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. It most naturally manifests as a [[2.3.5.11 subgroup]] temperament, tempering out [[4000/3993]] and [[5632/5625]]. | 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]]. | ||
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 | 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|(87 & 22)}}, 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 (aka 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. | ||
<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> | ||
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== 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 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. | ||
Subgroup: 2.3.5.11 | Subgroup: 2.3.5.11 | ||
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=== 2.3.5.11.31 subgroup === | === 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. | 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: 2.3.5.11.31 | ||
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=== Escaped === | === 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 19e & 27 temperament (sensi extension).'' | 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]]. | 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]]. | ||