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Lehmerismic temperaments: Difference between revisions

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* ''[[Spectacle]]'' (+225/224 or 243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Spectacle]]'' (+225/224 or 243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Portent]]'' (+385/384 or 441/440) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Portent]]'' (+385/384 or 441/440) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Indra]]'' (+540/539) → [[Mirkwai family #Indra|Mirkwai family]]
* ''[[Indra]]'' (+540/539) → [[Swetismic temperaments #Indra|Swetismic temperaments]]
* ''[[Tolerant]]'' (+896/891) → [[Pentacircle clan #Tolerant|Pentacircle clan]]
* ''[[Tolerant]]'' (+896/891) → [[Pentacircle clan #Tolerant|Pentacircle clan]]
* ''[[Freya]]'' (+2401/2400) → [[Breed family #Freya|Breed family]]
* ''[[Freya]]'' (+2401/2400) → [[Breed family #Freya|Breed family]]
* ''[[Triglav]]'' (+3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
* ''[[Triglav]]'' (+3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
* ''[[Thor]]'' (+4375/4374) → [[Ragismic family #Thor|Ragismic family]]
* ''[[Thor]]'' (+4375/4374) → [[Ragismic family #Thor|Ragismic family]]
* ''[[Kapo]]'' (+5120/5103) → [[Hemifamity family #Kapo|Hemifamity family]]
* ''[[Kapo]]'' (+5120/5103) → [[Aberschismic family #Kapo|Aberschismic family]]
* ''[[Ganesha]]'' (+5632/5625) → [[Vishdelismic clan #Ganesha|Vishdelismic clan]]
* ''[[Tyr]]'' (+102487/102400) → [[Landscape family #Tyr|Landscape family]]
* ''[[Lux]]'' (+131072/130977) → [[Olympic clan #Lux|Olympic clan]]


Considered below are skadi, ganesha, hanuman, lux, and galaxy, in the order of increasing badness. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].  
Considered below are skadi, hanuman, and galaxy, in the order of increasing [[badness]]. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].  


== Skadi ==
== Skadi ==
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[[Badness]] (Sintel): 0.419
[[Badness]] (Sintel): 0.419
== Ganesha ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Quasiorwellismic]].''
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 3025/3024, 5632/5625
{{Mapping|legend=1| 1 0 0 -22 -9 | 0 1 0 1 2 | 0 0 1 10 4 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9176{{c}}, ~3/2 = 702.1769{{c}}, ~5/4 = 386.6642{{c}}
: [[error map]]: {{val| -0.082 +0.139 +0.186 +0.075 -0.390 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1836{{c}}, ~5/4 = 386.6860{{c}}
: error map: {{val| 0.000 +0.229 +0.372 +0.218 -0.207 }}
{{Optimal ET sequence|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}
[[Badness]] (Sintel): 0.469


== Hanuman ==
== Hanuman ==
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Comma list: 625/624, 1575/1573, 2080/2079
Comma list: 625/624, 1575/1573, 2080/2079


Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}
{{Mapping|legend=0| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 0.610
Badness (Sintel): 0.610
== Lux ==
The last generator of lux, represented by 55/48, exceeds 8/7 by 385/384, which is equated with a number of important superparticular ratios in the 13-limit: 325/324, 352/351, 364/363, and 441/440.
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 3025/3024, 131072/130977
{{Mapping|legend=1| 1 0 -5 4 9 | 0 1 4 -1 -3 | 0 0 5 2 -4 }}
: mapping generators: ~2, ~3, ~55/48
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9605{{c}}, ~3/2 = 702.0886{{c}}, ~55/48 = 235.5706{{c}}
: [[error map]]: {{val| -0.039 +0.094 -0.067 +0.108 -0.103 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1120{{c}}, ~55/48 = 235.5763{{c}}
: error map: {{val| 0.000 +0.157 +0.016 +0.215 +0.041 }}
{{Optimal ET sequence|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}
[[Badness]] (Sintel): 0.611
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 3025/3024, 4096/4095
Mapping: {{mapping| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}
Optimal tunings:
* WE: ~2 = 1199.9727{{c}}, ~3/2 = 702.0946{{c}}, ~55/48 = 235.5597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1109{{c}}, ~55/48 = 235.5649{{c}}
{{Optimal ET sequence|legend=0| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
Badness (Sintel): 0.337


== Galaxy ==
== Galaxy ==
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Comma list: 1716/1715, 3025/3024, 4225/4224
Comma list: 1716/1715, 3025/3024, 4225/4224


Mapping: {{mapping| 1 0 -4 10 11 13 | 0 1 3 -3 -3 -4 | 0 0 9 -14 -16 -17 }}
{{Mapping|legend=0| 1 0 -4 10 11 13 | 0 1 3 -3 -3 -4 | 0 0 9 -14 -16 -17 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 715/714, 936/935, 1225/1224, 4225/4224
Comma list: 715/714, 936/935, 1225/1224, 4225/4224


Mapping: {{mapping| 1 0 -4 10 11 13 9 | 0 1 3 -3 -3 -4 -2 | 0 0 9 -14 -16 -17 -10 }}
{{Mapping|legend=0| 1 0 -4 10 11 13 9 | 0 1 3 -3 -3 -4 -2 | 0 0 9 -14 -16 -17 -10 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 0.888
Badness (Sintel): 0.888


[[Category:Lehmerismic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Lehmerismic temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
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