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{{Technical data page}}
{{Technical data page}}


The '''escapade family''' 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]] temperament, tempering out [[4000/3993]] and [[5632/5625]].
<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>


Extensions of escapade to incorporate prime 7 (and therefore the full [[11-limit]]) include escapist (21 & 22), tempering out [[225/224]] and mapping 7 to -4 generators; escaped (87 & 22), tempering out [[245/243]] and mapping 7 to -26 generators; alphaquarter (65d & 87), tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (aka biscapade) (22 & 86), tempering out [[6144/6125]], splitting the octave in half and mapping 7 to -15 generators; and arch (43 & 87), tempering out [[3136/3125]] and splitting the generator into two [[64/63]] intervals; all are considered below.
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]].


<div style="float:right;">
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.
[[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>


= Escapade (5-limit) =
== Escapade ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 15: Line 15:


{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}
: mapping generators: ~2, ~16875/16384
: mapping generators: ~2, ~16875/16384


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~16875/16384 = 55.3052
* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}
* [[POTE]]: ~2 = 1\1, ~16875/16384 = 55.293
: [[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 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}


[[Badness]]: 0.083778
[[Badness]] (Sintel): 1.965
 
{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation
|-
| 3/2 || 701.955 || 702.253 || +0.298
|-
| 5/4 || 386.314 || 387.136 || +0.823
|}
 
= 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]].


== 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 (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 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
Subgroup: 2.3.5.11
Line 48: Line 34:
Comma list: 4000/3993, 5632/5625
Comma list: 4000/3993, 5632/5625


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


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 55.2760
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=1| 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee }}
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e }}


Badness: 0.0107
Badness (Sintel): 0.335


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
=== 2.3.5.11.31 subgroup ===
|+ style="font-size: 105%;" | Harmonics
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.
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation
|-
| 3/2 || 701.955 || 702.516 || +0.561
|-
| 5/4 || 386.314 || 386.932 || +0.618
|-
| 11/8 || 551.318 || 552.760 || +1.442
|}


== Strong 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]].
 
=== 2.3.5.11.31 subgroup ===
One may also note that the generator represents the square root of [[16/15]] and therefore it would be logical to temper out S31 = [[961/960]] so that the generator is equated to [[32/31]]~[[31/30]] in addition to 33/32.


Subgroup: 2.3.5.11.31
Subgroup: 2.3.5.11.31
Line 79: Line 55:
Comma list: 496/495, 961/960, 4000/3993
Comma list: 496/495, 961/960, 4000/3993


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


Optimal tuning (CTE): ~2 = 1\1, ~32/31 = 55.276
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 }}


{{Optimal ET sequence|legend=1| 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 }}


Badness (Dirichlet): 0.251
Badness (Sintel): 0.251


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
= Strong extensions =
|+ style="font-size: 105%;" | Harmonics
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*
|-
|-
! CTE tuning !! Deviation
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
|-
|-
| 3/2 || 701.955 || 702.518 || +0.563
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br> ↓ [[87edo|87]]
|-
|-
| 5/4 || 386.314 || 386.931 || +0.617
| [[#Escaped|Escaped]] || -26 || ↑ 87
|-
| 11/8 || 551.318 || 552.758 || +1.440
|-
| 31/16 || 1145.036 || 1144.724 || -0.311
|}
|}
<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).''
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]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 245/243, 65625/65536
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
[[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 }}


=== Escapist ===
[[Badness]] (Sintel): 2.25
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]].


=== 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|legend=0| 1 2 2 4 3 | 0 -9 7 -26 10 }}
 
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 }}
 
{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}


Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}
Badness (Sintel): 1.18


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.354
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
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 }}


Badness: 0.036700
Optimal tunings:
* WE: ~2 = 1199.9926{{c}}, ~28/27 = 55.1378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1382{{c}}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation
|-
| 3/2 || 701.955 || 701.626 || -0.329
|-
| 5/4 || 386.314 || 387.624 || +1.310
|-
| 7/4 || 968.826 || 978.501 || +0.617
|-
| 11/8 || 551.318 || 553.749 || +2.431
|}


==== 13-limit ====
Badness (Sintel): 1.30
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 99/98, 176/175, 507/500
== 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.


Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 55.550
[[Comma list]]: 5120/5103, 29360128/29296875


{{Optimal ET sequence|legend=1| 21, 22, 43 }}
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}


Badness: 0.035261
[[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 }}


=== Escaped ===
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
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 19e &amp; 27 temperament (sensi extension).''


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]].
[[Badness]] (Sintel): 2.95


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


Comma list: 245/243, 385/384, 4000/3993
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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
Comma list: 352/351, 625/624, 847/845, 1575/1573


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126
{{Mapping|legend=0| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
Optimal tunings:
* WE: ~2 = 1199.6491{{c}}, ~33/32 = 55.2200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2328{{c}}


Badness: 0.035844
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Badness (Sintel): 1.047
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation
|-
| 3/2 || 701.955 || 703.831 || +1.876
|-
| 5/4 || 386.314 || 385.909 || -0.405
|-
| 7/4 || 968.826 || 966.624 || -2.202
|-
| 11/8 || 551.318 || 551.299 || -0.019
|}


==== 13-limit ====
== Escapist ==
Subgroup: 2.3.5.7.11.13
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, 352/351, 385/384, 625/624
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
[[Comma list]]: 225/224, 12288/12005


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138
{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}


{{Optimal ET sequence|legend=1| 22, 65, 87, 283 }}
[[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 }}


Badness: 0.031366
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}


=== Alphaquarter ===
[[Badness]] (Sintel): 1.97
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.


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


Comma list: 3025/3024, 4000/3993, 5120/5103
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: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
{{Mapping|legend=0| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.243
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=1| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


Badness: 0.029638
Badness (Sintel): 1.457


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
= Weak extensions =
|+ style="font-size: 105%;" | Harmonics
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to weak extensions
|-
|-
! rowspan="2" | Prime harmonic !! rowspan="2" | Just tuning !! colspan="2" | Tunings
! rowspan="2" | Extensions !! rowspan="2" | Periods per octave !! colspan="2" | Position of original generator
|-
|-
! CTE tuning !! Deviation
! Number of generators !! Number of periods
|-
|-
| 3/2 || 701.955 || 702.918 || +0.963
| [[#Septisuperfourth|Septisuperfourth]] || period = 1/2 octave || 1 generator || + 0 periods
|-
|-
| 5/4 || 386.314 || 386.620 || +0.306
| [[#Arch|Arch]] || period = octave || 2 generators || + 0 periods
|-
| 7/4 || 968.826 || 969.113 || +0.287
|-
| 11/8 || 551.318 || 552.314 || +0.996
|}
|}


==== 13-limit ====
== Septisuperfourth ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 352/351, 625/624, 847/845, 1575/1573
[[Comma list]]: 6144/6125, 118098/117649


Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}
: mapping generators: ~343/243, ~405/392


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
[[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| 65d, 87, 152f, 239f }}
{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
 
Badness: 0.025344


== Weak extensions ==
[[Badness]] (Sintel): 1.50


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


Comma list: 540/539, 4000/3993, 5632/5625
Comma list: 540/539, 4000/3993, 5632/5625


Mapping: {{mapping| 2 4 4 7 6 | 0 -9 7 -15 10 }}
{{Mapping|legend=0| 2 4 4 7 6 | 0 -9 7 -15 10 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~33/32 = 55.304
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=1| 22, 86, 108, 130, 152, 282 }}
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282, 434de, 716dee, 1150cdddeee }}


Badness: 0.024619
Badness (Sintel): 0.814


==== 13-limit ====
==== 13-limit ====
Line 257: Line 289:
Comma list: 540/539, 729/728, 1575/1573, 3584/3575
Comma list: 540/539, 729/728, 1575/1573, 3584/3575


Mapping: {{mapping| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}
{{Mapping|legend=0| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~33/32 = 55.325
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=1| 22f, 108f, 130, 282 }}
{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282, 976cddeeeff }}


Badness: 0.022887
Badness (Sintel): 0.946


==== Septisuperquad ====
==== Septisuperquad ====
This temperament is also known as "biscapade".
This temperament is also known as ''biscapade''.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 272: Line 306:
Comma list: 351/350, 364/363, 540/539, 4096/4095
Comma list: 351/350, 364/363, 540/539, 4096/4095


Mapping: {{mapping| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}
{{Mapping|legend=0| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~33/32 = 55.359
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=1| 22, 108, 130 }}
{{Optimal ET sequence|legend=0| 22, 108, 130 }}


Badness: 0.033038
Badness (Sintel): 1.37


=== Arch ===
== 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
Subgroup: 2.3.5.7.11


Comma list: 441/440, 3136/3125, 4000/3993
Comma list: 441/440, 3136/3125, 4000/3993


Mapping: {{mapping| 1 2 2 2 3 | 0 -18 14 35 20 }}
{{Mapping|legend=0| 1 2 2 2 3 | 0 -18 14 35 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~64/63 = 27.663
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=1| 43, 87, 130, 217, 347e, 911cde }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e }}


Badness: 0.036541
Badness (Sintel): 1.21


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


Comma list: 364/363, 441/440, 676/675, 3136/3125
Comma list: 364/363, 441/440, 676/675, 3136/3125


Mapping: {{mapping| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}
{{Mapping|legend=0| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~64/63 = 27.660
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=1| 43, 87, 130, 217, 347e, 564e }}
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}


Badness: 0.019504
Badness (Sintel): 0.806


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