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}}
* ''[[Oxpecker]]'', {121/120, 126/125} → [[Starling family #Oxpecker|Starling family]]
This is a collection of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] that [[temper out]] the lehmerisma, [[3025/3024]].
* ''[[Manwe]]'', {176/175, 1331/1323} → [[Valinorsmic temperaments #Manwe|Valinorsmic temperaments]]
* ''[[Spectacle]]'', {225/224, 243/242} → [[Marvel family #Spectacle|Marvel family]]
* ''[[Portent]]'', {385/384, 441/440} → [[Gamelismic family #Portent|Gamelismic family]]
* ''[[Indra]]'', {540/539, 1375/1372} → [[Mirkwai family #Indra|Mirkwai family]]
* ''[[Tolerant]]'', {896/891, 2200/2187} → [[Pentacircle temperaments #Tolerant|Pentacircle temperaments]]
* ''[[Freya]]'', {2401/2400, 3025/3024} → [[Breed family #Freya|Breed family]]
* ''[[Triglav]]'', {3025/3024, 3136/3125} → [[Hemimean family #Triglav|Hemimean family]]
* ''[[Thor]]'', {3025/3024, 4375/4374} → [[Ragismic family #Thor|Ragismic family]]
* ''[[Kapo]]'', {3025/3024, 5120/5103} → [[Hemifamity family #Kapo|Hemifamity family]]


Considered below are tyr, skadi, ganesha, hanuman, lux, and galaxy, but we can begin by looking at the rank-4 temperament.
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) → [[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]]
 
[[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 generators: ~2, ~3, ~5, ~55/36
 
{{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 }}
 
== Tyr ==
{{See also| Landscape family }}
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 3025/3024, 102487/102400
 
[[Mapping]]: [{{val| 3 0 0 4 8 }}, {{val| 0 2 0 -4 1 }}, {{val| 0 0 1 2 0 }}]
 
Mapping generators: ~63/50, ~400/231, ~5
 
{{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 }}
 
[[Badness]]: 0.259 × 10<sup>-3</sup>
 
=== Borneo ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 3025/3024
 
Mapping: [{{val| 3 0 0 4 8 -3 }}, {{val| 0 2 0 -4 1 3 }}, {{val| 0 0 1 2 0 1 }}]
 
Vals: {{Val list| 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 ==
Subgroup: 2.3.5.7.11
[[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 -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 }}


{{Val list|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 ==
Subgroup: 2.3.5.7.11
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Quasiorwellismic]].''
 
[[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]]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 }}


{{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]] (Sintel): 0.469


== 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
[[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| 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 96: Line 77:
Comma list: 625/624, 1575/1573, 2080/2079
Comma list: 625/624, 1575/1573, 2080/2079


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 2 0 -8 1 -6 | 0 -3 0 11 1 3 | 0 0 1 4 1 4 }}


Vals: {{Val list| 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}}


Badness: 0.653 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c }}
 
Badness (Sintel): 0.610


== Lux ==
== Lux ==
=== 7-limit ===
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]].  
Subgroup: 2.3.5.7
 
[[Comma list]]: 26843545600/26795786661
 
[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 4 -1 }}, {{val| 0 0 -5 -2 }}]
 
Mapping generators: ~2, ~3, ~40960/35721
 
{{Multival|legend=1|rank=3| 5 2 13 30 }}
 
[[POTE generator]]s: ~3/2 = 702.1152, ~40960/35721 = 235.5941


{{Val list|legend=1| 41, 87, 128, 142, 178, 183, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}
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.


[[Badness]]: 1.71 × 10<sup>-3</sup>
[[Subgroup]]: 2.3.5.7.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 0 6 5 }}, {{val| 0 1 4 -1 -3 }}, {{val| 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}}
: [[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 }}


[[POTE generator]]s: ~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 }}


{{Val list|legend=1| 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d }}
[[Badness]] (Sintel): 0.611
 
[[Badness]]: 0.508 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 140: Line 114:
Comma list: 2080/2079, 3025/3024, 4096/4095
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 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}


POTE generators: ~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}}


Vals: {{Val list| 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 ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [{{val| 1 0 5 -4 -5 }}, {{val| 0 1 3 -3 -3 }}, {{val| 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 }}


[[POTE generator]]s: ~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 }}


{{Val list|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 170: Line 147:
Comma list: 1716/1715, 3025/3024, 4225/4224
Comma list: 1716/1715, 3025/3024, 4225/4224


Mapping: [{{val| 1 0 5 -4 -5 -4 }}, {{val| 0 1 3 -3 -3 -4 }}, {{val| 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 }}


POTE generators: ~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}}


Vals: {{val list| 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 183: Line 162:
Comma list: 715/714, 936/935, 1225/1224, 4225/4224
Comma list: 715/714, 936/935, 1225/1224, 4225/4224


Mapping: [{{val| 1 0 5 -4 -5 -4 -1 }}, {{val| 0 1 3 -3 -3 -4 -2 }}, {{val| 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 }}


POTE generators: ~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}}


Vals: {{val list| 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: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]]

Latest revision as of 12:15, 6 June 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, ganesha, hanuman, lux, 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

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: [⟨1 0 0 -22 -9], ⟨0 1 0 1 2], ⟨0 0 1 10 4]]

Optimal tunings:

  • WE: ~2 = 1199.9176 ¢, ~3/2 = 702.1769 ¢, ~5/4 = 386.6642 ¢
error map: ⟨-0.082 +0.139 +0.186 +0.075 -0.390]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1836 ¢, ~5/4 = 386.6860 ¢
error map: ⟨0.000 +0.229 +0.372 +0.218 -0.207]

Optimal ET sequence: 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502

Badness (Sintel): 0.469

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

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.

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: [⟨1 0 -5 4 9], ⟨0 1 4 -1 -3], ⟨0 0 5 2 -4]]

mapping generators: ~2, ~3, ~55/48

Optimal tunings:

  • WE: ~2 = 1199.9605 ¢, ~3/2 = 702.0886 ¢, ~55/48 = 235.5706 ¢
error map: ⟨-0.039 +0.094 -0.067 +0.108 -0.103]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1120 ¢, ~55/48 = 235.5763 ¢
error map: ⟨0.000 +0.157 +0.016 +0.215 +0.041]

Optimal ET sequence: 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: [⟨1 0 -5 4 9 13], ⟨0 1 4 -1 -3 -5], ⟨0 0 5 2 -4 -7]]

Optimal tunings:

  • WE: ~2 = 1199.9727 ¢, ~3/2 = 702.0946 ¢, ~55/48 = 235.5597 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1109 ¢, ~55/48 = 235.5649 ¢

Optimal ET sequence: 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d

Badness (Sintel): 0.337

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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