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Lehmerismic temperaments: Difference between revisions

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{{Technical data page}}
These are rank-3 temperaments tempering out [[3025/3024]], but we can begin by looking at the rank-4 temperament.
These are [[rank-3 temperament]]s tempering out [[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 three clan #Spectacle|Rastmic rank-3 clan]]
* ''[[Portent]]'' (+385/384 or 441/440) → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Indra]]'' (+540/539) → [[Mirkwai family #Indra|Mirkwai family]]
* ''[[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) → [[Hemifamity family #Kapo|Hemifamity family]]


Temperaments discussed elsewhere are:
Considered below are tyr, skadi, ganesha, hanuman, lux, and galaxy. For the rank-4 temperament, see [[Rank-4 temperament #Lehmerismic (3025/3024)]].
* [[Starling family #Oxpecker|Oxpecker]]
* [[Valinorsmic temperaments #Manwe|Manwe]]
* [[Marvel family #Spectacle|Spectacle]]
* [[Gamelismic family #Portent|Portent]]
* [[Mirkwai family #Indra|Indra]]
* [[Tolermic family #Tolerant|Tolerant]]
* [[Breed family #Freya|Freya]]
* [[Hemimean family #Triglav|Triglav]]
* [[Ragismic family #Thor|Thor]]
* [[Hemifamity family #Kapo|Kapo]]


= Lehmerismic =
== Tyr ==
Subgroup: 2.3.5.7.11
{{See also| Landscape family }}


[[Comma list]]: 3025/3024
[[Subgroup]]: 2.3.5.7.11


[[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 }}]
[[Comma list]]: 3025/3024, 102487/102400


Mapping generators: ~2, ~3, ~5, ~55/36
{{Mapping|legend=1| 3 0 0 4 8 | 0 2 0 -4 1 | 0 0 1 2 0 }}


{{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 }}
: mapping generators: ~63/50, ~400/231, ~5


= Tyr =
{{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 }}
{{see also| Landscape family }}


Subgroup: 2.3.5.7.11
[[Badness]]: 0.259 × 10<sup>-3</sup>


[[Comma list]]: 3025/3024, 102487/102400
=== Borneo ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 3 0 0 4 8 }}, {{val| 0 2 0 -4 1 }}, {{val| 0 0 1 2 0 }}]
Comma list: 676/675, 1001/1000, 3025/3024


Mapping generators: ~63/50, ~400/231, ~5
Mapping: {{mapping| 3 0 0 4 8 -3 | 0 2 0 -4 1 3 | 0 0 1 2 0 1 }}


{{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 ET sequence|legend=1| 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270 }}


[[Badness]]: 0.259 × 10<sup>-3</sup>
Badness: 0.549 × 10<sup>-3</sup>


= Skadi =
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


Subgroup: 2.3.5.7.11
== Skadi ==
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 5 8 1 }}, {{val| 0 1 1 3 2 }}, {{val| 0 0 -6 -14 -1 }}]
{{Mapping|legend=1| 1 0 5 8 1 | 0 1 1 3 2 | 0 0 -6 -14 -1 }}


Mapping generators: ~2, ~3, ~18/11
: mapping generators: ~2, ~3, ~18/11


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


[[Badness]]: 0.349 × 10<sup>-3</sup>
[[Badness]]: 0.349 × 10<sup>-3</sup>


= Ganesha =
== Ganesha ==
=== 7-limit (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 }}


Subgroup: 2.3.5.7.11
[[Badness]]: 1.13 × 10<sup>-3</sup>
 
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 5632/5625
[[Comma list]]: 3025/3024, 5632/5625


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


Mapping generators: ~2, ~3, ~5
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.2251, ~5/4 = 386.6908


{{Val list|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]]: 0.390 × 10<sup>-3</sup>


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


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


[[Mapping]]: [{{val| 1 2 0 -8 1 }}, {{val| 0 -3 0 11 1 }}, {{val| 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


{{Val list|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]]: 0.500 × 10<sup>-3</sup>


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
{{Optimal ET sequence|legend=1| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
 
Badness: 0.653 × 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 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 0 6 5 | 0 1 4 -1 -3 | 0 0 -5 -2 4 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1/1, ~3/2 = 702.1117, ~55/48 = 235.5784
 
{{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>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0674, ~55/48 = 235.5870
 
{{Optimal ET sequence|legend=1| 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
 
Badness: 0.360 × 10<sup>-3</sup>
 
== Galaxy ==
[[Subgroup]]: 2.3.5.7.11
 
[[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 generators: ~2, ~3, ~6125/3456
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.9856, ~6125/3456 = 991.0688
 
{{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>


=== 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: 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 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9368, ~484/273 = 991.0597
 
{{Optimal ET sequence|legend=1| 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125 }}
 
Badness: 0.463 × 10<sup>-3</sup>
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 936/935, 1225/1224, 4225/4224


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 -5 -4 -1 | 0 1 3 -3 -3 -4 -2 | 0 0 -9 14 16 17 10 }}


Mapping generators: ~2, ~11/10, ~5
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1144, ~85/48 = 991.1035


Vals: {{Val list| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
{{Optimal ET sequence|legend=1| 46, 75e, 103, 121, 149, 167, 224, 270, 494g }}


Badness: 0.653 × 10<sup>-3</sup>
Badness: 0.933 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Pages with mostly numerical content]]
[[Category:Lehmerismic]]
[[Category:Lehmerismic temperaments| ]] <!-- main article -->
[[Category:Lehmerismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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