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Lehmerismic temperaments: Difference between revisions

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These are rank-3 temperaments tempering out [[3025/3024]]. Temperaments discussed elsewhere are:  
{{Technical data page}}
This is a collection of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] that [[temper out]] the lehmerisma, [[3025/3024]].  
 
Temperaments discussed elsewhere are:  
* ''[[Oxpecker]]'' (+121/120 or 126/125) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Oxpecker]]'' (+121/120 or 126/125) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Manwe]]'' (+176/175) → [[Valinorsmic clan #Manwe|Valinorsmic clan]]
* ''[[Manwe]]'' (+176/175) → [[Valinorsmic clan #Manwe|Valinorsmic clan]]
* ''[[Spectacle]]'' (+225/224 or 243/242) → [[Rastmic rank three 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]]
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* ''[[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) → [[Hemifamity family #Kapo|Hemifamity family]]
* ''[[Tyr]]'' (+102487/102400) → [[Landscape family #Tyr|Landscape family]]


Considered below are tyr, skadi, ganesha, hanuman, lux, and galaxy. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].  
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)]].  
 
== Tyr ==
{{See also| Landscape family }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 102487/102400
 
{{Mapping|legend=1| 3 0 0 4 8 | 0 2 0 -4 1 | 0 0 1 2 0 }}
 
: mapping generators: ~63/50, ~400/231, ~5
 
{{Optimal ET sequence|legend=1| 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}
 
[[Badness]]: 0.259 × 10<sup>-3</sup>
 
=== Borneo ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 3025/3024
 
Mapping: {{mapping| 3 0 0 4 8 -3 | 0 2 0 -4 1 3 | 0 0 1 2 0 1 }}
 
{{Optimal ET sequence|legend=1| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}
 
Badness: 0.549 × 10<sup>-3</sup>
 
Complexity spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7


== Skadi ==
== Skadi ==
Line 46: Line 22:
[[Comma list]]: 3025/3024, 703125/702464
[[Comma list]]: 3025/3024, 703125/702464


{{Mapping|legend=1| 1 0 5 8 1 | 0 1 1 3 2 | 0 0 -6 -14 -1 }}
{{Mapping|legend=1| 1 0 -1 -6 0 | 0 1 1 3 2 | 0 0 6 14 1 }}
: mapping generators: ~2, ~3, ~11/9


: mapping generators: ~2, ~3, ~18/11
[[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 }}


{{Optimal ET sequence|legend=1| 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}
{{Optimal ET sequence|legend=1| 24d, 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}


[[Badness]]: 0.349 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.419


== Ganesha ==
== Ganesha ==
=== 7-limit (quasiorwellismic) ===
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Quasiorwellismic]].''
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 29360128/29296875
 
{{Mapping|legend=1| 1 0 0 -22 | 0 1 0 1 | 0 0 1 10 }}
 
: mapping generators: ~2, ~3, ~5
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.0901, ~5/4 = 386.6931
 
{{Optimal ET sequence|legend=1| 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c }}
 
[[Badness]]: 1.13 × 10<sup>-3</sup>


=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 77: Line 44:
{{Mapping|legend=1| 1 0 0 -22 -9 | 0 1 0 1 2 | 0 0 1 10 4 }}
{{Mapping|legend=1| 1 0 0 -22 -9 | 0 1 0 1 2 | 0 0 1 10 4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2251, ~5/4 = 386.6908
[[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 }}
{{Optimal ET sequence|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}


[[Badness]]: 0.390 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.469


== Hanuman ==
== Hanuman ==
Line 89: Line 60:


{{Mapping|legend=1| 1 2 0 -8 1 | 0 -3 0 11 1 | 0 0 1 4 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


: mapping generators: ~2, ~11/10, ~5
[[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 }}


{{Optimal ET sequence|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}
{{Optimal ET sequence|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}


[[Badness]]: 0.500 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.600


=== 13-limit ===
=== 13-limit ===
Line 103: Line 79:
Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}
Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}


{{Optimal ET sequence|legend=1| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
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}}
 
{{Optimal ET sequence|legend=0| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}


Badness: 0.653 × 10<sup>-3</sup>
Badness (Sintel): 0.610


== Lux ==
== 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.  
The last generator of lux, represented by [[55/48]], exceeds [[8/7]] by [[385/384]], which is equated with a number of important [[superparticular ratio]]s in the [[13-limit]]: [[325/324]], [[352/351]], [[364/363]], and [[441/440]].
 
This ultra-efficient full 13-limit temperament was first considered by [[Flora Canou]] in 2021. After a few attempts to come up with a memorable name, ''lux'', suggested by [[Godtone]], who associated certain intervals of 13 with light, was adopted.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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[[Comma list]]: 3025/3024, 131072/130977
[[Comma list]]: 3025/3024, 131072/130977


{{Mapping|legend=1| 1 0 0 6 5 | 0 1 4 -1 -3 | 0 0 -5 -2 4 }}
{{Mapping|legend=1| 1 0 -5 4 9 | 0 1 4 -1 -3 | 0 0 5 2 -4 }}
: mapping generators: ~2, ~3, ~55/48


{{Multival|legend=1|rank=3| 5 2 -4 13 -1 10 30 25 34 71 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9605{{c}}, ~3/2 = 702.0886{{c}}, ~55/48 = 235.5706{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1/1, ~3/2 = 702.1117, ~55/48 = 235.5784
: [[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 }}
{{Optimal ET sequence|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}


[[Badness]]: 0.508 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.611


=== 13-limit ===
=== 13-limit ===
Line 129: Line 114:
Comma list: 2080/2079, 3025/3024, 4096/4095
Comma list: 2080/2079, 3025/3024, 4096/4095


Mapping: {{mapping| 1 0 0 6 5 6 | 0 1 4 -1 -3 -5 | 0 0 -5 -2 4 7 }}
Mapping: {{mapping| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0674, ~55/48 = 235.5870
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=1| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
{{Optimal ET sequence|legend=0| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}


Badness: 0.360 × 10<sup>-3</sup>
Badness (Sintel): 0.337


== Galaxy ==
== Galaxy ==
Line 142: Line 129:
[[Comma list]]: 3025/3024, 20614528/20588575
[[Comma list]]: 3025/3024, 20614528/20588575


{{Mapping|legend=1| 1 0 5 -4 -5 | 0 1 3 -3 -3 | 0 0 -9 14 16 }}
{{Mapping|legend=1| 1 0 -4 10 11 | 0 1 3 -3 -3 | 0 0 9 -14 -16 }}
 
: mapping generators: ~2, ~3, ~1936/1715
: mapping generators: ~2, ~3, ~6125/3456
 
{{Multival|legend=1| 9 -14 -16 -14 -20 5 34 35 6 21 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.9856, ~6125/3456 = 991.0688
[[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 }}


{{Optimal ET sequence|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}
{{Optimal ET sequence|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}


[[Badness]]: 0.841 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Line 159: Line 147:
Comma list: 1716/1715, 3025/3024, 4225/4224
Comma list: 1716/1715, 3025/3024, 4225/4224


Mapping: {{mapping| 1 0 5 -4 -5 -4 | 0 1 3 -3 -3 -4 | 0 0 -9 14 16 17 }}
Mapping: {{mapping| 1 0 -4 10 11 13 | 0 1 3 -3 -3 -4 | 0 0 9 -14 -16 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9368, ~484/273 = 991.0597
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=1| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}


Badness: 0.463 × 10<sup>-3</sup>
Badness (Sintel): 0.433


=== 17-limit ===
=== 17-limit ===
Line 172: Line 162:
Comma list: 715/714, 936/935, 1225/1224, 4225/4224
Comma list: 715/714, 936/935, 1225/1224, 4225/4224


Mapping: {{mapping| 1 0 5 -4 -5 -4 -1 | 0 1 3 -3 -3 -4 -2 | 0 0 -9 14 16 17 10 }}
Mapping: {{mapping| 1 0 -4 10 11 13 9 | 0 1 3 -3 -3 -4 -2 | 0 0 9 -14 -16 -17 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1144, ~85/48 = 991.1035
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=1| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}


Badness: 0.933 × 10<sup>-3</sup>
Badness (Sintel): 0.888


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