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{{Technical data page}}
{{Technical data page}}
These are [[rank-2 temperament|rank-2]] [[temperament]]s that [[tempering out|temper out]] the [[meter]], 703125/702464 = {{monzo| -11 2 7 -3 }}.
This is a collection of [[rank-2 temperament|rank-2]] '''metric microtemperaments''', which [[tempering out|temper out]] the [[meter]] ({{monzo|legend=1| -11 2 7 -3 }}, [[ratio]]: 703125/702464).


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Term (temperament)|Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Undim]]'' (+5120/5103) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quintupole]]'' (+4000/3969) → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Trinity]]'' (+1600000/1594323) → [[Amity family #Trinity|Amity family]]
* ''[[Marfifths]]'' (+10976/10935) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Enneadecal]]'' (+4375/4374) → [[Ragismic microtemperaments #Enneadecal|Ragismic microtemperaments]]
* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]
* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[Quintupole]]'' (+4000/3969) → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Enneadecal]]'' (+4375/4374) → [[Ragismic microtemperaments #Enneadecal|Ragismic microtemperaments]]
* ''[[Undim]]'' (+5120/5103) → [[Undim family #Septimal undim|Undim family]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Marfifths]]'' (+10976/10935) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]
* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]
* ''[[Trinity]]'' (+1600000/1594323) → [[Amity family #Trinity|Amity family]]


== Luminal ==
Considered below are geb, stockhausenic, decimetra, and luminal, in the order of increasing [[badness]].
Luminal is the intersection of [[Meter|metric]] and [[lux]], tempering out [[3025/3024]] in the 11-limit, [[2080/2079]], [[4096/4095]] in the 13-limit, and [[1156/1155]] and [[2601/2600]] in the 17-limit.  
 
== Geb ==
: ''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, ~15/8


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8505/8192 = 65.5883
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8908{{c}}, ~15/8 = 1088.4483{{c}}
: [[error map]]: {{val| -0.109 +0.103 -0.250 +0.427 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5477{{c}}
: 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]]: 0.134822
[[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 tuning (POTE): ~2 = 1\1, ~80/77 = 65.5881
Optimal tunings:
* WE: ~2 = 1199.9691{{c}}, ~15/8 = 1088.5116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5398{{c}}


{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}


Badness: 0.032816
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 tuning (POTE): ~2 = 1\1, ~27/26 = 65.5883
Optimal tunings:
* WE: ~2 = 1200.0267{{c}}, ~15/8 = 1088.5666{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5423{{c}}


{{Optimal ET sequence|legend=1| 128, 183, 311, 494, 1299, 1793 }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323 }}


Badness: 0.016849
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 tuning (POTE): ~2 = 1\1, ~27/26 = 65.5878
Optimal tunings:
* WE: ~2 = 1200.0454{{c}}, ~15/8 = 1088.5821{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5407{{c}}


{{Optimal ET sequence|legend=1| 128, 183, 311, 494 }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}


Badness: 0.014674
Badness (Sintel): 0.978


== Decimetra ==
== Decimetra ==
The ''decimetra'' temperament (50 & 140) tempers out the [[procyon comma]], 823543/820125 = {{monzo| 0 -8 -3 7 }} as well as [[linus comma]].
Named by [[Xenllium]] in 2021, decimetra tempers out the [[procyon comma]] as well as the [[linus comma]], and may be described as {{nowrap| 140 & 190 }}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 77: Line 112:


{{Mapping|legend=1| 10 3 20 12 | 0 4 1 5 }}
{{Mapping|legend=1| 10 3 20 12 | 0 4 1 5 }}
: mapping generators: ~15/14, ~5/4


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~5/4 = 385.495
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 120.0204{{c}}, ~5/4 = 385.5608{{c}}
: [[error map]]: {[val| +0.204 +0.349 -0.345 -0.777 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~5/4 = 385.5321{{c}}
: error map: {[val| 0.000 +0.173 -0.782 -1.166 }}


{{Optimal ET sequence|legend=1| 50, 90, 140, 330, 470, 800cd }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 330, 470, 800cd }}


[[Badness]]: 0.121818
[[Badness]] (Sintel): 3.08


=== 11-limit ===
=== 11-limit ===
Line 91: Line 131:
Mapping: {{mapping| 10 3 20 12 41 | 0 4 1 5 -2 }}
Mapping: {{mapping| 10 3 20 12 41 | 0 4 1 5 -2 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.396
Optimal tunings:
* WE: ~15/14 = 120.0379{{c}}, ~5/4 = 385.5181{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~5/4 = 385.4329{{c}}


{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e, 520de }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e, 520de }}


Badness: 0.056915
Badness (Sintel): 1.88


=== 13-limit ===
=== 13-limit ===
Line 104: Line 146:
Mapping: {{mapping| 10 3 20 12 41 37 | 0 4 1 5 -2 0 }}
Mapping: {{mapping| 10 3 20 12 41 37 | 0 4 1 5 -2 0 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.419
Optimal tunings:
* WE: ~15/14 = 120.0324{{c}}, ~5/4 = 385.5231{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~5/4 = 385.4445{{c}}


{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e, 520de }}


Badness: 0.029297
Badness (Sintel): 1.21


=== 17-limit ===
=== 17-limit ===
Line 117: Line 161:
Mapping: {{mapping| 10 3 20 12 41 37 28 | 0 4 1 5 -2 0 4 }}
Mapping: {{mapping| 10 3 20 12 41 37 28 | 0 4 1 5 -2 0 4 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.435
Optimal tunings:
* WE: ~15/14 = 120.0380{{c}}, ~5/4 = 385.5574{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~5/4 = 385.4670{{c}}


{{Optimal ET sequence|legend=1| 50, 90, 140, 190g, 330eg }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190g, 330eg }}


Badness: 0.021949
Badness (Sintel): 1.12


== Stockhausenic ==
== Luminal ==
The stockhausenic temperament (140 & 183, named after [[Wikipedia:Karlheinz Stockhausen|Karlheinz Stockhausen]]) is related to [[25ed5]] tuning, and tempers out the hemfiness comma (saquadru-atriyo), 4096000/4084101 = {{monzo| 15 -5 3 -5 }}.
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 4 0 -1 | 0 -26 25 41 }}
{{Mapping|legend=1| 1 0 5 8 | 0 29 -49 -95 }}
: mapping generators: ~2, ~8505/8192


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 111.453
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9728{{c}}, ~8505/8192 = 65.5868{{c}}
: [[error map]]: {{val| -0.027 +0.062 -0.202 +0.212 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8505/8192 = 65.5884{{c}}
: 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]]: 0.088753
[[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 4 0 -1 -1 | 0 -26 25 41 48 }}
Mapping: {{mapping| 1 0 5 8 1 | 0 29 -49 -95 45 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.460
Optimal tunings:
* WE: ~2 = 1199.9685{{c}}, ~80/77 = 65.5864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/77 = 65.5882{{c}}


{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}
{{Optimal ET sequence|legend=0| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}


Badness: 0.056043
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 4 0 -1 -1 5 | 0 -26 25 41 48 -14 }}
Mapping: {{mapping| 1 0 5 8 1 -1 | 0 29 -49 -95 45 86 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.458
Optimal tunings:
* WE: ~2 = 1199.9715{{c}}, ~27/26 = 65.5867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5883{{c}}


{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 829def, 1152deef }}
{{Optimal ET sequence|legend=0| 128, 183, 311, 494, 1299, 1793 }}


Badness: 0.029947
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 4 0 -1 -1 5 0 | 0 -26 25 41 48 -14 44 }}
Mapping: {{mapping| 1 0 5 8 1 -1 6 | 0 29 -49 -95 45 86 -35 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.459
 
{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 506, 829defg }}
 
Badness: 0.019205
 
== Geb ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Geb (5-limit)]].''
 
The geb temperament (171 & 506, named after the earth god of ancient Egyptian religion) tempers out the meter and the [[euzenius comma]] (78125000/78121827) as well as [[scheme comma]] (14348907/14336000) in the 7-limit. [[Osiris]] (171 & 670, named after the eldest son of Geb) tempers out the same 5-limit comma as the geb, but with the [[2401/2400|breedsma]] rather than the meter tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 703125/702464, 14348907/14336000
 
{{Mapping|legend=1| 1 -3 -10 -29 | 0 16 43 111 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8000/6561 = 343.863
Optimal tunings:
* WE: ~2 = 1199.9941{{c}}, ~27/26 = 65.5875{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5878{{c}}


{{Optimal ET sequence|legend=1| 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc }}
{{Optimal ET sequence|legend=0| 128, 183, 311, 494 }}


[[Badness]]: 0.048770
Badness (Sintel): 0.748


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