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Lehmerismic temperaments: Difference between revisions

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These are rank-3 temperaments tempering out [[3025/3024]], but we can begin by looking at the rank-4 temperament.  
{{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:  
Temperaments discussed elsewhere are:  
* ''[[oxpecker]]'', {121/120, 126/125} → [[Starling family #Oxpecker]]
* ''[[Oxpecker]]'' (+121/120 or 126/125) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[manwe]]'', {176/175, 1331/1323} → [[Valinorsmic temperaments #Manwe]]
* ''[[Manwe]]'' (+176/175) → [[Valinorsmic clan #Manwe|Valinorsmic clan]]
* ''[[spectacle]]'', {225/224, 243/242} → [[Marvel family #Spectacle]]
* ''[[Spectacle]]'' (+225/224 or 243/242) → [[Rastmic rank-3 clan #Spectacle|Rastmic rank-3 clan]]
* ''[[portent]]'', {385/384, 441/440} → [[Gamelismic family #Portent]]
* ''[[Portent]]'' (+385/384 or 441/440) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[indra]]'', {540/539, 1375/1372} → [[Mirkwai family #Indra]]
* ''[[Indra]]'' (+540/539) → [[Swetismic temperaments #Indra|Swetismic temperaments]]
* ''[[tolerant]]'', {896/891, 2200/2187} → [[Tolermic family #Tolerant]]
* ''[[Tolerant]]'' (+896/891) → [[Pentacircle clan #Tolerant|Pentacircle clan]]
* ''[[freya]]'', {2401/2400, 3025/3024} → [[Breed family #Freya]]
* ''[[Freya]]'' (+2401/2400) → [[Breed family #Freya|Breed family]]
* ''[[triglav]]'', {3025/3024, 3136/3125} → [[Hemimean family #Triglav]]
* ''[[Triglav]]'' (+3136/3125) → [[Hemimean family #Triglav|Hemimean family]]
* ''[[thor]]'', {3025/3024, 4375/4374} → [[Ragismic family #Thor]]
* ''[[Thor]]'' (+4375/4374) → [[Ragismic family #Thor|Ragismic family]]
* ''[[kapo]]'', {3025/3024, 5120/5103} → [[Hemifamity family #Kapo]]
* ''[[Kapo]]'' (+5120/5103) → [[Hemifamity family #Kapo|Hemifamity family]]
* ''[[Tyr]]'' (+102487/102400) → [[Landscape family #Tyr|Landscape family]]


== Lehmerismic ==
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)]].  
Subgroup: 2.3.5.7.11


[[Comma list]]: 3025/3024
== Skadi ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 703125/702464


[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 1 2 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 2 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, ~5, ~55/36
[[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 }}


{{Val list|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
{{Optimal ET sequence|legend=1| 24d, 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}


== Tyr ==
[[Badness]] (Sintel): 0.419
{{see also| Landscape family }}


Subgroup: 2.3.5.7.11
== Ganesha ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Quasiorwellismic]].''


[[Comma list]]: 3025/3024, 102487/102400
[[Subgroup]]: 2.3.5.7.11


[[Mapping]]: [{{val| 3 0 0 4 8 }}, {{val| 0 2 0 -4 1 }}, {{val| 0 0 1 2 0 }}]
[[Comma list]]: 3025/3024, 5632/5625


Mapping generators: ~63/50, ~400/231, ~5
{{Mapping|legend=1| 1 0 0 -22 -9 | 0 1 0 1 2 | 0 0 1 10 4 }}


{{Val list|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 }}
[[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 }}


[[Badness]]: 0.259 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}


== Skadi ==
[[Badness]] (Sintel): 0.469


Subgroup: 2.3.5.7.11
== Hanuman ==
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 5 8 1 }}, {{val| 0 1 1 3 2 }}, {{val| 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


Mapping 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 }}


{{Val list|legend=1| 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce }}
{{Optimal ET sequence|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}


[[Badness]]: 0.349 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.600


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


Subgroup: 2.3.5.7.11
Comma list: 625/624, 1575/1573, 2080/2079


[[Comma list]]: 3025/3024, 5632/5625
Mapping: {{mapping| 1 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}


[[Mapping]]: [{{val| 1 0 0 -22 -9 }}, {{val| 0 1 0 1 2 }}, {{val| 0 0 1 10 4 }}]
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}}


Mapping generators: ~2, ~3, ~5
{{Optimal ET sequence|legend=0| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}


{{Val list|legend=1| 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502 }}
Badness (Sintel): 0.610


[[Badness]]: 0.390 × 10<sup>-3</sup>
== 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 ratio]]s in the [[13-limit]]: [[325/324]], [[352/351]], [[364/363]], and [[441/440]].  


== Hanuman ==
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


[[Comma list]]: 3025/3024, 4000/3993
[[Comma list]]: 3025/3024, 131072/130977


[[Mapping]]: [{{val| 1 2 0 -8 1 }}, {{val| 0 -3 0 11 1 }}, {{val| 0 0 1 4 1 }}]
{{Mapping|legend=1| 1 0 -5 4 9 | 0 1 4 -1 -3 | 0 0 5 2 -4 }}
: mapping generators: ~2, ~3, ~55/48


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


{{Val list|legend=1| 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998 }}
{{Optimal ET sequence|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}


[[Badness]]: 0.500 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.611


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


Comma list: 625/624, 1575/1573, 2080/2079
Comma list: 2080/2079, 3025/3024, 4096/4095


Mapping: [{{val| 1 2 0 -8 1 -6 }}, {{val| 0 -3 0 11 1 3 }}, {{val| 0 0 1 4 1 4 }}]
Mapping: {{mapping| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}


Mapping generators: ~2, ~11/10, ~5
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}}


Vals: {{Val list| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
{{Optimal ET sequence|legend=0| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}


Badness: 0.653 × 10<sup>-3</sup>
Badness (Sintel): 0.337


== Lux ==
== Galaxy ==
=== 7-limit ===
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7


[[Comma list]]: 26843545600/26795786661
[[Comma list]]: 3025/3024, 20614528/20588575


[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 4 -1 }}, {{val| 0 0 -5 -2 }}]
{{Mapping|legend=1| 1 0 -4 10 11 | 0 1 3 -3 -3 | 0 0 9 -14 -16 }}
: mapping generators: ~2, ~3, ~1936/1715


{{Multival|legend=1|rank=3| 5 2 13 30 }}
[[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 }}


[[POTE generator]]s: ~3/2 = 702.1152, ~40960/35721 = 235.5941
{{Optimal ET sequence|legend=1| 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665 }}


{{Val list|legend=1| 41, 87, 128, 142, 178, 183, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
[[Badness]] (Sintel): 1.01


[[Badness]]: 1.709 × 10<sup>-3</sup>
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Comma list]]: 3025/3024, 131072/130977
Comma list: 1716/1715, 3025/3024, 4225/4224


[[Mapping]]: [{{val| 1 0 0 6 5 }}, {{val| 0 1 4 -1 -3 }}, {{val| 0 0 -5 -2 4 }}]
Mapping: {{mapping| 1 0 -4 10 11 13 | 0 1 3 -3 -3 -4 | 0 0 9 -14 -16 -17 }}


{{Multival|legend=1|rank=3| 5 2 -4 13 -1 10 30 25 34 71 }}
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}}


[[POTE generator]]s: ~3/2 = 702.1117, ~55/48 = 235.5784
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}


{{Val list|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}
Badness (Sintel): 0.433


[[Badness]]: 0.5083 × 10<sup>-3</sup>
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== 13-limit ===
Comma list: 715/714, 936/935, 1225/1224, 4225/4224
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 3025/3024, 4096/4095


Mapping: [{{val| 1 0 0 6 5 6 }}, {{val| 0 1 4 -1 -3 -5 }}, {{val| 0 0 -5 -2 4 7 }}]
Mapping: {{mapping| 1 0 -4 10 11 13 9 | 0 1 3 -3 -3 -4 -2 | 0 0 9 -14 -16 -17 -10 }}


POTE generators: ~3/2 = 702.0674, ~55/48 = 235.5870
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}}


Vals: {{Val list| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
{{Optimal ET sequence|legend=0| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}


Badness: 0.3602 × 10<sup>-3</sup>
Badness (Sintel): 0.888


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Lehmerismic temperaments| ]] <!-- main article -->
[[Category:Lehmerismic temperaments| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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