Escapade family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
<div style="float: right;">  
<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.]]
[[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>
</div>


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]].
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 &amp; 22)}}, tempering out [[225/224]] and mapping 7 to −4 generators; escaped {{nowrap|(22 &amp; 87)}}, tempering out [[245/243]] and mapping 7 to −26 generators; alphaquarter {{nowrap|(65d &amp; 87)}}, tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (aka biscapade) {{nowrap|(22 &amp; 86)}}, tempering out [[6144/6125]], splitting the octave in half and mapping 7 to −15 generators; and arch {{nowrap|(43 &amp; 87)}}, tempering out [[3136/3125]] and splitting the generator into two [[64/63]] intervals; all are considered below.
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 (5-limit) =
== Escapade ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 14: 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"
=== 2.3.5.11 subgroup ===
|+ style="font-size: 105%;" | Harmonics
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.
|-
 
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
Subgroup: 2.3.5.11
|-
 
! CTE tuning !! Deviation from just
Comma list: 4000/3993, 5632/5625
|-
 
| 3/2 || 702.253 || +0.298
{{Mapping|legend=2| 1 2 2 3 | 0 -9 7 10 }}
|-
 
| 5/4 || 387.136 || +0.823
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 }}


= Escapade =
Badness (Sintel): 0.335
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 ==
=== 2.3.5.11.31 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.
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.


Subgroup: 2.3.5.11
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]].


Comma list: 4000/3993, 5632/5625
Subgroup: 2.3.5.11.31


Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}
Comma list: 496/495, 961/960, 4000/3993


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 55.2760
{{Mapping|legend=2| 1 2 2 3 5 | 0 -9 7 10 -1 }}


{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee }}
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 }}


Badness: 0.0107
{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Badness (Sintel): 0.251
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.516 || +0.561
|-
| 5/4 || 386.932 || +0.618
|-
| 11/8 || 552.760 || +1.442
|}


== Strong extensions ==
= Strong extensions =
{| class="wikitable center-all"
{| 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*
! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*
Line 77: Line 75:
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
|-
|-
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br /> ↓ [[87edo|87]]  
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br> ↓ [[87edo|87]]  
|-
|-
| [[#Escaped|Escaped]] || -26 || ↑ 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
<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).''


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


Subgroup: 2.3.5.11.31
[[Subgroup]]: 2.3.5.7


Comma list: 496/495, 961/960, 4000/3993
[[Comma list]]: 245/243, 65625/65536


Mapping: {{Mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


Optimal tuning (CTE): ~2 = 1\1, ~32/31 = 55.276
[[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| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}


Badness (Dirichlet): 0.251
[[Badness]] (Sintel): 2.25


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
=== 11-limit ===
|+ style="font-size: 105%;" | Harmonics
Subgroup: 2.3.5.7.11
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.518 || +0.563
|-
| 5/4 || 386.931 || +0.617
|-
| 11/8 || 552.758 || +1.440
|-
| 31/16 || 1144.724 || -0.311
|}


=== Escapist ===
Comma list: 245/243, 385/384, 4000/3993
''[[#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]].
{{Mapping|legend=0| 1 2 2 4 3 | 0 -9 7 -26 10 }}


Subgroup: 2.3.5.7.11
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 }}


Comma list: 99/98, 176/175, 2560/2541
{{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


Badness: 0.036700
{{Mapping|legend=0| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Optimal tunings:
|+ style="font-size: 105%;" | Harmonics
* WE: ~2 = 1199.9926{{c}}, ~28/27 = 55.1378{{c}}
|-
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 55.1382{{c}}
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 701.626 || -0.329
|-
| 5/4 || 387.624 || +1.310
|-
| 7/4 || 978.501 || +9.675
|-
| 11/8 || 553.749 || +2.431
|}


==== 13-limit ====
{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 99/98, 176/175, 507/500
Badness (Sintel): 1.30


Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}
== 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.


Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 55.550
[[Subgroup]]: 2.3.5.7


{{Optimal ET sequence|legend=1| 21, 22, 43 }}
[[Comma list]]: 5120/5103, 29360128/29296875


Badness: 0.035261
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}


=== Escaped ===
[[Optimal tuning]]s:
''[[#Strong extensions|Return to the map]]''
* [[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 }}


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


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


Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126
Comma list: 352/351, 625/624, 847/845, 1575/1573


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
{{Mapping|legend=0| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


Badness: 0.035844
Optimal tunings:
* WE: ~2 = 1199.6491{{c}}, ~33/32 = 55.2200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2328{{c}}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 703.831 || +1.876
|-
| 5/4 || 385.909 || -0.405
|-
| 7/4 || 966.624 || -2.202
|-
| 11/8 || 551.299 || -0.019
|}


==== 13-limit ====
Badness (Sintel): 1.047
Subgroup: 2.3.5.7.11.13


Comma list: 245/243, 352/351, 385/384, 625/624
== 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]].


Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
[[Subgroup]]: 2.3.5.7


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138
[[Comma list]]: 225/224, 12288/12005


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


Badness: 0.031366
[[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 }}


=== Alphaquarter ===
{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}
''[[#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.
[[Badness]] (Sintel): 1.97


=== 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: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.243
{{Mapping|legend=0| 1 2 2 3 3 | 0 -9 7 -4 10 }}


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
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 }}


Badness: 0.029638
{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Badness (Sintel): 1.21
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.918 || +0.963
|-
| 5/4 || 386.620 || +0.306
|-
| 7/4 || 969.113 || +0.287
|-
| 11/8 || 552.314 || +0.996
|}


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


Comma list: 352/351, 625/624, 847/845, 1575/1573
Comma list: 78/77, 99/98, 176/175, 507/500


Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
{{Mapping|legend=0| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
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, 152f, 239f }}
{{Optimal ET sequence|legend=0| 21, 22, 43 }}


Badness: 0.025344
Badness (Sintel): 1.457


== Weak extensions ==
= Weak extensions =
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to weak extensions
|+ style="font-size: 105%;" | Map to weak extensions
Line 267: Line 249:
|}
|}


=== Septisuperfourth ===
== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''
[[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
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 ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}
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 }}


Badness: 0.024619
{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282, 434de, 716dee, 1150cdddeee }}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Badness (Sintel): 0.814
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.070 || +0.115
|-
| 5/4 || 387.279 || +0.965
|-
| 7/4 || 970.117 || +1.291
|-
| 11/8 || 553.255 || +1.937
|}


==== 13-limit ====
==== 13-limit ====
Line 303: 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 318: 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 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


Optimal tuning (POTE): ~55/39 = 1\2, ~33/32 = 55.359
{{Mapping|legend=1| 1 2 2 2 | 0 -18 14 35 }}
: mapping generators: ~2, ~64/63


{{Optimal ET sequence|legend=1| 22, 108, 130 }}
[[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 }}


Badness: 0.033038
{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347 }}


=== Arch ===
[[Badness]] (Sintel): 2.39
''[[#Weak extensions|Return to map]]''


=== 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 ET sequence|legend=1| 43, 87, 130, 217, 347e, 911cde }}
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 }}


Badness: 0.036541
{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e }}


{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
Badness (Sintel): 1.21
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.178 || +0.223
|-
| 5/4 || 387.195 || +0.881
|-
| 7/4 || 967.987 || -0.839
|-
| 11/8 || 553.135 || +1.817
|}


==== 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]]

Latest revision as of 09:34, 4 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
Escapade.png
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.

The escapade family of temperaments tempers out the escapade comma, [32 -7 -9⟩, of size 9.492 cents. 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.

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 (22 & 87), tempering out 245/243 and mapping 7 to −26 generators; alphaquarter (65d & 87), tempering out 5120/5103 and mapping 7 to 61 generators; septisuperfourth (a.k.a. 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.

Escapade

Subgroup: 2.3.5

Comma list: 4294967296/4271484375

Mapping: [⟨1 2 2], ⟨0 -9 7]]

mapping generators: ~2, ~16875/16384

Optimal tunings:

  • WE: ~2 = 1199.8082 ¢, ~16875/16384 = 55.2840 ¢
error map: ⟨-0.192 +0.105 +0.291]
  • CWE: ~2 = 1200.0000 ¢, ~16875/16384 = 55.2961 ¢
error map: ⟨0.000 +0.380 +0.759]

Optimal ET sequence: 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 (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

Subgroup-val mapping: [⟨1 2 2 3], ⟨0 -9 7 10]]

Optimal tunings:

  • WE: ~2 = 1199.7406 ¢, ~33/32 = 55.2653 ¢
error map: ⟨-0.259 +0.139 +0.024 +0.557]
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2770 ¢
error map: ⟨0.000 +0.552 +0.625 +1.452]

Optimal ET sequence: 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 (S31) so that the generator is equated to 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.

Subgroup: 2.3.5.11.31

Comma list: 496/495, 961/960, 4000/3993

Subgroup-val mapping: [⟨1 2 2 3 5], ⟨0 -9 7 10 -1]]

Optimal tunings:

  • WE: ~2 = 1199.8050 ¢, ~32/31 = 55.2669 ¢
error map: ⟨-0.195 +0.253 +0.165 +0.766 -1.277]
  • CWE: ~2 = 1200.0000 ¢, ~32/31 = 55.2759 ¢
error map: ⟨0.000 +0.562 +0.617 +1.441 -0.311]

Optimal ET sequence: 21, 22, 43, 65, 87, 152, 369, 521e, 673e

Badness (Sintel): 0.251

Strong extensions

Map to strong full 7- and 11-limit extensions
Extension Mapping of 7 Tuning range*
Escapist -4 ↓ 65
Alphaquarter +61 ↑ 65
↓ 87
Escaped -26 ↑ 87

* 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 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: [⟨1 2 2 4], ⟨0 -9 7 -26]]

Optimal tunings:

  • WE: ~2 = 1199.9190 ¢, ~28/27 = 55.1186 ¢
error map: ⟨-0.081 +1.816 -0.646 -2.232]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1242 ¢
error map: ⟨0.000 +1.927 -0.444 -2.056]

Optimal ET sequence: 22, 65, 87, 196, 283

Badness (Sintel): 2.25

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 4000/3993

Mapping: [⟨1 2 2 4 3], ⟨0 -9 7 -26 10]]

Optimal tunings:

  • WE: ~2 = 1199.9480 ¢, ~28/27 = 55.1241 ¢
error map: ⟨-0.052 +1.824 -0.549 -2.261 -0.233]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1271 ¢
error map: ⟨0.000 +1.901 -0.424 -2.131 -0.047]

Optimal ET sequence: 22, 65, 87, 196, 283

Badness (Sintel): 1.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 352/351, 385/384, 625/624

Mapping: [⟨1 2 2 4 3 2], ⟨0 -9 7 -26 10 37]]

Optimal tunings:

  • WE: ~2 = 1199.9926 ¢, ~28/27 = 55.1378 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 55.1382 ¢

Optimal ET sequence: 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: [⟨1 2 2 0], ⟨0 -9 7 61]]

Optimal tunings:

  • WE: ~2 = 1199.7349 ¢, ~16128/15625 = 55.2306 ¢
error map: ⟨-0.265 +0.439 -0.230 +0.242]
  • CWE: ~2 = 1200.0000 ¢, ~16128/15625 = 55.2405 ¢
error map: ⟨0.000 +0.880 +0.370 +0.846]

Optimal ET sequence: 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: [⟨1 2 2 0 3], ⟨0 -9 7 61 10]]

Optimal tunings:

  • WE: ~2 = 1199.7229 ¢, ~33/32 = 55.2303 ¢
error map: ⟨-0.277 +0.418 -0.256 +0.220 +0.153]
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2407 ¢
error map: ⟨0.000 +0.879 +0.371 +0.858 +1.089]

Optimal ET sequence: 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: [⟨1 2 2 0 3 2], ⟨0 -9 7 61 10 37]]

Optimal tunings:

  • WE: ~2 = 1199.6491 ¢, ~33/32 = 55.2200 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.2328 ¢

Optimal ET sequence: 65d, 87, 152f, 239f

Badness (Sintel): 1.047

Escapist

This temperament makes the identification of the 4-generator interval, representing (16/15)2, 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: [⟨1 2 2 3], ⟨0 -9 7 -4]]

Optimal tunings:

  • WE: ~2 = 1198.9926 ¢, ~49/48 = 55.2809 ¢
error map: ⟨-1.007 -1.498 -1.363 +7.028]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 55.3479 ¢
error map: ⟨0.000 -0.086 +1.122 +9.782]

Optimal ET sequence: 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: [⟨1 2 2 3 3], ⟨0 -9 7 -4 10]]

Optimal tunings:

  • WE: ~2 = 1199.0859 ¢, ~33/32 = 55.3117 ¢
error map: ⟨-0.914 -1.588 -0.960 +7.185 -0.944]
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.3574 ¢
error map: ⟨0.000 -0.172 +1.188 +9.745 +2.256]

Optimal ET sequence: 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: [⟨1 2 2 3 3 3], ⟨0 -9 7 -4 10 15]]

Optimal tunings:

  • WE: ~2 = 1199.5949 ¢, ~33/32 = 55.5317 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 55.5480 ¢

Optimal ET sequence: 21, 22, 43

Badness (Sintel): 1.457

Weak extensions

Map to weak extensions
Extensions Periods per octave Position of original generator
Number of generators Number of periods
Septisuperfourth period = 1/2 octave 1 generator + 0 periods
Arch period = octave 2 generators + 0 periods

Septisuperfourth

Subgroup: 2.3.5.7

Comma list: 6144/6125, 118098/117649

Mapping: [⟨2 4 4 7], ⟨0 -9 7 -15]]

mapping generators: ~343/243, ~405/392

Optimal tunings:

  • WE: ~343/243 = 599.8762 ¢, ~405/392 = 55.3089 ¢
error map: ⟨-0.248 -0.230 +0.353 +0.674]
  • CWE: ~343/243 = 600.0000 ¢, ~405/392 = 55.3273 ¢
error map: ⟨0.000 +0.100 +0.977 +1.265]

Optimal ET sequence: 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: [⟨2 4 4 7 6], ⟨0 -9 7 -15 10]]

Optimal tunings:

  • WE: ~99/70 = 599.8383 ¢, ~33/32 = 55.2895 ¢
error map: ⟨-0.323 -0.207 +0.066 +0.700 +0.606]
  • CWE: ~99/70 = 600.0000 ¢, ~33/32 = 55.3081 ¢
error map: ⟨0.000 +0.272 +0.843 +1.553 +1.763]

Optimal ET sequence: 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: [⟨2 4 4 7 6 11], ⟨0 -9 7 -15 10 -39]]

Optimal tunings:

  • WE: ~99/70 = 599.8331 ¢, ~33/32 = 55.3093 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/32 = 55.3295 ¢

Optimal ET sequence: 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: [⟨2 4 4 7 6 5], ⟨0 -9 7 -15 10 26]]

Optimal tunings:

  • WE: ~55/39 = 599.9152 ¢, ~33/32 = 55.3509 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~33/32 = 55.3584 ¢

Optimal ET sequence: 22, 108, 130

Badness (Sintel): 1.37

Arch

Subgroup: 2.3.5.7

Comma list: 3136/3125, 5250987/5242880

Mapping: [⟨1 2 2 2], ⟨0 -18 14 35]]

mapping generators: ~2, ~64/63

Optimal tunings:

  • WE: ~2 = 1199.9246 ¢, ~64/63 = 27.6662 ¢
error map: ⟨-0.075 -0.097 +0.862 -0.661]
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6676 ¢
error map: ⟨0.000 +0.029 +1.032 -0.461]

Optimal ET sequence: 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: [⟨1 2 2 2 3], ⟨0 -18 14 35 20]]

Optimal tunings:

  • WE: ~2 = 1199.8347 ¢, ~64/63 = 27.6590 ¢
error map: ⟨-0.165 -0.147 +0.581 -1.092 +1.366]
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6617 ¢
error map: ⟨0.000 +0.134 +0.950 -0.667 1.916]

Optimal ET sequence: 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: [⟨1 2 2 2 3 4], ⟨0 -18 14 35 20 -13]]

Optimal tunings:

  • WE: ~2 = 1199.8733 ¢, ~64/63 = 27.6569 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/63 = 27.6594 ¢

Optimal ET sequence: 43, 87, 130, 217, 347e, 564e

Badness (Sintel): 0.806

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