Fifive family: Difference between revisions

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m restore old badness for reference
Review (restore POTE tuning since the replacement was based on a wrong assessment: CWE is ~140.629 not .623)
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: mapping generators: ~78125/50421, ~27/25
: mapping generators: ~78125/50421, ~27/25


[[Optimal tuning]] ([[CWE]]): ~78125/50421 = 1\2, ~27/25 = 140.623
[[Optimal tuning]]s:
* [[CTE]]: ~78125/50421 = 1\2, ~27/25 = 140.6349
* [[POTE]]: ~78125/50421 = 1\2, ~27/25 = 140.624


{{Optimal ET sequence|legend=1| 8, 26, 34, 94, 128 }}
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}


[[Badness]]: 0.205812
[[Badness]]:
 
* Smith: 0.205812
Dirichlet badness: 4.828
* Dirichlet: 4.828


=== 2.3.5.13 subgroup ===
=== 2.3.5.13 subgroup ===
Subgroup: 2.3.5.13


[[Subgroup]]: 2.3.5.13
Comma list: 325/324, 20000/19773


[[Comma list]]: 9765625/9565938, 325/324
Mapping: {{mapping| 2 2 3 6 | 0 5 7 6 }}
 
{{Mapping|legend=1| 2 2 3 6 | 0 5 7 6 }}


: mapping generators: ~351/250, ~13/12
: mapping generators: ~351/250, ~13/12


[[Optimal tuning]] ([[CWE]]): ~351/250 = 1\2, ~13/12 = 140.623
Optimal tunings:
* CTE: ~351/250 = 1\2, ~13/12 = 140.5685
* CWE: ~351/250 = 1\2, ~13/12 = 140.6232


[[Patent val]] EDO tunings: 8, 26, 34, 60, 68, 94, 128
Optimal ET sequence: {{Optimal ET sequence| 8, 18bcf, 26, 34, 94, 128 }}


Dirichlet badness: 0.800
Badness:
* Smith: 0.0240
* Dirichlet: 0.800


=== 2.3.5.13.17 subgroup ===
=== 2.3.5.13.17 subgroup ===
Subgroup: 2.3.5.13.17


[[Subgroup]]: 2.3.5.13.17
Comma list: 170/169, 289/288, 325/324
 
[[Comma list]]: 9765625/9565938, 325/324, 289/288


{{Mapping|legend=1| 2 2 3 6 7 | 0 5 7 6 5 }}
Mapping: {{mapping| 2 2 3 6 7 | 0 5 7 6 5 }}


: mapping generators: ~17/12, ~13/12
: mapping generators: ~17/12, ~13/12


[[Optimal tuning]] ([[CWE]]): ~17/12 = 1\2, ~13/12 = 140.606
Optimal tunings:
* CTE: ~17/12 = 1\2, ~13/12 = 140.5958
* CWE: ~17/12 = 1\2, ~13/12 = 140.6057


[[Patent val]] EDO tunings: 8, 26, 34, 60, 68, 94, 128
Optimal ET sequence: {{Optimal ET sequence| 8, 18bcfg, 26, 34, 94, 128 }}


Dirichlet badness: 0.488  
Badness:
* Smith: 0.0110
* Dirichlet: 0.488


== Crepuscular ==
== Crepuscular ==

Revision as of 02:47, 19 April 2024

The fifive family tempers out the fifive comma, 9765625/9565938 = [-1 -14 10⟩.

Temperaments discussed elsewhere include:

Considered below are crepuscular and fifives.

Fifive

Subgroup: 2.3.5

Comma list: 9765625/9565938

Mapping: [⟨2 2 3], ⟨0 5 7]]

mapping generators: ~78125/50421, ~27/25

Optimal tunings:

  • CTE: ~78125/50421 = 1\2, ~27/25 = 140.6349
  • POTE: ~78125/50421 = 1\2, ~27/25 = 140.624

Optimal ET sequence: 8, 18bc, 26, 34, 94, 128

Badness:

  • Smith: 0.205812
  • Dirichlet: 4.828

2.3.5.13 subgroup

Subgroup: 2.3.5.13

Comma list: 325/324, 20000/19773

Mapping: [⟨2 2 3 6], ⟨0 5 7 6]]

mapping generators: ~351/250, ~13/12

Optimal tunings:

  • CTE: ~351/250 = 1\2, ~13/12 = 140.5685
  • CWE: ~351/250 = 1\2, ~13/12 = 140.6232

Optimal ET sequence: 8, 18bcf, 26, 34, 94, 128

Badness:

  • Smith: 0.0240
  • Dirichlet: 0.800

2.3.5.13.17 subgroup

Subgroup: 2.3.5.13.17

Comma list: 170/169, 289/288, 325/324

Mapping: [⟨2 2 3 6 7], ⟨0 5 7 6 5]]

mapping generators: ~17/12, ~13/12

Optimal tunings:

  • CTE: ~17/12 = 1\2, ~13/12 = 140.5958
  • CWE: ~17/12 = 1\2, ~13/12 = 140.6057

Optimal ET sequence: 8, 18bcfg, 26, 34, 94, 128

Badness:

  • Smith: 0.0110
  • Dirichlet: 0.488

Crepuscular

See also: Jubilismic clan § Crepuscular

Subgroup: 2.3.5.7

Comma list: 50/49, 4375/4374

Mapping: [⟨2 2 3 4], ⟨0 5 7 7]]

Wedgie: ⟨⟨ 10 14 14 -1 -6 -7 ]]

Optimal tuning (POTE): ~7/5 = 1\2, ~27/25 = 140.349

Optimal ET sequence: 8d, 26, 34d, 60d

Badness: 0.086669

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 99/98, 864/847

Mapping: [⟨2 2 3 4 6], ⟨0 5 7 7 4]]

Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.587

Optimal ET sequence: 8d, 26, 34d, 60d

Badness: 0.040758

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 99/98, 144/143

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

Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.554

Optimal ET sequence: 8d, 26, 34d, 60d

Badness: 0.024368

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 85/84, 99/98, 144/143

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

Optimal tuning (POTE): ~7/5 = 1\2, ~12/11 = 140.405

Optimal ET sequence: 8d, 26, 34d, 60d

Badness: 0.018567

Fifives

Subgroup: 2.3.5.7

Comma list: 875/864, 83349/81920

Mapping: [⟨2 2 3 7], ⟨0 5 7 -6]]

Optimal tuning (POTE): ~567/400 = 1\2, ~27/25 = 139.909

Optimal ET sequence: 8, 26, 34, 60

Badness: 0.130589

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 3969/3872

Mapping: [⟨2 2 3 7 6], ⟨0 5 7 -6 4]]

Optimal tuning (POTE): ~63/44 = 1\2, ~12/11 = 139.884

Optimal ET sequence: 8, 26, 34, 60

Badness: 0.080306

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 1352/1331

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

Optimal tuning (POTE): ~55/39 = 1\2, ~12/11 = 139.867

Optimal ET sequence: 8, 26, 34, 60

Badness: 0.044253

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 170/169, 221/220

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

Optimal tuning (POTE): ~17/12 = 1\2, ~12/11 = 139.868

Optimal ET sequence: 8, 26, 34, 60

Badness: 0.029429

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