Pentacircle clan: Difference between revisions

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+etypyth and terrapyth as proposed by Scott Dakota and found in the temperament finder
Try making it clearer and improve linking
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== Pentacircle ==
== Pentacircle ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 896/891
[[Comma list]]: 896/891
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[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 1 }}]


[[POTE generator]]s: ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722


{{Val list|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
{{Val list|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
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== Parapyth ==
== Parapyth ==
Subgroup: 2.3.7.11
Parapyth is the no-5 [[subgroup temperament]] tempering out [[352/351]] and [[364/363]] in the 2.3.7.11.13 subgroup.
 
[[Subgroup]]: 2.3.7.11


[[Comma list]]: 896/891
[[Comma list]]: 896/891
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Sval mapping generators: ~2, ~3, ~7
Sval mapping generators: ~2, ~3, ~7


[[POTE generator]]s: ~3/2 = 703.8345, ~7/4 = 969.8722
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8345, ~7/4 = 969.8722


{{Val list|legend=1| 12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee }}
{{Val list|legend=1| 12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee }}
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Sval mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 -4 -7 }}, {{val| 0 0 0 1 1 1 }}]
Sval mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 -4 -7 }}, {{val| 0 0 0 1 1 1 }}]


POTE generators: ~3/2 = 703.8563, ~7/4 = 969.9074
Optimal tuning (POTE): ~3/2 = 703.8563, ~7/4 = 969.9074


Optimal GPV sequence: {{Val list| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}
Optimal GPV sequence: {{Val list| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}
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Sval mapping: [{{val| 1 0 0 0 7 12 -13 }}, {{val| 0 1 0 0 -4 -7 9 }}, {{val| 0 0 0 1 1 1 1 }}]
Sval mapping: [{{val| 1 0 0 0 7 12 -13 }}, {{val| 0 1 0 0 -4 -7 9 }}, {{val| 0 0 0 1 1 1 1 }}]


POTE generators: ~3/2 = 704.0315, ~7/4 = 970.6051
Optimal tuning (POTE): ~3/2 = 704.0315, ~7/4 = 970.6051


Optimal GPV sequence: {{Val list| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}
Optimal GPV sequence: {{Val list| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}
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== Tolerant ==
== Tolerant ==
=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 179200/177147
[[Comma list]]: 179200/177147
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Mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


[[POTE generator]]s: ~3/2 = 703.9571, ~5/4 = 386.8863
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.9571, ~5/4 = 386.8863


{{Val list|legend=1| 41, 80, 87, 121, 167, 208, 329b, 375b, 537b, 583b, 704bd }}
{{Val list|legend=1| 41, 80, 87, 121, 167, 208, 329b, 375b, 537b, 583b, 704bd }}
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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Comma list: 896/891, 2200/2187
[[Comma list]]: 896/891, 2200/2187


Mapping: [{{val| 1 0 0 -10 -3 }}, {{val| 0 1 0 11 7 }}, {{val| 0 0 1 -2 -2 }}]
[[Mapping]]: [{{val| 1 0 0 -10 -3 }}, {{val| 0 1 0 11 7 }}, {{val| 0 0 1 -2 -2 }}]


[[POTE generator]]s: ~3/2 = 704.0412, ~5/4 = 387.2927
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.0412, ~5/4 = 387.2927


Vals: {{Val list| 41, 80, 87, 121, 167, 208, 334be, 375be, 542bce }}
{{Val list|legend=1| 41, 80, 87, 121, 167, 208, 334be, 375be, 542bce }}


Badness: 1.039 × 10<sup>-3</sup>
[[Badness]]: 1.039 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
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Mapping: [{{val| 1 0 0 -10 -3 2 }}, {{val| 0 1 0 11 7 4 }}, {{val| 0 0 1 -2 -2 -2 }}]
Mapping: [{{val| 1 0 0 -10 -3 2 }}, {{val| 0 1 0 11 7 4 }}, {{val| 0 0 1 -2 -2 -2 }}]


POTE generators: ~3/2 = 703.9605, ~5/4 = 386.9831
Optimal tuning (POTE): ~3/2 = 703.9605, ~5/4 = 386.9831


Vals: {{Val list| 41, 46, 80, 87, 121, 167, 208, 375be, 583bef }}
Vals: {{Val list| 41, 46, 80, 87, 121, 167, 208, 375be, 583bef }}
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Mapping: [{{val| 1 0 0 -10 -3 2 8 }}, {{val| 0 1 0 11 7 4 -1 }}, {{val| 0 0 1 -2 -2 -2 -1 }}]
Mapping: [{{val| 1 0 0 -10 -3 2 8 }}, {{val| 0 1 0 11 7 4 -1 }}, {{val| 0 0 1 -2 -2 -2 -1 }}]


[[POTE generator]]s: ~3/2 = 704.0831, ~5/4 = 387.3269
Optimal tuning (POTE): ~3/2 = 704.0831, ~5/4 = 387.3269


Vals: {{Val list| 41, 46, 75e, 80, 87, 121, 167, 288beg }}
Vals: {{Val list| 41, 46, 75e, 80, 87, 121, 167, 288beg }}
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Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.  
Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 896/891, 585640/583443
[[Comma list]]: 896/891, 585640/583443
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Mapping generators: ~2, ~3, ~7
Mapping generators: ~2, ~3, ~7


[[POTE generator]]s: ~3/2 = 704.1814, ~7/4 = 970.6217
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.1814, ~7/4 = 970.6217


{{Val list|legend=1| 17c, 29, 46, 92de, 121, 167, 288be }}
{{Val list|legend=1| 17c, 29, 46, 92de, 121, 167, 288be }}
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Mapping: [{{val| 1 0 -31 0 7 12 }}, {{val| 0 1 0 11 7 4 -7 }}, {{val| 0 0 1 -2 -2 -2 1 }}]
Mapping: [{{val| 1 0 -31 0 7 12 }}, {{val| 0 1 0 11 7 4 -7 }}, {{val| 0 0 1 -2 -2 -2 1 }}]


POTE generators: ~3/2 = 704.1691, ~7/4 = 970.8432
Optimal tuning (POTE): ~3/2 = 704.1691, ~7/4 = 970.8432


Vals: {{Val list| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
Vals: {{Val list| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
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Mapping: [{{val| 1 0 -31 0 7 12 -13 }}, {{val| 0 1 0 11 7 4 -7 9 }}, {{val| 0 0 1 -2 -2 -2 1 1 }}]
Mapping: [{{val| 1 0 -31 0 7 12 -13 }}, {{val| 0 1 0 11 7 4 -7 9 }}, {{val| 0 0 1 -2 -2 -2 1 1 }}]


POTE generators: ~3/2 = 704.1628, ~7/4 = 970.6620
Optimal tuning (POTE): ~3/2 = 704.1628, ~7/4 = 970.6620


Vals: {{Val list| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
Vals: {{Val list| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}

Revision as of 19:03, 14 September 2022

The pentacircle clan of rank-3 temperaments tempers out the pentacircle comma, 896/891. But we can start with the rank-4 temperament.

Temperaments discussed elsewhere are:

Considered below, in addition to the no-5 subgroup temperament parapyth, are tolerant and terrapyth.

Pentacircle

Subgroup: 2.3.5.7.11

Comma list: 896/891

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 -4], ⟨0 0 1 0 0], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722

Template:Val list

Badness: 0.0658 × 10-6

Parapyth

Parapyth is the no-5 subgroup temperament tempering out 352/351 and 364/363 in the 2.3.7.11.13 subgroup.

Subgroup: 2.3.7.11

Comma list: 896/891

Sval mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 -4], ⟨0 0 0 1 1]]

Sval mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~3/2 = 703.8345, ~7/4 = 969.8722

Template:Val list

Badness: 0.0205 × 10-3

2.3.7.11.13 subgroup

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

Sval mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 -4 -7], ⟨0 0 0 1 1 1]]

Optimal tuning (POTE): ~3/2 = 703.8563, ~7/4 = 969.9074

Optimal GPV sequence: Template:Val list

Badness: 0.101 × 10-3

Etypyth

Subgroup: 2.3.7.11.13.17

Comma list: 352/351, 364/363, 442/441

Sval mapping: [⟨1 0 0 0 7 12 -13], ⟨0 1 0 0 -4 -7 9], ⟨0 0 0 1 1 1 1]]

Optimal tuning (POTE): ~3/2 = 704.0315, ~7/4 = 970.6051

Optimal GPV sequence: Template:Val list

Badness: 0.325 × 10-3

Tolerant

7-limit

Subgroup: 2.3.5.7

Comma list: 179200/177147

Mapping: [⟨1 0 0 -10], ⟨0 1 0 11], ⟨0 0 1 -2]]

Mapping generators: ~2, ~3, ~5

Optimal tuning (POTE): ~3/2 = 703.9571, ~5/4 = 386.8863

Template:Val list

Badness: 0.165 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187

Mapping: [⟨1 0 0 -10 -3], ⟨0 1 0 11 7], ⟨0 0 1 -2 -2]]

Optimal tuning (POTE): ~3/2 = 704.0412, ~5/4 = 387.2927

Template:Val list

Badness: 1.039 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363

Mapping: [⟨1 0 0 -10 -3 2], ⟨0 1 0 11 7 4], ⟨0 0 1 -2 -2 -2]]

Optimal tuning (POTE): ~3/2 = 703.9605, ~5/4 = 386.9831

Vals: Template:Val list

Badness: 1.021 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363

Mapping: [⟨1 0 0 -10 -3 2 8], ⟨0 1 0 11 7 4 -1], ⟨0 0 1 -2 -2 -2 -1]]

Optimal tuning (POTE): ~3/2 = 704.0831, ~5/4 = 387.3269

Vals: Template:Val list

Badness: 0.982 × 10-3

Terrapyth

Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 585640/583443

Mapping: [⟨1 0 -31 0 7], ⟨0 1 21 0 -4], ⟨0 0 0 1 1]]

Mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~3/2 = 704.1814, ~7/4 = 970.6217

Template:Val list

Badness: 5.35 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 9295/9261

Mapping: [⟨1 0 -31 0 7 12], ⟨0 1 0 11 7 4 -7], ⟨0 0 1 -2 -2 -2 1]]

Optimal tuning (POTE): ~3/2 = 704.1691, ~7/4 = 970.8432

Vals: Template:Val list

Badness: 2.48 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 364/363, 442/441, 715/714

Mapping: [⟨1 0 -31 0 7 12 -13], ⟨0 1 0 11 7 4 -7 9], ⟨0 0 1 -2 -2 -2 1 1]]

Optimal tuning (POTE): ~3/2 = 704.1628, ~7/4 = 970.6620

Vals: Template:Val list

Badness: 1.52 × 10-3

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