Quintaleap family: Difference between revisions

The '''quintaleap family''' tempers out {{monzo| 37 -16 -5 }}, equating a stack of two [[Pythagorean comma]]s with a stack of five [[schisma]]s, making it a member of the [[schismic-Pythagorean equivalence continuum]]. It is also the temperament where [[4/3]] is identified by a stack of five [[135/128]]'s.  

[[POTE generator]]: ~135/128 = 99.267

{{Val list|legend=1| 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc }}

[[Badness]]: 0.444506

=== 2.3.5.17.19 subgroup ===

Comma list: 256/255, 361/360, 4624/4617

Sval mapping: [{{val| 1 2 1 5 4 }}, {{val| 0 -5 16 -11 3 }}]

Gencom: [2 18/17; 256/255 361/360 4624/4617]

POTE generator: ~18/17 = 99.276

Vals: {{Val list| 12, 109, 121, 133 }}

[[Badness]]: 0.014623

The ''quintupole'' temperament tempers out the octagar comma and the mistisma (458752/455625) in the 7-limit; 896/891 (pentacircle), 1375/1372 (moctdel), and 4375/4356 (fantares, luluzoquadyo) in the 11-limit. The word "quintupole" means ''five poles'', but also a play on the words "quintuple" and "polypyth". It is so named because the generator is 1/5 of the [[Porwell temperaments #Polypyth|polypyth]] fourth (~4/3, about 495.8 cents). Proper pronunciation of the word "quintupole" is /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". ''Not to be confused with [[Marvel temperaments #Quintapole|quint<u>'''a'''</u>pole]] temperament (12&amp;85).''

[[Mapping]]: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]

{{Multival|legend=1| 5 -16 -34 -37 -68 -34 }}

[[POTE generator]]: ~135/128 = 99.175

 

[[Badness]]: 0.111620

Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]

POTE generator: ~35/33 = 99.156

Vals: {{Val list| 12, 109, 121, 351bde, 472bdee }}

Badness: 0.056501

Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]

POTE generator: ~35/33 = 99.165

Vals: {{Val list| 12f, 109, 121 }}

Badness: 0.038431

Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]

POTE generator: ~18/17 = 99.172

Vals: {{Val list| 12f, 109, 121 }}

Badness: 0.028721

Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]

POTE generator: ~18/17 = 99.164

Vals: {{Val list| 12f, 109, 121 }}

Badness: 0.023818

 

Aside from the marvel comma, the ''quintapole'' temperament (12&amp;85) tempers out 7812500/7411887 (sepru-atritriyo). In the 11-limit, it tempers out the ptolemisma (100/99) as well as 85184/84035 (trilo-aquinru-agu). It is so named for the following reasons - it has the same commas as the [[Marvel family #Appolo|apollo temperament]], and its generator is a semitone five of which gives a flat fourth (~4/3, about 495 cents). Proper pronunciation of the word "quintapole" is /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". ''Not to be confused with [[Octagar temperaments #Quintupole|quint<u>'''u'''</u>pole]] temperament (12&amp;121).''

[[Comma list]]: 225/224, 7812500/7411887

[[Mapping]]: [{{val|1 2 1 1}}, {{val|0 -5 16 22}}]

[[POTE generator]]: ~21/20 = 98.994

{{Val list|legend=1| 12, 73c, 85, 97d }}

[[Badness]]: 0.192498

Comma list: 100/99, 225/224, 85184/84035

Mapping: [{{val|1 2 1 1 0}}, {{val|0 -5 16 22 42}}]

POTE generator: ~21/20 = 98.954

Vals: {{Val list| 12, 73ce, 85, 97d }}

Badness: 0.104353

== Quinticosiennic ==

The ''quinticosiennic'' temperament (12&amp;145) tempers out the hemifamity comma (5120/5103) and 395136/390625 (trizo-aquadbigu) in the 7-limit; 441/440 (werckisma), 896/891 (pentacircle), and 78408/78125 (lolosepgu) in the 11-limit. The word "quinticosiennic" means 5 (quintuple) × 29 (είκοσι εννέα) = 145, and so named because 1/5 of [[29edo|29EDO]] fourth, i.e. 12\145, is a possible generator.

[[Comma list]]: 5120/5103, 395136/390625

 

{{Multival|legend=1|5 -16 -46 -37 -87 -62}}

[[POTE generator]]: ~135/128 = 99.345

{{Val list|legend=1| 12, 133, 145, 157, 302c, 459bcc }}

[[Badness]]: 0.158041

Comma list: 441/440, 896/891, 78408/78125

Mapping: [{{val|1 2 1 -1 -2}}, {{val|0 -5 16 46 66}}]

POTE generator: ~35/33 = 99.318

Vals: {{Val list| 12, 133, 145 }}

Badness: 0.080674

Comma list: 196/195, 352/351, 364/363, 78408/78125

Mapping: [{{val|1 2 1 -1 -2 -3}}, {{val|0 -5 16 46 66 81}}]

POTE generator: ~35/33 = 99.307

Vals: {{Val list| 12f, 133, 145 }}

Badness: 0.052464

Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750

Mapping: [{{val|1 2 1 -1 -2 -3 5}}, {{val|0 -5 16 46 66 81 -11}}]

POTE generator: ~18/17 = 99.308

Vals: {{Val list| 12f, 133, 145 }}

Badness: 0.037108

Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475

Mapping: [{{val|1 2 1 -1 -2 -3 5 4}}, {{val|0 -5 16 46 66 81 -11 3}}]

POTE generator: ~18/17 = 99.303

Vals: {{Val list| 12f, 133, 145 }}

Badness: 0.028440