Escapade family: Difference between revisions

[[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.]]

The '''escapade family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[escapade comma]], {{monzo| 32 -7 -9 }}, of size 9.492 [[cent]]s. The defining feature of this comma is splitting [[5/3]] into sixteen quartertones of which [[5/4]] makes up seven and [[4/3]] makes up nine; therefore [[16/15]] is two generator steps. It most naturally manifests as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament, tempering out [[4000/3993]] and [[5632/5625]].

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

[[Comma list]]: 4294967296/4271484375

* [[WE]]: ~2 = 1199.8082{{c}}, ~16875/16384 = 55.2840{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~16875/16384 = 55.2961{{c}}

[[Badness]] (Sintel): 1.965

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.

Comma list: 4000/3993, 5632/5625

 

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

 

* 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 }}

 

 

 

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.

 

 

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

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

* 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=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }}

Badness (Sintel): 0.251

= Strong extensions =

|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions

| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 <br> ↓ [[87edo|87]]  

<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions

 

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}}

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

[[Badness]] (Sintel): 2.25

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

Optimal tunings:

: error map: {{val| -0.052 +1.824 -0.549 -2.261 -0.233 }}

: error map: {{val| 0.000 +1.901 -0.424 -2.131 -0.047 }}

{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}

Badness (Sintel): 1.18

Subgroup: 2.3.5.7.11.13

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 }}

* 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

Given the slightly sharp ~[[3/2]] in ideal tunings of escapade (between [[65edo]] and [[87edo]]), it does very little damage to temper out [[5120/5103]] to extend it to prime 7; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.

[[Subgroup]]: 2.3.5.7

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

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

* [[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}}

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

[[Badness]] (Sintel): 2.95

Comma list: 3025/3024, 4000/3993, 5120/5103

 

 

* WE: ~2 = 1199.7229{{c}}, ~33/32 = 55.2303{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2407{{c}}

 

 

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:

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 55.2328{{c}}

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

Badness (Sintel): 1.047

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|legend=1| 1 2 2 3 | 0 -9 7 -4 }}

[[Optimal tuning]]s:

: error map: {{val| 0.000 -0.086 +1.122 +9.782 }}

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

[[Badness]] (Sintel): 1.97

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

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

* 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 ===

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 =

== Septisuperfourth ==

[[Subgroup]]: 2.3.5.7

 

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

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

* 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

{{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

This temperament is also known as ''biscapade''.

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

 

* CWE: ~55/39 = 600.0000{{c}}, ~33/32 = 55.3584{{c}}

{{Optimal ET sequence|legend=0| 22, 108, 130 }}

Badness (Sintel): 1.37

[[Subgroup]]: 2.3.5.7

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

{{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}}

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

Badness (Sintel): 1.21

=== 13-limit ===

{{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:Catalogs of rank-2 temperaments]]