Xenharmonic Wiki

Quintaleap family: Difference between revisions

← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,134 edits
m Update links
 
(9 intermediate revisions by 6 users not shown)
Line 1: Line 1:
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.  


== Quintaleap ==
== Quintaleap ==
Line 17: Line 18:


=== Subgroup temperament ===
=== 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]].
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]].


==== 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 }}]
[[Tp tuning|POL2 generator]]: ~18/17 = 99.272
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3051 cents
==== 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


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


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


POTE generator: ~18/17 = 99.276
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 4 }}, {{val| 0 -5 16 -11 3 }}]
 
[[Tp tuning|POL2 generator]]: ~18/17 = 99.276


{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3427 cents


Badness: 0.014623
Badness: 0.014623


== Quintupole ==
== Quintupole ==
{{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). [[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).''


Line 43: Line 67:


[[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 }}


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


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


[[Badness]]: 0.111620
[[Badness]]: 0.111620
Line 61: Line 83:
POTE generator: ~35/33 = 99.156
POTE generator: ~35/33 = 99.156


{{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: 0.056501
Line 74: Line 96:
POTE generator: ~35/33 = 99.165
POTE generator: ~35/33 = 99.165


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


Badness: 0.038431
Badness: 0.038431
Line 87: Line 109:
POTE generator: ~18/17 = 99.172
POTE generator: ~18/17 = 99.172


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


Badness: 0.028721
Badness: 0.028721
Line 100: Line 122:
POTE generator: ~18/17 = 99.164
POTE generator: ~18/17 = 99.164


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


Badness: 0.023818
Badness: 0.023818


== Quintapole ==
== 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).''
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 #Apollo|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
Line 112: Line 134:


[[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 130: Line 150:
POTE generator: ~21/20 = 98.954
POTE generator: ~21/20 = 98.954


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


Badness: 0.104353
Badness: 0.104353
==== Galileic ====
The name ''galileic'' comes from "{{w|Vincenzo Galilei}}", because this temperament is strongly related to [[Galilei's tuning]].
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 225/224, 275/273, 12168/12005
Mapping: [{{val| 1 2 1 1 0 -1 }}, {{val| 0 -5 16 22 42 57 }}]
POTE generator: ~21/20 = 98.992
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceff, 85f, 97d }}
Badness: 0.078183
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 120/119, 225/224, 275/273, 2431/2401
Mapping: [{{val| 1 2 1 1 0 -1 5 }}, {{val| 0 -5 16 22 42 57 -11 }}]
POTE generator: ~18/17 = 99.002
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceffg, 85fg, 97dg }}
Badness: 0.058826
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 120/119, 225/224, 247/245, 275/273, 361/357
Mapping: [{{val| 1 2 1 1 0 -1 5 4 }}, {{val| 0 -5 16 22 42 57 -11 3 }}]
POTE generator: ~18/17 = 98.987
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceffg, 85fg, 97dg }}
Badness: 0.045459
==== Catagali ====
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 225/224, 847/845, 1040/1029
Mapping: [{{val| 1 2 1 1 0 0 }}, {{val| 0 -5 16 22 42 45 }}]
POTE generator: ~21/20 = 98.882
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ce, 85 }}
Badness: 0.074180
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 120/119, 225/224, 442/441, 847/845
Mapping: [{{val| 1 2 1 1 0 0 5 }}, {{val| 0 -5 16 22 42 45 -11 }}]
POTE generator: ~18/17 = 98.905
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceg, 85g }}
Badness: 0.057924
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 120/119, 209/208, 225/224, 361/357, 442/441
Mapping: [{{val| 1 2 1 1 0 0 5 4 }}, {{val| 0 -5 16 22 42 45 -11 3 }}]
POTE generator: ~18/17 = 98.888
Optimal ET sequence: {{Optimal ET sequence| 12f, 73ceg, 85g }}
Badness: 0.044714
=== Quintain ===
Subgroup: 2.3.5.7.11
Comma list: 225/224, 245/242, 5000/4851
Mapping: [{{val| 1 2 1 1 1 }}, {{val| 0 -5 16 22 30 }}]
POTE generator: ~21/20 = 98.816
Optimal ET sequence: {{Optimal ET sequence| 12, 61c, 73c, 85e }}
Badness: 0.112830
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 245/242, 275/273, 1040/1029
Mapping: [{{val| 1 2 1 1 1 0 }}, {{val| 0 -5 16 22 30 45 }}]
POTE generator: ~21/20 = 98.796
Optimal ET sequence: {{Optimal ET sequence| 12f, 61cf, 73c, 85e }}
Badness: 0.079900


== Quinticosiennic ==
== Quinticosiennic ==
Line 142: Line 268:


[[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 }}


[[POTE generator]]: ~135/128 = 99.345
[[POTE generator]]: ~135/128 = 99.345
Line 160: Line 284:
POTE generator: ~35/33 = 99.318
POTE generator: ~35/33 = 99.318


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


Badness: 0.080674
Badness: 0.080674
Line 173: Line 297:
POTE generator: ~35/33 = 99.307
POTE generator: ~35/33 = 99.307


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


Badness: 0.052464
Badness: 0.052464
Line 186: Line 310:
POTE generator: ~18/17 = 99.308
POTE generator: ~18/17 = 99.308


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


Badness: 0.037108
Badness: 0.037108
Line 199: Line 323:
POTE generator: ~18/17 = 99.303
POTE generator: ~18/17 = 99.303


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


Badness: 0.028440
Badness: 0.028440
Line 211: Line 335:


[[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 }}


[[POTE generator]]: ~36/35 = 49.621
[[POTE generator]]: ~36/35 = 49.621
Line 229: Line 351:
POTE generator: ~36/35 = 49.622
POTE generator: ~36/35 = 49.622


{{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: 0.084612
Line 242: Line 364:
POTE generator: ~36/35 = 49.620
POTE generator: ~36/35 = 49.620


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


Badness: 0.041763
Badness: 0.041763
Line 255: Line 377:
POTE generator: ~34/33 = 49.621
POTE generator: ~34/33 = 49.621


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


Badness: 0.025552
Badness: 0.025552
Line 268: Line 390:
POTE generator: ~34/33 = 49.624
POTE generator: ~34/33 = 49.624


{{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: 0.021208


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[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"