Quintaleap family: Difference between revisions

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Add the decimaleap temperament
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== Quintaleap ==
== Quintaleap ==
The name ''quintaleap'' comes from "quintans" (Latin for "one fifth") and "leapday", because the generator is 1/5 of the [[Hemifamity temperaments #Leapday|leapday]] fourth (~4/3, about 495.8 cents).
The name ''quintaleap'' comes from "quintans" (Latin for "one fifth") and "leapday", because the generator is 1/5 of the [[Hemifamity temperaments #Leapday|leapday]] fourth (~4/3, about 496 cents).


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 36: Line 36:


== Quintupole ==
== Quintupole ==
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). [[User:Xenllium|Xenllium]] proposes the pronunciation of the word "quintupole" as /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". ''Not to be confused with [[#Quintapole|quint<u>'''a'''</u>pole]] temperament (12&amp;85).''
The ''quintupole'' temperament tempers out the octagar comma (4000/3969) 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). [[User:Xenllium|Xenllium]] proposes the pronunciation of the word "quintupole" as /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". ''Not to be confused with [[#Quintapole|quint<u>'''a'''</u>pole]] temperament (12&amp;85).''


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 105: Line 105:


== Quintapole ==
== Quintapole ==
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). [[User:Xenllium|Xenllium]] proposes the pronunciation of the word "quintapole" as /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". ''Not to be confused with [[#Quintupole|quint<u>'''u'''</u>pole]] temperament (12&amp;121).''
The ''quintapole'' temperament (12&amp;85) tempers out the marvel comma (225/224) and 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). [[User:Xenllium|Xenllium]] proposes the pronunciation of the word "quintapole" as /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". ''Not to be confused with [[#Quintupole|quint<u>'''u'''</u>pole]] temperament (12&amp;121).''


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
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[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -5 16 22 }}]
[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -5 16 22 }}]
{{Multival|legend=1| 5 -16 -22 -37 -49 -6 }}


[[POTE generator]]: ~21/20 = 98.994
[[POTE generator]]: ~21/20 = 98.994
Line 200: Line 202:


Badness: 0.028440
Badness: 0.028440
== Decimaleap ==
The ''decimaleap'' temperament (24&amp;121) has a quarter-tone generator and tempers out the porwell comma (6144/6125) and 393379840/387420489 (sasaquadzo-ayo) in the 7-limit; 896/891 and 14700/14641 in the 11-limit. The name ''decimaleap'' comes from "decima" (Latin for "one tenth") and "leapday", because the generator is 1/10 of the leapday fourth.
Subgroup: 2.3.5.7
[[Comma list]]: 6144/6125, 393379840/387420489
[[Mapping]]: [{{val|1 2 1 5}}, {{val|0 -10 32 -53}}]
{{Multival|legend=1| 10 -32 53 -74 56 213 }}
[[POTE generator]]: ~36/35 = 49.621
{{Val list|legend=1| 24, 97d, 121, 145, 266 }}
[[Badness]]: 0.314227
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 896/891, 6144/6125, 14700/14641
Mapping: [{{val|1 2 1 5 4}}, {{val|0 -10 32 -53 -13}}]
POTE generator: ~36/35 = 49.622
Vals: {{Val list| 24, 97d, 121, 145, 266e }}
Badness: 0.084612
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 676/675, 6144/6125
Mapping: [{{val|1 2 1 5 4 3}}, {{val|0 -10 32 -53 -13 17}}]
POTE generator: ~36/35 = 49.620
Vals: {{Val list| 24, 97d, 121, 266ef }}
Badness: 0.041763
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 352/351, 364/363, 676/675, 1156/1155
Mapping: [{{val|1 2 1 5 4 3 5}}, {{val|0 -10 32 -53 -13 17 -22}}]
POTE generator: ~34/33 = 49.621
Vals: {{Val list| 24, 97dg, 121, 266efg }}
Badness: 0.025552
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 256/255, 352/351, 361/360, 364/363, 456/455, 665/663
Mapping: [{{val|1 2 1 5 4 3 5 4}}, {{val|0 -10 32 -53 -13 17 -22 6}}]
POTE generator: ~34/33 = 49.624
Vals: {{Val list| 24, 97dg, 121, 145, 266efg }}
Badness: 0.021208


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 23:59, 27 November 2021

The quintaleap family tempers out [37 -16 -5⟩, equating a stack of two Pythagorean commas with a stack of five schismas, 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.

Quintaleap

The name quintaleap comes from "quintans" (Latin for "one fifth") and "leapday", because the generator is 1/5 of the leapday fourth (~4/3, about 496 cents).

Subgroup: 2.3.5

Comma list: [37 -16 -5⟩ = 137438953472/134521003125

Mapping: [⟨1 2 1], ⟨0 -5 16]]

POTE generator: ~135/128 = 99.267

Template:Val list

Badness: 0.444506

Subgroup temperament

See also: No-sevens subgroup temperaments § Quintaleap

The quintaleap temperament works well for the 2.3.5.17.19 subgroup, tempering out 256/255 (equating 16/15 with 17/16), 361/360 (equating 19/18 with 20/19), and 4624/4617. An obvious 17-limit interpretation of the generator is ~18/17, equating three 18/17s with 19/16, five 18/17s with 4/3, and sixteen 18/17s with 5/2.

Subgroup: 2.3.5.17.19

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

Sval mapping: [⟨1 2 1 5 4], ⟨0 -5 16 -11 3]]

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

POTE generator: ~18/17 = 99.276

Vals: Template:Val list

Badness: 0.014623

Quintupole

The quintupole temperament tempers out the octagar comma (4000/3969) 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 polypyth fourth (~4/3, about 495.8 cents). Xenllium proposes the pronunciation of the word "quintupole" as /'kwɪntʊpəʊl/ or /'kwɪntʊpoʊl/, like as "quin-to-pole". Not to be confused with quintapole temperament (12&85).

Subgroup: 2.3.5.7

Comma list: 4000/3969, 458752/455625

Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]

Wedgie: ⟨⟨ 5 -16 -34 -37 -68 -34 ]]

POTE generator: ~135/128 = 99.175

Optimal ET sequence: 12, 97, 109, 121

Badness: 0.111620

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 1375/1372, 4375/4356

Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]

POTE generator: ~35/33 = 99.156

Vals: Template:Val list

Badness: 0.056501

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 625/624, 2704/2695

Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]

POTE generator: ~35/33 = 99.165

Vals: Template:Val list

Badness: 0.038431

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 375/374, 442/441

Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]

POTE generator: ~18/17 = 99.172

Vals: Template:Val list

Badness: 0.028721

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374

Mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]

POTE generator: ~18/17 = 99.164

Vals: Template:Val list

Badness: 0.023818

Quintapole

The quintapole temperament (12&85) tempers out the marvel comma (225/224) and 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 apollo temperament, and its generator is a semitone five of which gives a flat fourth (~4/3, about 495 cents). Xenllium proposes the pronunciation of the word "quintapole" as /'kwɪntəpəʊl/ or /'kwɪntəpoʊl/, like as "quint-a-pole". Not to be confused with quintupole temperament (12&121).

Subgroup: 2.3.5.7

Comma list: 225/224, 7812500/7411887

Mapping: [⟨1 2 1 1], ⟨0 -5 16 22]]

Wedgie: ⟨⟨ 5 -16 -22 -37 -49 -6 ]]

POTE generator: ~21/20 = 98.994

Template:Val list

Badness: 0.192498

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 1 1 0], ⟨0 -5 16 22 42]]

POTE generator: ~21/20 = 98.954

Vals: Template:Val list

Badness: 0.104353

Quinticosiennic

The quinticosiennic temperament (12&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 fourth, i.e. 12\145, is a possible generator.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 395136/390625

Mapping: [⟨1 2 1 -1], ⟨0 -5 16 46]]

Wedgie: ⟨⟨ 5 -16 -46 -37 -87 -62 ]]

POTE generator: ~135/128 = 99.345

Template:Val list

Badness: 0.158041

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 1 -1 -2], ⟨0 -5 16 46 66]]

POTE generator: ~35/33 = 99.318

Vals: Template:Val list

Badness: 0.080674

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 1 -1 -2 -3], ⟨0 -5 16 46 66 81]]

POTE generator: ~35/33 = 99.307

Vals: Template:Val list

Badness: 0.052464

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 1 -1 -2 -3 5], ⟨0 -5 16 46 66 81 -11]]

POTE generator: ~18/17 = 99.308

Vals: Template:Val list

Badness: 0.037108

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 1 -1 -2 -3 5 4], ⟨0 -5 16 46 66 81 -11 3]]

POTE generator: ~18/17 = 99.303

Vals: Template:Val list

Badness: 0.028440

Decimaleap

The decimaleap temperament (24&121) has a quarter-tone generator and tempers out the porwell comma (6144/6125) and 393379840/387420489 (sasaquadzo-ayo) in the 7-limit; 896/891 and 14700/14641 in the 11-limit. The name decimaleap comes from "decima" (Latin for "one tenth") and "leapday", because the generator is 1/10 of the leapday fourth.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 393379840/387420489

Mapping: [⟨1 2 1 5], ⟨0 -10 32 -53]]

Wedgie: ⟨⟨ 10 -32 53 -74 56 213 ]]

POTE generator: ~36/35 = 49.621

Template:Val list

Badness: 0.314227

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 6144/6125, 14700/14641

Mapping: [⟨1 2 1 5 4], ⟨0 -10 32 -53 -13]]

POTE generator: ~36/35 = 49.622

Vals: Template:Val list

Badness: 0.084612

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 676/675, 6144/6125

Mapping: [⟨1 2 1 5 4 3], ⟨0 -10 32 -53 -13 17]]

POTE generator: ~36/35 = 49.620

Vals: Template:Val list

Badness: 0.041763

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 676/675, 1156/1155

Mapping: [⟨1 2 1 5 4 3 5], ⟨0 -10 32 -53 -13 17 -22]]

POTE generator: ~34/33 = 49.621

Vals: Template:Val list

Badness: 0.025552

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 361/360, 364/363, 456/455, 665/663

Mapping: [⟨1 2 1 5 4 3 5 4], ⟨0 -10 32 -53 -13 17 -22 6]]

POTE generator: ~34/33 = 49.624

Vals: Template:Val list

Badness: 0.021208

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