Pentacircle clan: Difference between revisions
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== Pentacircle == | == Pentacircle == | ||
{{See also| Rank-4 temperament #Pentacircle (896/891) }} | |||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
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== Parapyth == | == Parapyth == | ||
{{Main| Parapyth }} | |||
Parapyth is the no-5 [[subgroup temperament]] tempering out [[352/351]] and [[364/363]] in the 2.3.7.11.13 subgroup. | Parapyth is the no-5 [[subgroup temperament]] tempering out [[352/351]] and [[364/363]] in the 2.3.7.11.13 subgroup. | ||
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Optimal tuning (POTE): ~3/2 = 703.9605, ~5/4 = 386.9831 | Optimal tuning (POTE): ~3/2 = 703.9605, ~5/4 = 386.9831 | ||
Optimal GPV sequence: {{Val list| 41, 46, 80, 87, 121, 167, 208, 375be, 583bef }} | |||
Badness: 1.021 × 10<sup>-3</sup> | Badness: 1.021 × 10<sup>-3</sup> | ||
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Optimal tuning (POTE): ~3/2 = 704.0831, ~5/4 = 387.3269 | Optimal tuning (POTE): ~3/2 = 704.0831, ~5/4 = 387.3269 | ||
Optimal GPV sequence: {{Val list| 41, 46, 75e, 80, 87, 121, 167, 288beg }} | |||
Badness: 0.982 × 10<sup>-3</sup> | Badness: 0.982 × 10<sup>-3</sup> | ||
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Optimal tuning (POTE): ~3/2 = 704.1691, ~7/4 = 970.8432 | Optimal tuning (POTE): ~3/2 = 704.1691, ~7/4 = 970.8432 | ||
Optimal GPV sequence: {{Val list| 17c, 29, 46, 75e, 92def, 121, 167, 288be }} | |||
Badness: 2.48 × 10<sup>-3</sup> | Badness: 2.48 × 10<sup>-3</sup> | ||
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Optimal tuning (POTE): ~3/2 = 704.1628, ~7/4 = 970.6620 | Optimal tuning (POTE): ~3/2 = 704.1628, ~7/4 = 970.6620 | ||
Optimal GPV sequence: {{Val list| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }} | |||
Badness: 1.52 × 10<sup>-3</sup> | Badness: 1.52 × 10<sup>-3</sup> | ||
Revision as of 15:30, 9 December 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:
- Melpomene, {56/55, 81/80} → Didymus rank-3 family
- Apollo, {100/99, 225/224} → Marvel family
- Julius or varda, {176/175, 896/891} → Diaschismic rank-3 family
- Parahemif, {243/242, 896/891} → Rastmic rank-3 clan
- Sensamagic, {245/243, 385/384} → Sensamagic family
- Pele, {441/440, 896/891} → Hemifamity family
- Uni, {540/539, 896/891} → Hemimage family
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
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
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
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
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
Optimal GPV sequence: 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
Optimal GPV sequence: 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
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
Optimal GPV sequence: 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
Optimal GPV sequence: Template:Val list
Badness: 1.52 × 10-3