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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|}

== Strong extensions ==

= Strong extensions =

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Map to strong full 11-limit extensions

|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions

|-

! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*

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<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions

=== Escaped ===

== Escaped ==

''[[#Strong extensions|Return to the map]]''

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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]].

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

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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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[[Badness]] (Smith): 0.088746

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

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Badness (Smith): 0.031366

=== Alphaquarter ===

== Alphaquarter ==

''[[#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 ===

[[Subgroup]]: 2.3.5.7

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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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[[Badness]] (Smith): 0.116594

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

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Badness (Smith): 0.025344

=== Escapist ===

== Escapist ==

''[[#Strong extensions|Return to the map]]''

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

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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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[[Badness]] (Smith): 0.077950

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

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Badness (Smith): 0.035261

== Weak extensions ==

= Weak extensions =

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Map to weak extensions

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|}

=== Septisuperfourth ===

== Septisuperfourth ==

''[[#Weak extensions|Return to map]]''

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

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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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[[Badness]] (Smith): 0.059241

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

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Badness (Smith): 0.033038

=== Arch ===

== Arch ==

''[[#Weak extensions|Return to map]]''

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

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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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[[Badness]] (Smith): 0.094345

==== 11-limit ====

=== 11-limit ===

Subgroup: 2.3.5.7.11

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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
@@ Line 81: / Line 83: @@
 {{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 99: / Line 101: @@
 |}
-== Strong extensions ==
+= Strong extensions =
 {| class="wikitable center-all"
-|+ style="font-size: 105%;" | Map to strong full 11-limit extensions
+|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
 |-
 ! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*
@@ Line 113: / Line 115: @@
 <nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions
-=== Escaped ===
+== Escaped ==
 ''[[#Strong extensions|Return to the map]]''
@@ Line 120: / Line 122: @@
 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
@@ Line 125: / 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 134: / Line 135: @@
 [[Badness]] (Smith): 0.088746
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 176: / Line 177: @@
 Badness (Smith): 0.031366
-=== Alphaquarter ===
+== Alphaquarter ==
 ''[[#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 ===
 [[Subgroup]]: 2.3.5.7
@@ Line 186: / 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 195: / Line 195: @@
 [[Badness]] (Smith): 0.116594
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 237: / Line 237: @@
 Badness (Smith): 0.025344
-=== Escapist ===
+== Escapist ==
 ''[[#Strong extensions|Return to the map]]''
 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
@@ Line 247: / 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
@@ Line 256: / Line 255: @@
 [[Badness]] (Smith): 0.077950
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 298: / Line 297: @@
 Badness (Smith): 0.035261
-== Weak extensions ==
+= Weak extensions =
 {| class="wikitable center-all"
 |+ style="font-size: 105%;" | Map to weak extensions
@@ Line 311: / Line 310: @@
 |}
-=== Septisuperfourth ===
+== Septisuperfourth ==
 ''[[#Weak extensions|Return to map]]''
+=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 321: / Line 321: @@
 : 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 330: / Line 328: @@
 [[Badness]] (Smith): 0.059241
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 387: / Line 385: @@
 Badness (Smith): 0.033038
-=== Arch ===
+== Arch ==
 ''[[#Weak extensions|Return to map]]''
+=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 397: / 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 406: / Line 403: @@
 [[Badness]] (Smith): 0.094345
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 449: / Line 446: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Escapade family| ]] <!-- main article -->
 [[Category:Rank 2]]