|
|
| (2 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| {{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]] |
| | * ''[[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) → [[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]] | | * ''[[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 == |
| 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:Wizmic| ]] <!-- key article -->
| |
| [[Category:Rank 2]] | | [[Category:Rank 2]] |
| [[Category:Microtemperaments]]
| |