Escapade family: Difference between revisions

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== 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 ===

[[Subgroup]]: 2.3.5

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=== 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~~. 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]]~~.

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

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{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}

Badness (~~Dirichlet~~): 0.251

Badness (Sintel): 0.251

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"

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~~{{Multival|legend=1| 9 -7 26 -32 16 80 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122

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''[[#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 ~~the [[~~7~~-limit]]~~; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.

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 ===

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~~{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243

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~~{{Multival|legend=1| 9 -7 4 -32 -19 29 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327

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: mapping generators: ~343/243, ~16875/16384

~~{{Multival|legend=1| 18 -14 30 -64 -3 109 }}~~

[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320

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: mapping generators: ~2, ~64/63

~~{{Multival|legend=1| 18 -14 -35 -64 -106 -42 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Escapade family| ]]

[[Category:Rank 2]]

@@ Line 10: / Line 10: @@
 == 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 ===
 [[Subgroup]]: 2.3.5
@@ Line 40: / Line 42: @@
 === 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. 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]].
+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
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 {{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}
-Badness (Dirichlet): 0.251
+Badness (Sintel): 0.251
 {| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
@@ Line 126: / Line 128: @@
 {{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
-{{Multival|legend=1| 9 -7 26 -32 16 80 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122
@@ Line 180: / Line 180: @@
 ''[[#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 the [[7-limit]]; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.
+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 ===
@@ Line 188: / Line 188: @@
 {{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
-{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243
@@ Line 250: / Line 248: @@
 {{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
-{{Multival|legend=1| 9 -7 4 -32 -19 29 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327
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 : mapping generators: ~343/243, ~16875/16384
-{{Multival|legend=1| 18 -14 30 -64 -3 109 }}
 [[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320
@@ Line 402: / Line 396: @@
 : mapping generators: ~2, ~64/63
-{{Multival|legend=1| 18 -14 -35 -64 -106 -42 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668
@@ Line 454: / Line 446: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Escapade family| ]] <!-- main article -->
 [[Category:Rank 2]]