Escapade family: Difference between revisions

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.

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 & 44), 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.]]

= Escapade =

== 5-limit escapade ==

[[Comma list]]: 4294967296/4271484375

* [[CTE]]: ~2 = 1\1, ~16875/16384 = 55.3052

* [[POTE]]: ~2 = 1\1, ~16875/16384 = 55.293

[[Badness]]: 0.083778

== 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.

Comma list: 4000/3993, 5632/5625

Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 55.2760

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.

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

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

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

{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}

Badness (Dirichlet): 0.251

=== Escapist ===

This temperament makes the identification of the 4-generator interval, representing (16/15)², with [[8/7]] by tempering out [[225/224]]; 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: 2.3.5.7.11

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

Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.354

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

Badness: 0.036700

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

 

 

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

 

 

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 tuning (POTE): ~2 = 1\1, ~28/27 = 55.126

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

Badness: 0.035844

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138

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

Badness: 0.031366

=== 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 the [[7-limit]]; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.

Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.243

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

Badness: 0.029638

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

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

Badness: 0.025344

=== Septisuperfourth ===

Optimal tuning (POTE): ~99/70 = 1\2, ~33/32 = 55.304

Badness: 0.024619

Optimal tuning (POTE): ~99/70 = 1\2, ~33/32 = 55.325

{{Optimal ET sequence|legend=1| 22f, 108f, 130, 282 }}

Badness: 0.022887

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

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

Badness: 0.033038

=== Arch ===

Optimal tuning (POTE): ~2 = 1\1, ~64/63 = 27.663

Badness: 0.036541

Optimal tuning (POTE): ~2 = 1\1, ~64/63 = 27.660

{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347e, 564e }}

Badness: 0.019504