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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 '''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, alphaquarter, escaped, septisuperfourth (aka biscapade) and arch, all considered below.  
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.


== Escapade ==
== Escapade ==
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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.
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.
Subgroup: 2.3.5.11.31
Subgroup: 2.3.5.11.31


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


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 55.276
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 }}
{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e, 673e, 1194bcee, 1867bceeee }}
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== Escapist ==
== 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]]; 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
[[Subgroup]]: 2.3.5.7


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Badness: 0.035261
Badness: 0.035261


== Alphaquarter ==
== 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 &amp; 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 29360128/29296875
[[Comma list]]: 245/243, 65625/65536


{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}
{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}


{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}
{{Multival|legend=1| 9 -7 26 -32 16 80 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16128/15625 = 55.243
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 55.122


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}


[[Badness]]: 0.116594
[[Badness]]: 0.088746


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 4000/3993, 5120/5103
Comma list: 245/243, 385/384, 4000/3993


Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}
Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.243
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126


{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}
{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}


Badness: 0.029638
Badness: 0.035844


=== 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: 245/243, 352/351, 385/384, 625/624


Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}
Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138


{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}
{{Optimal ET sequence|legend=1| 22, 65, 87, 283 }}


Badness: 0.025344
Badness: 0.031366


== Escaped ==
== Alphaquarter ==
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).''
Given the slightly sharp fifth 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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 65625/65536
[[Comma list]]: 5120/5103, 29360128/29296875


{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}
{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}


{{Multival|legend=1| 9 -7 26 -32 16 80 }}
{{Multival|legend=1| 9 -7 -61 -32 -122 -122 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 55.122
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16128/15625 = 55.243


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}


[[Badness]]: 0.088746
[[Badness]]: 0.116594


=== 11-limit ===
=== 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: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.126
Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.243


{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}


Badness: 0.035844
Badness: 0.029638


=== 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: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}
Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.138
Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 55.236


{{Optimal ET sequence|legend=1| 22, 65, 87, 283 }}
{{Optimal ET sequence|legend=1| 65d, 87, 152f, 239f }}


Badness: 0.031366
Badness: 0.025344


== Septisuperfourth ==
== Septisuperfourth ==
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