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Quintaleap family: Difference between revisions

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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.  
{{Technical data page}}
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, making it a member of [[Ploidacot/Omega-pentacot|omega-pentacot]] temperaments.  


== Quintaleap ==
== Quintaleap ==
Line 10: Line 11:
[[Mapping]]: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}]
[[Mapping]]: [{{val| 1 2 1 }}, {{val| 0 -5 16 }}]


[[POTE generator]]: ~135/128 = 99.267
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.359¢, ~135/128 = 99.214¢
* [[CWE]]: ~2 = 1200.000¢, ~135/128 = 99.256¢


{{Optimal ET sequence|legend=1| 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc }}
{{Optimal ET sequence|legend=1| 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc }}


[[Badness]]: 0.444506
[[Badness]] (Sintel): 10.427


=== Subgroup temperament ===
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]].
{{see also| No-sevens subgroup temperaments #Quintaleap }}
 
==== 2.3.5.17 ====
Subgroup: 2.3.5.17
 
[[Comma list]]: 256/255, 1419857/1417176
 
[[Gencom]]: [2 18/17; 256/255 1419857/1417176]
 
[[Gencom|Gencom mapping]]: [{{val| 1 2 1 0 0 0 5 }}, {{val| 0 -5 16 0 0 0 -11 }}]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 }}, {{val| 0 -5 16 -11 }}]
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.328¢, ~18/17 = 99.217¢
* [[CWE]]: ~2 = 1200.000¢, ~18/17 = 99.270¢
 
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}


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]].
[[Badness]] (Sintel): 1.305


==== 2.3.5.17.19 ====
Subgroup: 2.3.5.17.19
Subgroup: 2.3.5.17.19


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


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


Gencom: [2 18/17; 256/255 361/360 4624/4617]
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 4 }}, {{val| 0 -5 16 -11 3 }}]


POTE generator: ~18/17 = 99.276
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.446¢, ~18/17 = 99.230¢
* [[CWE]]: ~2 = 1200.000¢, ~18/17 = 99.273¢


{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}


Badness: 0.014623
[[Badness]] (Sintel): 0.712


== Quintupole ==
== 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 [[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).''
{{Main| 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 [[Porwell temperaments #Polypyth|polypyth]] fourth (~4/3, about 495.8 cents).  


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 44: Line 70:
[[Mapping]]: [{{val| 1 2 1 0 }}, {{val| 0 -5 16 34 }}]
[[Mapping]]: [{{val| 1 2 1 0 }}, {{val| 0 -5 16 34 }}]


{{Multival|legend=1| 5 -16 -34 -37 -68 -34 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.324¢, ~135/128 = 99.120¢
[[POTE generator]]: ~135/128 = 99.175
* [[CWE]]: ~2 = 1200.000¢, ~135/128 = 99.162¢


{{Optimal ET sequence|legend=1| 12, 97, 109, 121 }}
{{Optimal ET sequence|legend=1| 12, 97, 109, 121 }}


[[Badness]]: 0.111620
[[Badness]] (Sintel): 2.825


=== 11-limit ===
=== 11-limit ===
Line 59: Line 85:
Mapping: [{{val| 1 2 1 0 -1 }}, {{val| 0 -5 16 34 54 }}]
Mapping: [{{val| 1 2 1 0 -1 }}, {{val| 0 -5 16 34 54 }}]


POTE generator: ~35/33 = 99.156
Optimal tunings:
* WE: ~2 = 1199.342¢, ~35/33 = 99.101¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.144¢


{{Optimal ET sequence|legend=1| 12, 109, 121, 351bde, 472bdee }}
Optimal ET sequence: {{Optimal ET sequence| 12, 109, 121, 351bde, 472bdee }}


Badness: 0.056501
Badness (Sintel): 1.868


=== 13-limit ===
=== 13-limit ===
Line 72: Line 100:
Mapping: [{{val| 1 2 1 0 -1 -2 }}, {{val| 0 -5 16 34 54 69 }}]
Mapping: [{{val| 1 2 1 0 -1 -2 }}, {{val| 0 -5 16 34 54 69 }}]


POTE generator: ~35/33 = 99.165
Optimal tunings:
* WE: ~2 = 1199.322¢, ~35/33 = 99.109¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.154¢


{{Optimal ET sequence|legend=1| 12f, 109, 121 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}


Badness: 0.038431
Badness (Sintel): 1.588


=== 17-limit ===
=== 17-limit ===
Line 85: Line 115:
Mapping: [{{val| 1 2 1 0 -1 -2 5 }}, {{val| 0 -5 16 34 54 69 -11 }}]
Mapping: [{{val| 1 2 1 0 -1 -2 5 }}, {{val| 0 -5 16 34 54 69 -11 }}]


POTE generator: ~18/17 = 99.172
Optimal tunings:
* WE: ~2 = 1199.223¢, ~18/17 = 99.108¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.163¢


{{Optimal ET sequence|legend=1| 12f, 109, 121 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}


Badness: 0.028721
Badness (Sintel): 1.463


=== 19-limit ===
=== 19-limit ===
Line 98: Line 130:
Mapping: [{{val| 1 2 1 0 -1 -2 5 4 }}, {{val| 0 -5 16 34 54 69 -11 3 }}]
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
Optimal tunings:
 
* WE: ~2 = 1199.368¢, ~18/17 = 99.112¢
{{Optimal ET sequence|legend=1| 12f, 109, 121 }}
* CWE: ~2 = 1200.000¢, ~18/17 = 99.158¢
 
Badness: 0.023818
 
== Quintapole ==
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
 
[[Comma list]]: 225/224, 7812500/7411887
 
[[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
 
{{Optimal ET sequence|legend=1| 12, 73c, 85, 97d }}
 
[[Badness]]: 0.192498
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
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


{{Optimal ET sequence|legend=1| 12, 73ce, 85, 97d }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 109, 121 }}


Badness: 0.104353
Badness (Sintel): 1.449


== Quinticosiennic ==
== Quinticosiennic ==
Line 143: Line 147:
[[Mapping]]: [{{val| 1 2 1 -1 }}, {{val| 0 -5 16 46 }}]
[[Mapping]]: [{{val| 1 2 1 -1 }}, {{val| 0 -5 16 46 }}]


{{Multival|legend=1| 5 -16 -46 -37 -87 -62 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.352¢, ~135/128 = 99.291¢
[[POTE generator]]: ~135/128 = 99.345
* [[CWE]]: ~2 = 1200.000¢, ~135/128 = 99.334¢


{{Optimal ET sequence|legend=1| 12, 133, 145, 157, 302c, 459bcc }}
{{Optimal ET sequence|legend=1| 12, 133, 145, 157, 302c, 459bcc }}


[[Badness]]: 0.158041
[[Badness]] (Sintel): 4.000


=== 11-limit ===
=== 11-limit ===
Line 158: Line 162:
Mapping: [{{val| 1 2 1 -1 -2 }}, {{val| 0 -5 16 46 66 }}]
Mapping: [{{val| 1 2 1 -1 -2 }}, {{val| 0 -5 16 46 66 }}]


POTE generator: ~35/33 = 99.318
Optimal tunings:
* WE: ~2 = 1199.380¢, ~35/33 = 99.266¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.308¢


{{Optimal ET sequence|legend=1| 12, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12, 133, 145 }}


Badness: 0.080674
Badness (Sintel): 2.667


=== 13-limit ===
=== 13-limit ===
Line 171: Line 177:
Mapping: [{{val| 1 2 1 -1 -2 -3 }}, {{val| 0 -5 16 46 66 81 }}]
Mapping: [{{val| 1 2 1 -1 -2 -3 }}, {{val| 0 -5 16 46 66 81 }}]


POTE generator: ~35/33 = 99.307
Optimal tunings:
* WE: ~2 = 1199.406¢, ~35/33 = 99.258¢
* CWE: ~2 = 1200.000¢, ~35/33 = 99.299¢


{{Optimal ET sequence|legend=1| 12f, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}


Badness: 0.052464
Badness (Sintel): 2.168


=== 17-limit ===
=== 17-limit ===
Line 184: Line 192:
Mapping: [{{val| 1 2 1 -1 -2 -3 5 }}, {{val| 0 -5 16 46 66 81 -11 }}]
Mapping: [{{val| 1 2 1 -1 -2 -3 5 }}, {{val| 0 -5 16 46 66 81 -11 }}]


POTE generator: ~18/17 = 99.308
Optimal tunings:
* WE: ~2 = 1199.389¢, ~18/17 = 99.257¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.302¢


{{Optimal ET sequence|legend=1| 12f, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}


Badness: 0.037108
Badness (Sintel): 1.890


=== 19-limit ===
=== 19-limit ===
Line 197: Line 207:
Mapping: [{{val| 1 2 1 -1 -2 -3 5 4 }}, {{val| 0 -5 16 46 66 81 -11 3 }}]
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
Optimal tunings:
* WE: ~2 = 1199.488¢, ~18/17 = 99.261¢
* CWE: ~2 = 1200.000¢, ~18/17 = 99.299¢


{{Optimal ET sequence|legend=1| 12f, 133, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 12f, 133, 145 }}


Badness: 0.028440
Badness (Sintel): 1.730


== Decimaleap ==
== Decimaleap ==
Line 212: Line 224:
[[Mapping]]: [{{val|1 2 1 5}}, {{val|0 -10 32 -53}}]
[[Mapping]]: [{{val|1 2 1 5}}, {{val|0 -10 32 -53}}]


{{Multival|legend=1| 10 -32 53 -74 56 213 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.430¢, ~36/35 = 49.598¢
[[POTE generator]]: ~36/35 = 49.621
* [[CWE]]: ~2 = 1200.000¢, ~36/35 = 49.623¢


{{Optimal ET sequence|legend=1| 24, 97d, 121, 145, 266 }}
{{Optimal ET sequence|legend=1| 24, 97d, 121, 145, 266 }}


[[Badness]]: 0.314227
[[Badness]] (Sintel): 7.952


=== 11-limit ===
=== 11-limit ===
Line 227: Line 239:
Mapping: [{{val|1 2 1 5 4}}, {{val|0 -10 32 -53 -13}}]
Mapping: [{{val|1 2 1 5 4}}, {{val|0 -10 32 -53 -13}}]


POTE generator: ~36/35 = 49.622
Optimal tunings:
* WE: ~2 = 1199.337¢, ~36/35 = 49.594¢
* CWE: ~2 = 1200.000¢, ~36/35 = 49.625¢


{{Optimal ET sequence|legend=1| 24, 97d, 121, 145, 266e }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97d, 121, 145, 266e }}


Badness: 0.084612
Badness (Sintel): 2.797


=== 13-limit ===
=== 13-limit ===
Line 240: Line 254:
Mapping: [{{val|1 2 1 5 4 3}}, {{val|0 -10 32 -53 -13 17}}]
Mapping: [{{val|1 2 1 5 4 3}}, {{val|0 -10 32 -53 -13 17}}]


POTE generator: ~36/35 = 49.620
Optimal tunings:
* WE: ~2 = 1199.308¢, ~36/35 = 49.591¢
* CWE: ~2 = 1200.000¢, ~36/35 = 49.622¢


{{Optimal ET sequence|legend=1| 24, 97d, 121, 266ef }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97d, 121, 266ef }}


Badness: 0.041763
Badness (Sintel): 1.726


=== 17-limit ===
=== 17-limit ===
Line 253: Line 269:
Mapping: [{{val|1 2 1 5 4 3 5}}, {{val|0 -10 32 -53 -13 17 -22}}]
Mapping: [{{val|1 2 1 5 4 3 5}}, {{val|0 -10 32 -53 -13 17 -22}}]


POTE generator: ~34/33 = 49.621
Optimal tunings:
* WE: ~2 = 1199.289¢, ~34/33 = 49.591¢
* CWE: ~2 = 1200.000¢, ~34/33 = 49.623¢


{{Optimal ET sequence|legend=1| 24, 97dg, 121, 266efg }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97dg, 121, 266efg }}


Badness: 0.025552
Badness (Sintel): 1.302


=== 19-limit ===
=== 19-limit ===
Line 266: Line 284:
Mapping: [{{val|1 2 1 5 4 3 5 4}}, {{val|0 -10 32 -53 -13 17 -22 6}}]
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
Optimal tunings:
* WE: ~2 = 1199.378¢, ~34/33 = 49.598¢
* CWE: ~2 = 1200.000¢, ~34/33 = 49.626¢


{{Optimal ET sequence|legend=1| 24, 97dg, 121, 145, 266efg }}
Optimal ET sequence: {{Optimal ET sequence| 24, 97dg, 121, 145, 266efg }}


Badness: 0.021208
Badness (Sintel): 1.290


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Quintaleap family| ]] <!-- main article -->
[[Category:Quintaleap family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Quintaleap_family"