Xenharmonic Wiki

Lehmerismic temperaments: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>FREEZE
No edit summary
- ganesha, - lux (more properly addressed in other pages)
 
(33 intermediate revisions by 6 users not shown)
Line 1: Line 1:
__FORCETOC__
{{Technical data page}}
These are rank three temperaments tempering out 3025/3024, but we can began by looking at the rank four temperament.
This is a collection of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] that [[temper out]] the lehmerisma, [[3025/3024]].  


Comma: 3025/3024
Temperaments discussed elsewhere are:  
* ''[[Oxpecker]]'' (+121/120 or 126/125) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Manwe]]'' (+176/175) → [[Valinorsmic clan #Manwe|Valinorsmic 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]]
* ''[[Indra]]'' (+540/539) → [[Swetismic temperaments #Indra|Swetismic temperaments]]
* ''[[Tolerant]]'' (+896/891) → [[Pentacircle clan #Tolerant|Pentacircle clan]]
* ''[[Freya]]'' (+2401/2400) → [[Breed family #Freya|Breed family]]
* ''[[Triglav]]'' (+3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
* ''[[Thor]]'' (+4375/4374) → [[Ragismic family #Thor|Ragismic 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]]


Map: [<1 0 0 0 2|, <0 1 0 1 2|, <0 0 1 0 -1|, <0 0 0 2 1|]
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)]].


Generators: 2, 3, 5, 55/36
== Skadi ==
[[Subgroup]]: 2.3.5.7.11


EDOs: 5, 10, 15, 26, 31, 41, 72, 118, 224, 270, 342, 612, 836, 1106, 1448, 2554
[[Comma list]]: 3025/3024, 703125/702464


=Tyr=
{{Mapping|legend=1| 1 0 -1 -6 0 | 0 1 1 3 2 | 0 0 6 14 1 }}
Commas: 3025/3024, 102487/102400
: mapping generators: ~2, ~3, ~11/9


Map: [<3 0 0 4 8|, <0 2 0 -4 1|, <0 0 1 2 0|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0399{{c}}, ~3/2 = 701.9113{{c}}, ~11/9 = 347.3783{{c}}
: [[error map]]: {{val| +0.040 -0.004 -0.132 +0.085 -0.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9144{{c}}, ~11/9 = 347.3654{{c}}
: error map: {{val| 0.000 -0.041 -0.207 +0.033 -0.124 }}


Generators: 63/50, 400/231, 5
{{Optimal ET sequence|legend=1| 24d, 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}


EDOs: 15, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566
[[Badness]] (Sintel): 0.419


Badness: 0.000259
== Hanuman ==
[[Subgroup]]: 2.3.5.7.11


=Skadi=
[[Comma list]]: 3025/3024, 4000/3993
Commas: 3025/3024, 703125/702464


Map: [<1 0 5 8 1|, <0 1 1 3 2|, <0 0 -6 -14 -1|]
{{Mapping|legend=1| 1 2 0 -8 1 | 0 -3 0 11 1 | 0 0 1 4 1 }}
: mapping generators: ~2, ~11/10, ~5


Generators: 2, 3, 18/11
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9539{{c}}, ~11/10 = 165.9317{{c}}, ~5/4 = 385.9220{{c}}
: [[error map]]: {{val| -0.046 +0.158 -0.484 +0.111 +0.398 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/10 = 165.9501{{c}}, ~5/4 = 385.8882{{c}}
: error map: {{val| 0.000 +0.195 -0.426 +0.178 +0.520 }}


EDOs: 31, 97, 121, 152, 311, 342, 836, 1178, 2014
{{Optimal ET sequence|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}


Badness: 0.000349
[[Badness]] (Sintel): 0.600


=Ganesha=
=== 13-limit ===
Commas: 3025/3024, 5632/5625
Subgroup: 2.3.5.7.11.13


Map: [<1 0 0 -22 -9|, <0 1 0 1 2|, <0 0 1 10 4|]
Comma list: 625/624, 1575/1573, 2080/2079


Generators: 2, 3, 5
Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}


EDOs: 31, 87, 118, 152, 239, 270, 962, 1232, 1502
Optimal tunings:  
* WE: ~2 = 1200.0188{{c}}, ~11/10 = 166.0060{{c}}, ~5/4 = 385.6865{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0007{{c}}, ~5/4 = 385.6918{{c}}


Badness: 0.000390
{{Optimal ET sequence|legend=0| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}


=Hanuman=
Badness (Sintel): 0.610
Commas: 4000/3993, 3025/3024


Map: [<1 2 0 -8 1|, <0 -3 0 11 1|, <0 0 1 4 1|]
== Galaxy ==
[[Subgroup]]: 2.3.5.7.11


Generators: 2, 11/10, 5
[[Comma list]]: 3025/3024, 20614528/20588575


EDOs: 15, 57, 72, 152, 224, 311, 463, 535, 998, 1533ce, 2307ce, 2531cde, 2842bce
{{Mapping|legend=1| 1 0 -4 10 11 | 0 1 3 -3 -3 | 0 0 9 -14 -16 }}
: mapping generators: ~2, ~3, ~1936/1715


Badness: 0.000500
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9912{{c}}, ~3/2 = 701.9805{{c}}, ~1936/1715 = 208.9296{{c}}
: [[error map]]: {{val| -0.009 +0.017 +0.003 +0.156 -0.204 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9856{{c}}, ~1936/1715 = 208.9333{{c}}
: error map: {{val| 0.000 +0.023 +0.020 +0.174 -0.184 }}


==13-limit==
{{Optimal ET sequence|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}
Commas: 625/624, 1575/1573, 2080/2079


Map: [<1 2 0 -8 1 -6|, <0 -3 0 11 1 3|, <0 0 1 4 1 4|]
[[Badness]] (Sintel): 1.01


Generators: 2, 11/10, 5
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 15, 72, 87, 137, 159, 224, 311, 535, 918c, 1142c, 1453c
Comma list: 1716/1715, 3025/3024, 4225/4224


Badness: 0.000653
Mapping: {{mapping| 1 0 -4 10 11 13 | 0 1 3 -3 -3 -4 | 0 0 9 -14 -16 -17 }}
 
Optimal tunings:
* WE: ~2 = 1200.0131{{c}}, ~3/2 = 701.9445{{c}}, ~44/39 = 208.9426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9368{{c}}, ~44/39 = 208.9373{{c}}
 
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}
 
Badness (Sintel): 0.433
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1199.9094{{c}}, ~3/2 = 702.0614{{c}}, ~44/39 = 208.8807{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0462{{c}}, ~44/39 = 208.9167{{c}}
 
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}
 
Badness (Sintel): 0.888
 
[[Category:Lehmerismic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Catalogs of rank-3 temperaments]]
Retrieved from "https://en.xen.wiki/w/Lehmerismic_temperaments"