 63 | 0 -52 -3 -82 -116 -81 -66 }}
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 | 0 52 3 82 116 81 66 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.891
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=1| 121, 149, 270 }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.019107
Badness (Sintel): 0.973


=== 19-limit ===
=== 19-limit ===
Line 114: Line 135:
Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125
Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125


Mapping: {{mapping| 1 48 5 76 107 76 63 -2 | 0 -52 -3 -82 -116 -81 -66 7 }}
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 5 | 0 52 3 82 116 81 66 -7 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.890
 
{{Optimal ET sequence|legend=1| 121, 149, 270 }}
 
Badness: 0.015621
 
== Witcher ==
{{See also| Amity family }}
 
The witcher 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|legend=1| 1 18 45 12 | 0 -25 -65 -14 }}
 
: mapping generators: ~2, ~63/40
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~80/63 = 412.098
 
{{Optimal ET sequence|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: {{mapping| 1 18 45 12 77 | 0 -25 -65 -14 -112 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~80/63 = 412.079
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 18 45 12 77 93 | 0 -25 -65 -14 -112 -136 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~33/26 = 412.070
 
{{Optimal ET sequence|legend=1| 99ef, 166, 265, 431c, 696cc }}
 
Badness: 0.057980
 
=== Semiwitcher ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 41503/41472, 585640/583443
 
Mapping: {{mapping| 2 11 25 10 16 | 0 -25 -65 -14 -29 }}
 
: mapping generators: ~99/70, ~49/44
 
Optimal tuning (POTE): ~99/70 = 1\2, ~49/44 = 187.904
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 11 25 10 16 24 | 0 -25 -65 -14 -29 -53 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~39/35 = 187.904
 
{{Optimal ET sequence|legend=1| 166, 198, 364, 562, 926 }}
 
Badness: 0.038009
 
=== Hitch ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 117649/117612, 1600000/1594323
 
Mapping: {{mapping| 1 18 45 12 -10 | 0 -50 -130 -28 41 }}
 
: mapping generators: ~2, ~1120/891
 
Optimal tuning (POTE): ~2 = 1\1, ~1120/891 = 393.950
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 18 45 12 -10 71 | 0 -50 -130 -28 41 -205 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~1120/891 = 393.948
 
{{Optimal ET sequence|legend=1| 198, 463, 661 }}
 
Badness: 0.060716
 
==== Hitcher ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561
 
Mapping: {{mapping| 1 18 45 12 -10 6 | 0 -50 -130 -28 41 -7 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/39 = 393.949
 
{{Optimal ET sequence|legend=1| 198, 463f, 661ff }}
 
Badness: 0.059591
 
== Gariwizmic ==
Gariwizmic (94&176) tempers out the wizma (420175/419904) and the garischisma (33554432/33480783). It assumes a semioctave period and a perfect fifth generator that is slightly sharp of just. Unlike schismic, it finds 5/4 at the ''trans''diminished fourth (+45 fifths) minus a half pythagorean comma, or equivalently, at the ''trans''diminished octave (+39 fifths) minus a semioctave.
 
[[Subgroup]]: 2.3.5.7
 
Comma list: 420175/419904, [[33554432/33480783]]
 
{{Mapping|legend=1| 2 2 -41 22|0 1 39 -14}}
 
* mapping generators: ~46305/32768, ~3
[[Optimal tuning|Optimal tunings]]:
 
* [[WE]]: ~2 = 599.996{{c}}, ~3/2 = 702.177{{c}}
 
: [[error map]]: {{val|-0.069, 0.153, -0.021, -0.053}}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}
 
: error map: {{val|0.000 0.261 0.114 0.149}}
 
{{Optimal ET sequence|legend=1|94, 176, 270, 364, 446, 634, 904}}
 
[[Badness]]: 2.225 (Sintel)
 
=== Cassawizmic ===
Subgroup: 2.3.5.7.11
 
Comma list: [[9801/9800]], [[19712/19683]], [[41503/41472]]
 
{{Mapping|legend=1| 2 2 -41 22 -20|0 1 39 -14 23}}
 
* mapping generators: ~99/70, ~3
 
* WE: ~99/70 = 599.979, ~3/2 = 702.194
: error map: {{val|-0.042 0.197 0.106 -0.001 -0.440}}
* CWE: ~99/70 = 1\2, ~3/2 = 702.218
: error map: {{val|0.000 0.263 0.185 0.123 -0.306}}
 
{{Optimal ET sequence|legend=0|94, 176, 270, 364, 446, 634, 904}}
 
[[Badness]]: 1.008 (Sintel)
 
==== 13-limit ====
Comma list: [[2080/2079]], [[1716/1715]], [[4096/4095]], [[35035/34992]]
 
{{Mapping|legend=1| 2 2 -41 22 -20 39|0 1 39 -14 23 -27}}
 
* mapping generators: ~99/70, ~3
 
[[Optimal tuning|Optimal tunings]]:
* WE: ~99/70 = 599.996, ~3/2 = 702.210
: error map: {{val|-0.008 0.246 0.035 0.146 -0.412 -0.353}}
* CWE: ~99/70 = 1\2, ~3/2 = 702.215
: error map: {{val|-0.000 0.260 0.054 0.170 -0.383 -0.321}}
 
{{Optimal ET sequence|legend=0|94, 176, 270, 364, 446, 634, 904}}
 
[[Badness]]: 0.822 (Sintel)
 
==== Subgroup: 2.3.5.7.11.13.19 ====
Comma list: [[2080/2079]], [[1540/1539]], [[1216/1215]], [[1716/1715]], [[1729/1728]]
 
{{Mapping|legend=1| 2 2 -41 22 -20 39 -43|0 1 39 -14 23 -27 44}}
 
* mapping generators: ~99/70, ~3


[[Optimal tuning|Optimal tunings]]:
Optimal tunings:  
* WE: ~99/70 = 599.996, ~3/2 = 702.210
* WE: ~2 = 1199.8964{{c}}, ~14/13 = 128.8793{{c}}
: error map: {{val|-0.006 0.250 0.057 0.147 -0.394 -0.355 -0.079}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}
* CWE: ~99/70 = 1\2, ~3/2 = 702.215
: 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, 364, 446, 634, 904}}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


[[Badness]]: 0.655 (Sintel)
Badness (Sintel): 0.950


[[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]]
Retrieved from "https://en.xen.wiki/w/Wizmic_microtemperaments"