Mint temperaments: Difference between revisions

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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Lafayette]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Lafayette]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 392/375
[[Comma list]]: 36/35, 392/375


[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 0 5 3 7 }}]
{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}
 
: mapping generators: ~2, ~15/14


{{Multival|legend=1| 5 3 7 -7 -3 8 }}
{{Multival|legend=1| 5 3 7 -7 -3 8 }}


[[POTE generator]]: ~15/14 = 140.927
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 140.927


{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
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Comma list: 36/35, 56/55, 77/75
Comma list: 36/35, 56/55, 77/75


Mapping: [{{val| 1 1 2 2 3 }}, {{val| 0 5 3 7 4 }}]
Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}


POTE generators: ~12/11 = 140.747
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 140.747


{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
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Comma list: 26/25, 36/35, 56/55, 66/65
Comma list: 26/25, 36/35, 56/55, 66/65


Mapping: [{{val| 1 1 2 2 3 3 }}, {{val| 0 5 3 7 4 6 }}]
Mapping: {{mapping| 1 1 2 2 3 3 | 0 5 3 7 4 6 }}


POTE generators: ~13/12 = 140.751
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.751


{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
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Comma list: 26/25, 36/35, 51/50, 56/55, 66/65
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65


Mapping: [{{val| 1 1 2 2 3 3 4 }}, {{val| 0 5 3 7 4 6 1 }}]
Mapping: {{mapping| 1 1 2 2 3 3 4 | 0 5 3 7 4 6 1 }}


POTE generators: ~13/12 = 141.404
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 141.404


{{Optimal ET sequence|legend=1| 8d, 9, 17cg }}
{{Optimal ET sequence|legend=1| 8d, 9, 17cg }}
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Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65


Mapping: [{{val| 1 1 2 2 3 3 4 4 }}, {{val| 0 5 3 7 4 6 1 2 }}]
Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 0 5 3 7 4 6 1 2 }}


POTE generators: ~13/12 = 140.479
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.479


{{Optimal ET sequence|legend=1| 8d, 9, 17cg }}
{{Optimal ET sequence|legend=1| 8d, 9, 17cg }}
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{{see also| Ripple family #Ripple }}
{{see also| Ripple family #Ripple }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 2560/2401
[[Comma list]]: 36/35, 2560/2401


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


{{Multival|legend=1| 5 8 2 1 -11 -18 }}
{{Multival|legend=1| 5 8 2 1 -11 -18 }}


[[POTE generator]]: ~21/20 = 99.483
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 99.483


{{Optimal ET sequence|legend=1| 12 }}
{{Optimal ET sequence|legend=1| 12 }}
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Comma list: 36/35, 80/77, 126/121
Comma list: 36/35, 80/77, 126/121


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


POTE generator: ~21/20 = 99.385
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.385


{{Optimal ET sequence|legend=1| 12 }}
{{Optimal ET sequence|legend=1| 12 }}
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Comma list: 36/35, 40/39, 66/65, 147/143
Comma list: 36/35, 40/39, 66/65, 147/143


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


POTE generator: ~21/20 = 98.572
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 98.572


{{Optimal ET sequence|legend=1| 12f }}
{{Optimal ET sequence|legend=1| 12f }}
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== Smate ==
== Smate ==
{{see also| Smate family #Smate }}
{{See also| Smate family #Smate }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 2048/1875
[[Comma list]]: 36/35, 2048/1875


[[Mapping]]: [{{val| 1 3 2 6 }}, {{val| 0 -4 1 -9 }}]
{{Mapping|legend=1| 1 3 2 6 | 0 -4 1 -9 }}


{{Multival|legend=1| 4 -1 9 -11 3 24 }}
{{Multival|legend=1| 4 -1 9 -11 3 24 }}


[[POTE generator]]: ~5/4 = 422.275
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 422.275


{{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37ccdd }}
{{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37ccdd }}
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Comma list: 36/35, 56/55, 243/242
Comma list: 36/35, 56/55, 243/242


Mapping: [{{val| 1 3 2 6 7 }}, {{val| 0 -4 1 -9 -10 }}]
Mapping: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }}


POTE generator: ~5/4 = 422.217
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217


{{Optimal ET sequence|legend=1| 3de, 14, 17c, 37ccddee }}
{{Optimal ET sequence|legend=1| 3de, 14, 17c, 37ccddee }}
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Comma list: 26/25, 36/35, 56/55, 243/242  
Comma list: 26/25, 36/35, 56/55, 243/242  


Mapping: [{{val| 1 3 2 6 7 3 }}, {{val| 0 -4 1 -9 -10 2 }}]
Mapping: {{mapping| 1 3 2 6 7 3 | 0 -4 1 -9 -10 2 }}


POTE generator: ~5/4 = 423.020
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020


{{Optimal ET sequence|legend=1| 3de, 14, 17c }}
{{Optimal ET sequence|legend=1| 3de, 14, 17c }}
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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Delorean]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Delorean]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 1029/1000
[[Comma list]]: 36/35, 1029/1000


[[Mapping]]: [{{val| 1 3 4 4 }}, {{val| 0 -6 -7 -5 }}]
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}


{{Multival|legend=1| 6 7 5 -3 -9 -8 }}
{{Multival|legend=1| 6 7 5 -3 -9 -8 }}


[[POTE generator]]: ~6/5 = 284.219
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 284.219


{{Optimal ET sequence|legend=1| 4, 17c, 21, 38c }}
{{Optimal ET sequence|legend=1| 4, 17c, 21, 38c }}
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Comma list: 36/35, 77/75, 352/343
Comma list: 36/35, 77/75, 352/343


Mapping: [{{val| 1 3 4 4 7 }}, {{val| 0 -6 -7 -5 -15 }}]
Mapping: {{mapping| 1 3 4 4 7 | 0 -6 -7 -5 -15 }}


POTE generator: ~6/5 = 283.253
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.253


{{Optimal ET sequence|legend=1| 17c, 55ccd }}
{{Optimal ET sequence|legend=1| 17c, 55ccd }}
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Comma list: 26/25, 36/35, 66/65, 352/343  
Comma list: 26/25, 36/35, 66/65, 352/343  


POTE generator: ~6/5 = 282.856
Mapping: {{mapping| 1 3 4 4 7 7 | 0 -6 -7 -5 -15 -14 }}
 
Mapping: [{{val| 1 3 4 4 7 7 }}, {{val| 0 -6 -7 -5 -15 -14 }}]


POTE generator: ~6/5 = 282.856
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 282.856


{{Optimal ET sequence|legend=1| 17c }}
{{Optimal ET sequence|legend=1| 17c }}
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Comma list: 36/35, 56/55, 1029/1000
Comma list: 36/35, 56/55, 1029/1000


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


POTE generator: ~6/5 = 283.822
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.822


{{Optimal ET sequence|legend=1| 4, 17c, 21, 38ce, 55ccde }}
{{Optimal ET sequence|legend=1| 4, 17c, 21, 38ce, 55ccde }}
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Comma list: 36/35, 56/55, 66/65, 640/637
Comma list: 36/35, 56/55, 66/65, 640/637


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


POTE generator: ~6/5 = 283.882
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.882


{{Optimal ET sequence|legend=1| 4, 17c, 21, 38ce, 55ccde }}
{{Optimal ET sequence|legend=1| 4, 17c, 21, 38ce, 55ccde }}
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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Slurpee]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Slurpee]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 51200/50421
[[Comma list]]: 36/35, 51200/50421


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -7 -11 -3 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}


{{Multival|legend=1| 7 11 3 1 -15 -24 }}
{{Multival|legend=1| 7 11 3 1 -15 -24 }}


[[POTE generator]]: ~21/20 = 72.303
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 72.303


{{Optimal ET sequence|legend=1| 16, 17c, 33, 50cd, 83bccd }}
{{Optimal ET sequence|legend=1| 16, 17c, 33, 50cd, 83bccd }}
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Comma list: 36/35, 121/120, 352/343  
Comma list: 36/35, 121/120, 352/343  


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


POTE generator: ~21/20 = 72.297
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.297


{{Optimal ET sequence|legend=1| 16, 17c, 33, 50cd, 83bccd }}
{{Optimal ET sequence|legend=1| 16, 17c, 33, 50cd, 83bccd }}
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Comma list: 36/35, 66/65, 143/140, 352/343
Comma list: 36/35, 66/65, 143/140, 352/343


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


POTE generator: ~21/20 = 72.409
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.409


{{Optimal ET sequence|legend=1| 16, 17c, 33, 83bccd }}
{{Optimal ET sequence|legend=1| 16, 17c, 33, 83bccd }}
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== Penta ==
== Penta ==
{{see also| Gamelismic clan #Gidorah }}
{{See also| Gamelismic clan #Gidorah }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 28/25, 36/35
[[Comma list]]: 28/25, 36/35


[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 0 3 2 4 }}]
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}


{{Multival|legend=1| 3 2 4 -4 -2 4 }}
{{Multival|legend=1| 3 2 4 -4 -2 4 }}


[[POTE generator]]: ~6/5 = 239.977
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 239.977


{{Optimal ET sequence|legend=1| 1bd, 5 }}
{{Optimal ET sequence|legend=1| 1bd, 5 }}
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Mint]]
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 10:25, 6 August 2023

These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. 36 is both a square and a triangular number, and this helps make 36/35 a septimal interval of considerable significance. Temperaments tempering it out include beep, dicot, dominant, father, diminished, august, hystrix, penta, hexadecimal, ripple and gorgo.

Progression

For the 5-limit version of this temperament, see High badness temperaments #Lafayette.

Subgroup: 2.3.5.7

Comma list: 36/35, 392/375

Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]

mapping generators: ~2, ~15/14

Wedgie: ⟨⟨ 5 3 7 -7 -3 8 ]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 140.927

Optimal ET sequence: 8d, 9, 17c

Badness: 0.048356

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 77/75

Mapping: [⟨1 1 2 2 3], ⟨0 5 3 7 4]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 140.747

Optimal ET sequence: 8d, 9, 17c

Badness: 0.026050

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 66/65

Mapping: [⟨1 1 2 2 3 3], ⟨0 5 3 7 4 6]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.751

Optimal ET sequence: 8d, 9, 17c

Badness: 0.018158

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 26/25, 36/35, 51/50, 56/55, 66/65

Mapping: [⟨1 1 2 2 3 3 4], ⟨0 5 3 7 4 6 1]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 141.404

Optimal ET sequence: 8d, 9, 17cg

Badness: 0.016743

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65

Mapping: [⟨1 1 2 2 3 3 4 4], ⟨0 5 3 7 4 6 1 2]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.479

Optimal ET sequence: 8d, 9, 17cg

Badness: 0.016841

Ripple

See also: Ripple family § Ripple

Subgroup: 2.3.5.7

Comma list: 36/35, 2560/2401

Mapping: [⟨1 2 3 3], ⟨0 -5 -8 -2]]

Wedgie: ⟨⟨ 5 8 2 1 -11 -18 ]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.483

Optimal ET sequence: 12

Badness: 0.059735

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 80/77, 126/121

Mapping: [⟨1 2 3 3 4], ⟨0 -5 -8 -2 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.385

Optimal ET sequence: 12

Badness: 0.038811

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 40/39, 66/65, 147/143

Mapping: [⟨1 2 3 3 4 4], ⟨0 -5 -8 -2 -6 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 98.572

Optimal ET sequence: 12f

Badness: 0.031639

Smate

See also: Smate family § Smate

Subgroup: 2.3.5.7

Comma list: 36/35, 2048/1875

Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]

Wedgie: ⟨⟨ 4 -1 9 -11 3 24 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.275

Optimal ET sequence: 3d, 11d, 14, 17c, 37ccdd

Badness: 0.077871

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 243/242

Mapping: [⟨1 3 2 6 7], ⟨0 -4 1 -9 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217

Optimal ET sequence: 3de, 14, 17c, 37ccddee

Badness: 0.042518

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 243/242

Mapping: [⟨1 3 2 6 7 3], ⟨0 -4 1 -9 -10 2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020

Optimal ET sequence: 3de, 14, 17c

Badness: 0.036836

Subklei

For the 5-limit version of this temperament, see High badness temperaments #Delorean.

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1000

Mapping: [⟨1 3 4 4], ⟨0 -6 -7 -5]]

Wedgie: ⟨⟨ 6 7 5 -3 -9 -8 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 284.219

Optimal ET sequence: 4, 17c, 21, 38c

Badness: 0.061101

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 77/75, 352/343

Mapping: [⟨1 3 4 4 7], ⟨0 -6 -7 -5 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.253

Optimal ET sequence: 17c, 55ccd

Badness: 0.044651

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 66/65, 352/343

Mapping: [⟨1 3 4 4 7 7], ⟨0 -6 -7 -5 -15 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 282.856

Optimal ET sequence: 17c

Badness: 0.032324

Subkla

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 1029/1000

Mapping: [⟨1 3 4 4 3], ⟨0 -6 -7 -5 2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.822

Optimal ET sequence: 4, 17c, 21, 38ce, 55ccde

Badness: 0.047247

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 66/65, 640/637

Mapping: [⟨1 3 4 4 3 3], ⟨0 -6 -7 -5 2 3]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.882

Optimal ET sequence: 4, 17c, 21, 38ce, 55ccde

Badness: 0.036701

Slurpee

For the 5-limit version of this temperament, see High badness temperaments #Slurpee.

Subgroup: 2.3.5.7

Comma list: 36/35, 51200/50421

Mapping: [⟨1 2 3 3], ⟨0 -7 -11 -3]]

Wedgie: ⟨⟨ 7 11 3 1 -15 -24 ]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.303

Optimal ET sequence: 16, 17c, 33, 50cd, 83bccd

Badness: 0.114869

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 121/120, 352/343

Mapping: [⟨1 2 3 3 4], ⟨0 -7 -11 -3 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.297

Optimal ET sequence: 16, 17c, 33, 50cd, 83bccd

Badness: 0.050514

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 66/65, 143/140, 352/343

Mapping: [⟨1 2 3 3 4 4], ⟨0 -7 -11 -3 -9 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.409

Optimal ET sequence: 16, 17c, 33, 83bccd

Badness: 0.033102

Penta

See also: Gamelismic clan § Gidorah

Subgroup: 2.3.5.7

Comma list: 28/25, 36/35

Mapping: [⟨1 1 2 2], ⟨0 3 2 4]]

Wedgie: ⟨⟨ 3 2 4 -4 -2 4 ]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 239.977

Optimal ET sequence: 1bd, 5

Badness: 0.046882

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