Pentacircle clan: Difference between revisions
m FloraC moved page Pentacircle temperaments to Pentacircle clan: By definition |
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The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. But we can start with the rank-4 temperament. | |||
Temperaments discussed elsewhere are: | Temperaments discussed elsewhere are: | ||
* ''[[Melpomene]]'', {56/55, 81/80} → [[Didymus rank three family #Melpomene]] | * ''[[Melpomene]]'', {56/55, 81/80} → [[Didymus rank three family #Melpomene|Didymus rank-3 family]] | ||
* ''[[Apollo]]'', {100/99, 225/224} → [[Marvel family #Apollo]] | * ''[[Apollo]]'', {100/99, 225/224} → [[Marvel family #Apollo|Marvel family]] | ||
* ''[[Julius]]'' or ''[[varda]]'', {176/175, 896/891} → [[Diaschismic rank three family #Julius aka varda]] | * ''[[Julius]]'' or ''[[varda]]'', {176/175, 896/891} → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]] | ||
* [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic]] | * [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]] | ||
* ''[[Pele]]'', {441/440, 896/891} → [[Hemifamity family #Pele]] | * ''[[Pele]]'', {441/440, 896/891} → [[Hemifamity family #Pele|Hemifamity family]] | ||
* ''[[Uni]]'', {540/539, 896/891} → [[Hemimage family #Uni]] | * ''[[Uni]]'', {540/539, 896/891} → [[Hemimage family #Uni|Hemimage family]] | ||
Considered below, in addition to the no-5 subgroup temperament ''parapyth'', is ''tolerant''. | Considered below, in addition to the no-5 subgroup temperament ''parapyth'', is ''tolerant''. | ||
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[[Badness]]: 0.0205 × 10<sup>-3</sup> | [[Badness]]: 0.0205 × 10<sup>-3</sup> | ||
=== 2.3.7.11.13 === | === 2.3.7.11.13 subgroup === | ||
Subgroup: 2.3.7.11.13 | Subgroup: 2.3.7.11.13 | ||
Revision as of 09:47, 1 March 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
- 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, is tolerant.
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]]
POTE generators: ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
Badness: 0.0658 × 10-6
Parapyth
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
POTE generators: ~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]]
POTE generators: ~3/2 = 703.8563, ~7/4 = 969.9074
Badness: 0.1099 × 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
POTE generators: ~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]]
POTE generators: ~3/2 = 704.0412, ~5/4 = 387.2927
Vals: 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]]
POTE generators: ~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]]
POTE generators: ~3/2 = 704.0831, ~5/4 = 387.3269
Vals: Template:Val list
Badness: 0.982 × 10-3