Lehmerismic temperaments: Difference between revisions

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- ganesha, - lux (more properly addressed in other pages)
 
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* ''[[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]]
* ''[[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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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 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
[[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 ==

Latest revision as of 12:44, 8 September 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-3 temperaments that temper out the lehmerisma, 3025/3024.

Temperaments discussed elsewhere are:

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

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 703125/702464

Mapping: [⟨1 0 -1 -6 0], ⟨0 1 1 3 2], ⟨0 0 6 14 1]]

mapping generators: ~2, ~3, ~11/9

Optimal tunings:

  • WE: ~2 = 1200.0399 ¢, ~3/2 = 701.9113 ¢, ~11/9 = 347.3783 ¢
error map: ⟨+0.040 -0.004 -0.132 +0.085 -0.037]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9144 ¢, ~11/9 = 347.3654 ¢
error map: ⟨0.000 -0.041 -0.207 +0.033 -0.124]

Optimal ET sequence: 24d, 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce

Badness (Sintel): 0.419

Hanuman

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993

Mapping: [⟨1 2 0 -8 1], ⟨0 -3 0 11 1], ⟨0 0 1 4 1]]

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

Optimal tunings:

  • WE: ~2 = 1199.9539 ¢, ~11/10 = 165.9317 ¢, ~5/4 = 385.9220 ¢
error map: ⟨-0.046 +0.158 -0.484 +0.111 +0.398]
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9501 ¢, ~5/4 = 385.8882 ¢
error map: ⟨0.000 +0.195 -0.426 +0.178 +0.520]

Optimal ET sequence: 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998

Badness (Sintel): 0.600

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 0 -8 1 -6], ⟨0 -3 0 11 1 3], ⟨0 0 1 4 1 4]]

Optimal tunings:

  • WE: ~2 = 1200.0188 ¢, ~11/10 = 166.0060 ¢, ~5/4 = 385.6865 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0007 ¢, ~5/4 = 385.6918 ¢

Optimal ET sequence: 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c

Badness (Sintel): 0.610

Galaxy

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 20614528/20588575

Mapping: [⟨1 0 -4 10 11], ⟨0 1 3 -3 -3], ⟨0 0 9 -14 -16]]

mapping generators: ~2, ~3, ~1936/1715

Optimal tunings:

  • WE: ~2 = 1199.9912 ¢, ~3/2 = 701.9805 ¢, ~1936/1715 = 208.9296 ¢
error map: ⟨-0.009 +0.017 +0.003 +0.156 -0.204]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9856 ¢, ~1936/1715 = 208.9333 ¢
error map: ⟨0.000 +0.023 +0.020 +0.174 -0.184]

Optimal ET sequence: 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

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

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 ¢, ~3/2 = 701.9445 ¢, ~44/39 = 208.9426 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9368 ¢, ~44/39 = 208.9373 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 702.0614 ¢, ~44/39 = 208.8807 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0462 ¢, ~44/39 = 208.9167 ¢

Optimal ET sequence: 46, 75e, 103, 121, 149, 167, 224, 270, 494g

Badness (Sintel): 0.888

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