Escapade family: Difference between revisions

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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.
{{Technical data page}}


Extensions of escapade 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.
<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>


<div style="float:right;">
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]].
[[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 ({{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 13: 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
 
= 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 34: 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


== Strong extensions ==
=== 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.


=== 2.3.5.11.31 subgroup ===
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]].
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 51: 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


=== Escapist ===
= Strong extensions =
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]].
{| 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).''
 
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 }}
 
[[Badness]] (Sintel): 2.25


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


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 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 }}


{{Optimal ET sequence|legend=1| 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


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:
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).''
* [[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 }}


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]].
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
 
[[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: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}
{{Mapping|legend=0| 1 2 2 0 3 | 0 -9 7 61 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126
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=1| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}


Badness: 0.035844
Badness (Sintel): 0.980


==== 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: 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


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


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


Badness: 0.029638
Badness (Sintel): 1.21


==== 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|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 }}


Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
Badness (Sintel): 1.457


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
= 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
|}


{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}
== Septisuperfourth ==
[[Subgroup]]: 2.3.5.7


Badness: 0.025344
[[Comma list]]: 6144/6125, 118098/117649


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


=== Septisuperfourth ===
[[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 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 165: 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 180: 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 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~33/32 = 55.359
Badness (Sintel): 1.37


{{Optimal ET sequence|legend=1| 22, 108, 130 }}
== Arch ==
[[Subgroup]]: 2.3.5.7


Badness: 0.033038
[[Comma list]]: 3136/3125, 5250987/5242880


=== Arch ===
{{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]]

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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