Escapade family: Difference between revisions
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== Escapade == | == Escapade == | ||
For intervals along the chain of generators in the 2.3.5.11.31 subgroup temperament, out to 22 generators up, see the third column of [[16ed5/3#Intervals]]. | |||
=== 5-limit === | === 5-limit === | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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=== 2.3.5.11 subgroup === | === 2.3.5.11 subgroup === | ||
Since (an ideally slightly flat) 4/3 is split in three by the interval of 3 generators, it makes sense to equate that interval to [[11/10]] by tempering out 4000/3993, and therefore the generator to {{nowrap|(11/10)/(16/15) {{=}} [[33/32]]}}; this does minimal damage to the temperament. This structure in 2.3.5.11 occurs in all extensions of escapade to include prime 7, and therefore will be considered the fount of all further extensions | Since (an ideally slightly flat) 4/3 is split in three by the interval of 3 generators, it makes sense to equate that interval to [[11/10]] by tempering out 4000/3993, and therefore the generator to {{nowrap|(11/10)/(16/15) {{=}} [[33/32]]}}; this does minimal damage to the temperament. This structure in 2.3.5.11 occurs in all extensions of escapade to include prime 7, and therefore will be considered the fount of all further extensions. | ||
Subgroup: 2.3.5.11 | Subgroup: 2.3.5.11 | ||
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{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }} | {{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }} | ||
Badness ( | Badness (Sintel): 0.251 | ||
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed" | {| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed" | ||
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Here, [[245/243]] is tempered out so that [[9/7]] is equated to the square root of 5/3 (at 8 generators) present in the temperament. This works best where 5/3 is slightly flat, therefore on the end of the spectrum approaching [[22edo]]. | Here, [[245/243]] is tempered out so that [[9/7]] is equated to the square root of 5/3 (at 8 generators) present in the temperament. This works best where 5/3 is slightly flat, therefore on the end of the spectrum approaching [[22edo]]. | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }} | {{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122 | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122 | ||
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''[[#Strong extensions|Return to the map]]'' | ''[[#Strong extensions|Return to the map]]'' | ||
Given the slightly sharp ~[[3/2]] in ideal tunings of escapade (between [[65edo]] and [[87edo]]), it does very little damage to temper out [[5120/5103]] to extend it to | Given the slightly sharp ~[[3/2]] in ideal tunings of escapade (between [[65edo]] and [[87edo]]), it does very little damage to temper out [[5120/5103]] to extend it to prime 7; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up. | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }} | {{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243 | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243 | ||
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This temperament makes the identification of the 4-generator interval, representing (16/15)<sup>2</sup>, with [[8/7]] by tempering out [[225/224]] (along with [[12288/12005]]); however, this is somewhat inaccurate as the ~16/15 in escapade is slightly flat, while for a good marvel tuning it needs to be tempered sharpward to equate it with [[15/14]]. | This temperament makes the identification of the 4-generator interval, representing (16/15)<sup>2</sup>, with [[8/7]] by tempering out [[225/224]] (along with [[12288/12005]]); however, this is somewhat inaccurate as the ~16/15 in escapade is slightly flat, while for a good marvel tuning it needs to be tempered sharpward to equate it with [[15/14]]. | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }} | {{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327 | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327 | ||
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''[[#Weak extensions|Return to map]]'' | ''[[#Weak extensions|Return to map]]'' | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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: mapping generators: ~343/243, ~16875/16384 | : mapping generators: ~343/243, ~16875/16384 | ||
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320 | [[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320 | ||
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''[[#Weak extensions|Return to map]]'' | ''[[#Weak extensions|Return to map]]'' | ||
=== 7-limit === | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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: mapping generators: ~2, ~64/63 | : mapping generators: ~2, ~64/63 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668 | ||
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[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Escapade family| ]] <!-- main article --> | [[Category:Escapade family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||