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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.  


== Quintaleap ==
== Quintaleap ==
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=== 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
 
[[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 }}]


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


Gencom: [2 18/17; 256/255 361/360 4624/4617]
[[Tp tuning|POL2 generator]]: ~18/17 = 99.276


POTE generator: ~18/17 = 99.276
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}


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


Badness: 0.014623
Badness: 0.014623
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[[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
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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
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==== Galileic ====
==== Galileic ====
The name ''galileic'' comes from "[[wikipedia:Vincenzo Galilei|Vincenzo Galilei]]", because this temperament is strongly related to [[18/17 equal-step tuning|Galilei tuning]].
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
Subgroup: 2.3.5.7.11.13
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Badness: 0.044714
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 ==
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[[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
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[[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
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[[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"