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Metric microtemperaments: Difference between revisions

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* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]
* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]


Considered below are luminal, decimetra, stockhausenic, and geb.  
Considered below are geb, stockhausenic, decimetra, and luminal, in the order of increasing [[badness]].  


== Luminal ==
== Geb ==
Luminal may be described as the {{nowrap| 128 & 183 }} temperament. In the [[11-limit]] it extends via [[skadi]], the metric extension tempering out 3025/3024, the [[lehmerisma]], and joins with [[lux]] so that it also tempers out 131072/130977, the [[olympia]]. It was named by [[Flora Canou]] in 2021 following the naming of lux, whose name is the SI unit for illuminance, and lx⋅m<sup>2</sup> is lumen, the SI unit for luminous flux.  
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Geb (5-limit)]].''
 
Geb tempers out the [[scheme comma]] as well as the [[euzenius comma]], and may be described as {{nowrap| 164d & 171 }} temperament. It was named by [[Xenllium]] in 2021 after the earth god of ancient Egyptian religion and the father of Osiris; the latter being the name of a temperament that tempers out the same 5-limit comma as the geb (→ [[Breedsmic temperaments #Osiris]]).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 703125/702464, 14348907/14336000
 
{{Mapping|legend=1| 1 -3 -10 -29 | 0 16 43 111 }}
: mapping generators: ~2, ~8000/6561
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0532{{c}}, ~8000/6561 = 343.8779{{c}}
: [[error map]]: {{val| +0.053 -0.067 -0.094 +0.084 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8000/6561 = 343.8634{{c}}
: error map: {{val| 0.000 -0.141 -0.189 +0.009 }}
 
{{Optimal ET sequence|legend=1| 7d, …, 164d, 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc }}
 
[[Badness]] (Sintel): 1.23
 
== Stockhausenic ==
Stockhausenic may be described as the {{nowrap| 140 & 183 }} temperament, and has a generator chain of ~16/15 (or ~15/8, shown below), relating it to [[25ed5]]. It was named by Xenllium in 2021 after {{w|Karlheinz Stockhausen}} for his experimental use of 25ed5.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 703125/702464, 26843545600/26795786661
[[Comma list]]: 703125/702464, 4096000/4084101


{{Mapping|legend=1| 1 0 5 8 | 0 29 -49 -95 }}
{{Mapping|legend=1| 1 -22 25 40 | 0 26 -25 -41 }}
: mapping generators: ~2, ~8505/8192
: mapping generators: ~2, ~15/8


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9728{{c}}, ~8505/8192 = 65.5868{{c}}
* [[WE]]: ~2 = 1199.8908{{c}}, ~15/8 = 1088.4483{{c}}
: [[error map]]: {{val| -0.027 +0.062 -0.202 +0.212 }}
: [[error map]]: {{val| -0.109 +0.103 -0.250 +0.427 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8505/8192 = 65.5884{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5477{{c}}
: error map: {{val| 0.000 +0.108 -0.145 +0.277 }}
: error map: {{val| 0.000 +0.284 -0.005 +0.720 }}


{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793 }}
{{Optimal ET sequence|legend=1| 43, 97, 140, 183, 323, 786, 1109d, 1432d }}


[[Badness]] (Sintel): 3.41
[[Badness]] (Sintel): 2.25


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 131072/130977, 703125/702464
Comma list: 1375/1372, 5632/5625, 35937/35840


Mapping: {{mapping| 1 0 5 8 1 | 0 29 -49 -95 45 }}
Mapping: {{mapping| 1 -22 25 40 47 | 0 26 -25 -41 -48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9685{{c}}, ~80/77 = 65.5864{{c}}
* WE: ~2 = 1199.9691{{c}}, ~15/8 = 1088.5116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/77 = 65.5882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5398{{c}}


{{Optimal ET sequence|legend=0| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}


Badness (Sintel): 1.08
Badness (Sintel): 1.85


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 3025/3024, 4096/4095, 31250/31213
Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095


Mapping: {{mapping| 1 0 5 8 1 -1 | 0 29 -49 -95 45 86 }}
Mapping: {{mapping| 1 -22 25 40 47 -9 | 0 26 -25 -41 -48 14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9715{{c}}, ~27/26 = 65.5867{{c}}
* WE: ~2 = 1200.0267{{c}}, ~15/8 = 1088.5666{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5883{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5423{{c}}


{{Optimal ET sequence|legend=0| 128, 183, 311, 494, 1299, 1793 }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323 }}


Badness (Sintel): 0.696
Badness (Sintel): 1.24


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095
Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095


Mapping: {{mapping| 1 0 5 8 1 -1 6 | 0 29 -49 -95 45 86 -35 }}
Mapping: {{mapping| 1 -22 25 40 47 -9 44 | 0 26 -25 -41 -48 14 -44 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9941{{c}}, ~27/26 = 65.5875{{c}}
* WE: ~2 = 1200.0454{{c}}, ~15/8 = 1088.5821{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5407{{c}}


{{Optimal ET sequence|legend=0| 128, 183, 311, 494 }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}


Badness (Sintel): 0.748
Badness (Sintel): 0.978


== Decimetra ==
== Decimetra ==
Line 147: Line 169:
Badness (Sintel): 1.12
Badness (Sintel): 1.12


== Stockhausenic ==
== Luminal ==
Stockhausenic may be described as the {{nowrap| 140 & 183 }} temperament, and has a generator chain of ~16/15 (or ~15/8, shown below), relating it to [[25ed5]]. It was named by Xenllium in 2021 after {{w|Karlheinz Stockhausen}} for his experimental use of 25ed5.  
Luminal may be described as the {{nowrap| 128 & 183 }} temperament. In the [[11-limit]] it extends via [[skadi]], the metric extension tempering out 3025/3024, the [[lehmerisma]], and joins with [[lux]] so that it also tempers out 131072/130977, the [[olympia]]. It was named by [[Flora Canou]] in 2021 following the naming of lux, whose name is the SI unit for illuminance, and lx⋅m<sup>2</sup> is lumen, the SI unit for luminous flux.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 703125/702464, 4096000/4084101
[[Comma list]]: 703125/702464, 26843545600/26795786661


{{Mapping|legend=1| 1 -22 25 40 | 0 26 -25 -41 }}
{{Mapping|legend=1| 1 0 5 8 | 0 29 -49 -95 }}
: mapping generators: ~2, ~15/8
: mapping generators: ~2, ~8505/8192


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8908{{c}}, ~15/8 = 1088.4483{{c}}
* [[WE]]: ~2 = 1199.9728{{c}}, ~8505/8192 = 65.5868{{c}}
: [[error map]]: {{val| -0.109 +0.103 -0.250 +0.427 }}
: [[error map]]: {{val| -0.027 +0.062 -0.202 +0.212 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5477{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8505/8192 = 65.5884{{c}}
: error map: {{val| 0.000 +0.284 -0.005 +0.720 }}
: error map: {{val| 0.000 +0.108 -0.145 +0.277 }}


{{Optimal ET sequence|legend=1| 43, 97, 140, 183, 323, 786, 1109d, 1432d }}
{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793 }}


[[Badness]] (Sintel): 2.25
[[Badness]] (Sintel): 3.41


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 5632/5625, 35937/35840
Comma list: 3025/3024, 131072/130977, 703125/702464


Mapping: {{mapping| 1 -22 25 40 47 | 0 26 -25 -41 -48 }}
Mapping: {{mapping| 1 0 5 8 1 | 0 29 -49 -95 45 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9691{{c}}, ~15/8 = 1088.5116{{c}}
* WE: ~2 = 1199.9685{{c}}, ~80/77 = 65.5864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/77 = 65.5882{{c}}


{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}
{{Optimal ET sequence|legend=0| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}


Badness (Sintel): 1.85
Badness (Sintel): 1.08


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095
Comma list: 2080/2079, 3025/3024, 4096/4095, 31250/31213


Mapping: {{mapping| 1 -22 25 40 47 -9 | 0 26 -25 -41 -48 14 }}
Mapping: {{mapping| 1 0 5 8 1 -1 | 0 29 -49 -95 45 86 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0267{{c}}, ~15/8 = 1088.5666{{c}}
* WE: ~2 = 1199.9715{{c}}, ~27/26 = 65.5867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5883{{c}}


{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323 }}
{{Optimal ET sequence|legend=0| 128, 183, 311, 494, 1299, 1793 }}


Badness (Sintel): 1.24
Badness (Sintel): 0.696


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095
Comma list: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095


Mapping: {{mapping| 1 -22 25 40 47 -9 44 | 0 26 -25 -41 -48 14 -44 }}
Mapping: {{mapping| 1 0 5 8 1 -1 6 | 0 29 -49 -95 45 86 -35 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0454{{c}}, ~15/8 = 1088.5821{{c}}
* WE: ~2 = 1199.9941{{c}}, ~27/26 = 65.5875{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5407{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5878{{c}}


{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}
{{Optimal ET sequence|legend=0| 128, 183, 311, 494 }}


Badness (Sintel): 0.978
Badness (Sintel): 0.748
 
== Geb ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Geb (5-limit)]].''
 
Geb tempers out the [[euzenius comma]] as well as the [[scheme comma]], and may be described as {{nowrap| 164d & 171 }} temperament. It was named by [[Xenllium]] in 2021 after the earth god of ancient Egyptian religion and the father of Osiris; the latter being the name of a temperament that tempers out the same 5-limit comma as the geb (→ [[Osiris]]).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 703125/702464, 14348907/14336000
 
{{Mapping|legend=1| 1 -3 -10 -29 | 0 16 43 111 }}
: mapping generators: ~2, ~8000/6561
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0532{{c}}, ~8000/6561 = 343.8779{{c}}
: [[error map]]: {{val| +0.053 -0.067 -0.094 +0.084 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8000/6561 = 343.8634{{c}}
: error map: {{val| 0.000 -0.141 -0.189 +0.009 }}
 
{{Optimal ET sequence|legend=1| 7d, …, 164d, 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc }}
 
[[Badness]] (Sintel): 1.23


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