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The '''escapade family''' tempers out the [[escapade_comma|escapade comma]], |32 -7 -9>, of size 9.492 [[cent]]s. It includes the [[Hemifamity temperaments|alphaquarter temperament]] and the [[Hemimean clan|arch temperament]].
{{Technical data page}}


=Escapade=
<div style="float: right;">
Comma: 4294967296/4271484375
[[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>


[[POTE_tuning|POTE generator]]: ~16875/16384 = 55.293
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]].


Map: [&lt;1 2 2|, &lt;0 -9 7|]
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.


EDOs: [[21edo|21]], [[22edo|22]], [[43edo|43]], [[65edo|65]], 152, 217, 586, 803
== Escapade ==
[[Subgroup]]: 2.3.5


Badness: 0.0838
[[Comma list]]: 4294967296/4271484375


==7-limit==
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
Commas: 225/224, 12288/12005
: mapping generators: ~2, ~16875/16384


POTE generator: ~49/48 = 55.327
[[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 }}


Map: [&lt;1 2 2 3|, &lt;0 -9 7 -4|]
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}


Wedgie: &lt;&lt;9 -7 4 -32 -19 29||
[[Badness]] (Sintel): 1.965


EDOs: 21, 22, 43, 65d
=== 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.


Badness: 0.0779
Subgroup: 2.3.5.11


==11-limit==
Comma list: 4000/3993, 5632/5625
Commas: 99/98, 176/175, 2560/2541


POTE generator: ~49/48 = 55.354
{{Mapping|legend=2| 1 2 2 3 | 0 -9 7 10 }}


Map: [&lt;1 2 2 3 3|, &lt;0 -9 7 -4 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 }}


EDOs: 21, 22, 43, 65d
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }}


Badness: 0.0367
Badness (Sintel): 0.335


==13-limit==
=== 2.3.5.11.31 subgroup ===
Commas: 78/77, 99/98, 176/175, 507/500
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.


POTE generator: ~26/25 = 55.550
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]].


Map: [&lt;1 2 2 3 3 3|, &lt;0 -9 7 -4 10 15|]
Subgroup: 2.3.5.11.31


EDOs: 21, 22, 43
Comma list: 496/495, 961/960, 4000/3993


Badness: 0.0353
{{Mapping|legend=2| 1 2 2 3 5 | 0 -9 7 10 -1 }}


=Escaped=
Optimal tunings:
Commas: 245/243, 65625/65536
* 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 }}


[[POTE_tuning|POTE generator]]: ~28/27 = 55.122
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }}


Map: [&lt;1 2 2 4|, &lt;0 -9 7 -26|]
Badness (Sintel): 0.251


Wedgie: &lt;&lt;9 -7 26 -32 16 80||
= 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


EDOs: 22, 65, [[87edo|87]], 196, 283
== 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).''


Badness: 0.0887
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]].


==11-limit (aka [[Sensamagic clan|Sensa]])==
[[Subgroup]]: 2.3.5.7
Commas: 245/243, 385/384, 3388/3375


POTE generator: ~28/27 = 55.126
[[Comma list]]: 245/243, 65625/65536


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


EDOs: 22, 65, 87, 196, 283
[[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 }}


Badness: 0.0358
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}


==13-limit==
[[Badness]] (Sintel): 2.25
Commas: 245/243, 352/351, 364/363, 625/624


POTE generator: ~28/27 = 55.138
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 2 4 3 2|, &lt;0 -9 7 -26 10 37|]
Comma list: 245/243, 385/384, 4000/3993


EDOs: 22, 65, 87, 196, 283
{{Mapping|legend=0| 1 2 2 4 3 | 0 -9 7 -26 10 }}


Badness: 0.0314
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 }}


=Biscapade=
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}
Commas: 6144/6125, 118098/117649


[[POTE_tuning|POTE generator]]: ~16875/16384 = 55.230
Badness (Sintel): 1.18


Map: [&lt;2 4 4 7|, &lt;0 -9 7 -15|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;18 -14 30 -64 -3 109||
Comma list: 245/243, 352/351, 385/384, 625/624


EDOs: 22, [[86edo|86]], 108, 130, 152, 282
{{Mapping|legend=0| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}


Badness: 0.0592
Optimal tunings:
* WE: ~2 = 1199.9926{{c}}, ~28/27 = 55.1378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1382{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}
Commas: 540/539, 4000/3993, 5632/5625


POTE generator: ~33/32 = 55.304
Badness (Sintel): 1.30


Map: [&lt;2 4 4 7 6|, &lt;0 -9 7 -15 10|]
== 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.


EDOs: 22, 86, 108, 130, 152, 282
[[Subgroup]]: 2.3.5.7


Badness: 0.0246
[[Comma list]]: 5120/5103, 29360128/29296875


==13-limit==
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
Commas: 351/350, 364/363, 540/539, 4096/4095


POTE generator: ~33/32 =55.359
[[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 }}


Map: [&lt;2 4 4 7 6 5|, &lt;0 -9 7 -15 10 26|]
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}


EDOs: 22, 108, 130
[[Badness]] (Sintel): 2.95


Badness: 0.0330
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Category:Escapade]]
Comma list: 3025/3024, 4000/3993, 5120/5103
[[Category:family]]
 
[[Category:list]]
{{Mapping|legend=0| 1 2 2 0 3 | 0 -9 7 61 10 }}
[[Category:overview]]
 
[[Category:Theory]]
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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 625/624, 847/845, 1575/1573
 
{{Mapping|legend=0| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
 
Optimal tunings:
* WE: ~2 = 1199.6491{{c}}, ~33/32 = 55.2200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2328{{c}}
 
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}
 
Badness (Sintel): 1.047
 
== 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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 12288/12005
 
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
 
[[Badness]] (Sintel): 1.97
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 2560/2541
 
{{Mapping|legend=0| 1 2 2 3 3 | 0 -9 7 -4 10 }}
 
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 }}
 
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}
 
Badness (Sintel): 1.21
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1199.5949{{c}}, ~33/32 = 55.5317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.5480{{c}}
 
{{Optimal ET sequence|legend=0| 21, 22, 43 }}
 
Badness (Sintel): 1.457
 
= 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
|}
 
== Septisuperfourth ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 118098/117649
 
{{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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 5632/5625
 
{{Mapping|legend=0| 2 4 4 7 6 | 0 -9 7 -15 10 }}
 
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 ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 4000/3993
 
{{Mapping|legend=0| 1 2 2 2 3 | 0 -18 14 35 20 }}
 
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 }}
 
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e }}
 
Badness (Sintel): 1.21
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 3136/3125
 
{{Mapping|legend=0| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}
 
Optimal tunings:
* WE: ~2 = 1199.8733{{c}}, ~64/63 = 27.6569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/63 = 27.6594{{c}}
 
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}
 
Badness (Sintel): 0.806
 
[[Category:Escapade family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Escapade_family"