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The '''escapade family''' tempers out the [[escapade comma]], {{monzo|32 -7 -9}}, of size 9.492 [[cent]]s.
{{Technical data page}}
 
<div style="float: right;">
[[File:Escapade.png|alt=Escapade.png|thumb|600x560px|An image of the tuning spectrum of 2.3.5.11 escapade, in terms of the generator; [[Edo]] [[patent val]] tunings are marked with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled.]]
</div>
 
The '''escapade family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[escapade comma]], {{monzo| 32 -7 -9 }}, of size 9.492 [[cent]]s. The defining feature of this comma is splitting [[5/3]] into sixteen quartertones of which [[5/4]] makes up seven and [[4/3]] makes up nine; therefore [[16/15]] is two generator steps. It most naturally manifests as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament, tempering out [[4000/3993]] and [[5632/5625]].
 
Extensions of escapade to incorporate prime 7 (and therefore the full [[11-limit]]) include escapist ({{nowrap| 21 & 22 }}), tempering out [[225/224]] and mapping 7 to −4 generators; escaped ({{nowrap| 22 & 87 }}), tempering out [[245/243]] and mapping 7 to −26 generators; alphaquarter ({{nowrap| 65d & 87 }}), tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (a.k.a. biscapade) ({{nowrap| 22 & 86 }}), tempering out [[6144/6125]], splitting the octave in half and mapping 7 to −15 generators; and arch ({{nowrap| 43 & 87 }}), tempering out [[3136/3125]] and splitting the generator into two [[64/63]] intervals; all are considered below.


== Escapade ==
== Escapade ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 4294967296/4271484375
[[Comma list]]: 4294967296/4271484375


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -9 7 }}]
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
: mapping generators: ~2, ~16875/16384
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}
: [[error map]]: {{val| -0.192 +0.105 +0.291 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16875/16384 = 55.2961{{c}}
: error map: {{val| 0.000 +0.380 +0.759 }}
 
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}
 
[[Badness]] (Sintel): 1.965
 
=== 2.3.5.11 subgroup ===
Since (an ideally slightly flat) 4/3 is split into 9 generators, it makes sense to equate the 3-generator 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
 
Comma list: 4000/3993, 5632/5625
 
{{Mapping|legend=2| 1 2 2 3 | 0 -9 7 10 }}
 
Optimal tunings:
* WE: ~2 = 1199.7406{{c}}, ~33/32 = 55.2653{{c}}
: error map: {{val| -0.259 +0.139 +0.024 +0.557 }}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2770{{c}}
: error map: {{val| 0.000 +0.552 +0.625 +1.452 }}
 
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }}
 
Badness (Sintel): 0.335
 
=== 2.3.5.11.31 subgroup ===
One may note that the generator represents the square root of [[16/15]] and therefore it would be logical to also temper out [[961/960]] ({{S|31}}) so that the generator is equated to {{nowrap| [[32/31]][[~]][[31/30]] }} in addition to 33/32.
 
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]].


[[POTE generator]]: ~16875/16384 = 55.293
Subgroup: 2.3.5.11.31


{{Val list|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}
Comma list: 496/495, 961/960, 4000/3993


[[Badness]]: 0.083778
{{Mapping|legend=2| 1 2 2 3 5 | 0 -9 7 10 -1 }}


=== Extensions ===
Optimal tunings:
* WE: ~2 = 1199.8050{{c}}, ~32/31 = 55.2669{{c}}
: error map: {{val| -0.195 +0.253 +0.165 +0.766 -1.277 }}
* CWE: ~2 = 1200.0000{{c}}, ~32/31 = 55.2759{{c}}
: error map: {{val| 0.000 +0.562 +0.617 +1.441 -0.311 }}


Extensions of escapade include (septimal) escapade, escaped and biscapade, considered below, as well as these temperaments discussed elsewhere:
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }}
* ''[[arch]]'', {3136/3125, 5250987/5242880} → [[Hemimean clan #Arch]]
* ''[[alphaquarter]]'', {5120/5103, 29360128/29296875} → [[Hemifamity temperaments #Alphaquarter]]


== Septimal escapade ==
Badness (Sintel): 0.251
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 12288/12005
= Strong extensions =
{| class="wikitable center-all"
|+ 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*
|-
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
|-
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br> ↓ [[87edo|87]]
|-
| [[#Escaped|Escaped]] || -26 || ↑ 87
|}
<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions
 
== Escaped ==
This temperament was also known as ''sensa'' in earlier materials because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with the {{nowrap| 19e & 27 }} temperament (sensi extension).''


[[Mapping]]: [{{val|1 2 2 3}}, {{val|0 -9 7 -4}}]
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]].


{{Multival|legend=1|9 -7 4 -32 -19 29}}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~49/48 = 55.327
[[Comma list]]: 245/243, 65625/65536


{{Val list|legend=1| 21, 22, 43, 65d }}
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


[[Badness]]: 0.077950
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9190{{c}}, ~28/27 = 55.1186{{c}}
: [[error map]]: {{val| -0.081 +1.816 -0.646 -2.232 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 55.1242{{c}}
: error map: {{val| 0.000 +1.927 -0.444 -2.056 }}
 
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
 
[[Badness]] (Sintel): 2.25


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 2560/2541
Comma list: 245/243, 385/384, 4000/3993


Mapping: [{{val|1 2 2 3 3}}, {{val|0 -9 7 -4 10}}]
{{Mapping|legend=0| 1 2 2 4 3 | 0 -9 7 -26 10 }}


POTE generator: ~49/48 = 55.354
Optimal tunings:
* WE: ~2 = 1199.9480{{c}}, ~28/27 = 55.1241{{c}}
: error map: {{val| -0.052 +1.824 -0.549 -2.261 -0.233 }}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1271{{c}}
: error map: {{val| 0.000 +1.901 -0.424 -2.131 -0.047 }}


Vals: {{Val list| 21, 22, 43, 65d }}
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}


Badness: 0.036700
Badness (Sintel): 1.18


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 99/98, 176/175, 507/500
Comma list: 245/243, 352/351, 385/384, 625/624
 
{{Mapping|legend=0| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
 
Optimal tunings:
* WE: ~2 = 1199.9926{{c}}, ~28/27 = 55.1378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1382{{c}}
 
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}
 
Badness (Sintel): 1.30
 
== Alphaquarter ==
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 29360128/29296875
 
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7349{{c}}, ~16128/15625 = 55.2306{{c}}
: [[error map]]: {{val| -0.265 +0.439 -0.230 +0.242 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16128/15625 = 55.2405{{c}}
: error map: {{val| 0.000 +0.880 +0.370 +0.846 }}
 
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
 
[[Badness]] (Sintel): 2.95
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 5120/5103
 
{{Mapping|legend=0| 1 2 2 0 3 | 0 -9 7 61 10 }}
 
Optimal tunings:
* WE: ~2 = 1199.7229{{c}}, ~33/32 = 55.2303{{c}}
: error map: {{val| -0.277 +0.418 -0.256 +0.220 +0.153 }}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2407{{c}}
: error map: {{val| 0.000 +0.879 +0.371 +0.858 +1.089 }}
 
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}
 
Badness (Sintel): 0.980


Mapping: [{{val|1 2 2 3 3 3}}, {{val|0 -9 7 -4 10 15}}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~26/25 = 55.550
Comma list: 352/351, 625/624, 847/845, 1575/1573


Vals: {{Val list| 21, 22, 43 }}
{{Mapping|legend=0| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


Badness: 0.035261
Optimal tunings:
* WE: ~2 = 1199.6491{{c}}, ~33/32 = 55.2200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2328{{c}}


== Escaped ==
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}
{{See also|Sensamagic clan #Escaped}}


This temperament is also known as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with 19e&amp;27 temperament (sensi extension).''
Badness (Sintel): 1.047


Subgroup: 2.3.5.7
== Escapist ==
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]].


[[Comma list]]: 245/243, 65625/65536
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 2 4 }}, {{val| 0 -9 7 -26 }}]
[[Comma list]]: 225/224, 12288/12005


{{Multival|legend=1| 9 -7 26 -32 16 80 }}
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}


[[POTE generator]]: ~28/27 = 55.122
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9926{{c}}, ~49/48 = 55.2809{{c}}
: [[error map]]: {{val| -1.007 -1.498 -1.363 +7.028 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 55.3479{{c}}
: error map: {{val| 0.000 -0.086 +1.122 +9.782 }}


{{Val list|legend=1| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}


[[Badness]]: 0.088746
[[Badness]] (Sintel): 1.97


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 245/243, 385/384, 4000/3993
Comma list: 99/98, 176/175, 2560/2541


Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]
{{Mapping|legend=0| 1 2 2 3 3 | 0 -9 7 -4 10 }}


POTE generator: ~28/27 = 55.126
Optimal tunings:
* WE: ~2 = 1199.0859{{c}}, ~33/32 = 55.3117{{c}}
: error map: {{val| -0.914 -1.588 -0.960 +7.185 -0.944 }}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.3574{{c}}
: error map: {{val| 0.000 -0.172 +1.188 +9.745 +2.256 }}


Vals: {{Val list| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


Badness: 0.035844
Badness (Sintel): 1.21


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 245/243, 352/351, 385/384, 625/624
Comma list: 78/77, 99/98, 176/175, 507/500
 
{{Mapping|legend=0| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}


Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]
Optimal tunings:
* WE: ~2 = 1199.5949{{c}}, ~33/32 = 55.5317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.5480{{c}}


POTE generator: ~28/27 = 55.138
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


Vals: {{Val list| 22, 65, 87, 283 }}
Badness (Sintel): 1.457


Badness: 0.031366
= Weak extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to weak extensions
|-
! rowspan="2" | Extensions !! rowspan="2" | Periods per octave !! colspan="2" | Position of original generator
|-
! Number of generators !! Number of periods
|-
| [[#Septisuperfourth|Septisuperfourth]] || period = 1/2 octave || 1 generator || + 0 periods
|-
| [[#Arch|Arch]] || period = octave || 2 generators || + 0 periods
|}


== Biscapade ==
== Septisuperfourth ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 118098/117649
[[Comma list]]: 6144/6125, 118098/117649


[[Mapping]]: [{{val|2 4 4 7}}, {{val|0 -9 7 -15}}]
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}
: mapping generators: ~343/243, ~405/392
 
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8762{{c}}, ~405/392 = 55.3089{{c}}
: [[error map]]: {{val| -0.248 -0.230 +0.353 +0.674 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~405/392 = 55.3273{{c}}
: error map: {{val| 0.000 +0.100 +0.977 +1.265 }}
 
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
 
[[Badness]] (Sintel): 1.50


{{Multival|legend=1|18 -14 30 -64 -3 109}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~16875/16384 = 55.320
Comma list: 540/539, 4000/3993, 5632/5625


{{Val list|legend=1| 22, 86, 108, 130, 152, 282 }}
{{Mapping|legend=0| 2 4 4 7 6 | 0 -9 7 -15 10 }}


[[Badness]]: 0.059241
Optimal tunings:
* WE: ~99/70 = 599.8383{{c}}, ~33/32 = 55.2895{{c}}
: error map: {{val| -0.323 -0.207 +0.066 +0.700 +0.606 }}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3081{{c}}
: error map: {{val| 0.000 +0.272 +0.843 +1.553 +1.763 }}
 
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282, 434de, 716dee, 1150cdddeee }}
 
Badness (Sintel): 0.814
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1575/1573, 3584/3575
 
{{Mapping|legend=0| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8331{{c}}, ~33/32 = 55.3093{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/32 = 55.3295{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282, 976cddeeeff }}
 
Badness (Sintel): 0.946
 
==== Septisuperquad ====
This temperament is also known as ''biscapade''.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 540/539, 4096/4095
 
{{Mapping|legend=0| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}
 
Optimal tunings:
* WE: ~55/39 = 599.9152{{c}}, ~33/32 = 55.3509{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~33/32 = 55.3584{{c}}
 
{{Optimal ET sequence|legend=0| 22, 108, 130 }}
 
Badness (Sintel): 1.37
 
== Arch ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 5250987/5242880
 
{{Mapping|legend=1| 1 2 2 2 | 0 -18 14 35 }}
: mapping generators: ~2, ~64/63
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9246{{c}}, ~64/63 = 27.6662{{c}}
: [[error map]]: {{val| -0.075 -0.097 +0.862 -0.661 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/63 = 27.6676{{c}}
: error map: {{val| 0.000 +0.029 +1.032 -0.461 }}
 
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347 }}
 
[[Badness]] (Sintel): 2.39


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4000/3993, 5632/5625
Comma list: 441/440, 3136/3125, 4000/3993


Mapping: [{{val|2 4 4 7 6}}, {{val|0 -9 7 -15 10}}]
{{Mapping|legend=0| 1 2 2 2 3 | 0 -18 14 35 20 }}


POTE generator: ~33/32 = 55.304
Optimal tunings:
* WE: ~2 = 1199.8347{{c}}, ~64/63 = 27.6590{{c}}
: error map: {{val| -0.165 -0.147 +0.581 -1.092 +1.366 }}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6617{{c}}
: error map: {{val| 0.000 +0.134 +0.950 -0.667 1.916 }}


Vals: {{Val list| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e }}


Badness: 0.024619
Badness (Sintel): 1.21


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 364/363, 540/539, 4096/4095
Comma list: 364/363, 441/440, 676/675, 3136/3125


Mapping: [{{val|2 4 4 7 6 5}}, {{val|0 -9 7 -15 10 26}}]
{{Mapping|legend=0| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}


POTE generator: ~33/32 = 55.359
Optimal tunings:
* WE: ~2 = 1199.8733{{c}}, ~64/63 = 27.6569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6594{{c}}


Vals: {{Val list| 22, 108, 130 }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}


Badness: 0.033038
Badness (Sintel): 0.806


[[Category:Theory]]
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Escapade]]
[[Category:Catalogs of rank-2 temperaments]]
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