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 ==

Revision as of 02:27, 16 March 2025

The escapade family 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.

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

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

Comma list: 4294967296/4271484375

Mapping: [⟨1 2 2], ⟨0 -9 7]]

mapping generators: ~2, ~16875/16384

Optimal tunings:

  • CTE: ~2 = 1\1, ~16875/16384 = 55.3052
  • POTE: ~2 = 1\1, ~16875/16384 = 55.293

Optimal ET sequence: 21, 22, 43, 65, 152, 217, 586, 803

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.

Subgroup: 2.3.5.11

Comma list: 4000/3993, 5632/5625

Mapping: [⟨1 2 2 3], ⟨0 -9 7 10]]

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

Optimal ET sequence: 21, 22, 43, 65, 87, 152, 369, 521e, 1194bcee, 1715bceeee

Badness: 0.0107

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

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

Mapping: [⟨1 2 2 3 5], ⟨0 -9 7 10 -1]]

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

Optimal ET sequence: 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 16/152, 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

Comma list: 225/224, 12288/12005

Mapping: [⟨1 2 2 3], ⟨0 -9 7 -4]]

Wedgie: ⟨⟨ 9 -7 4 -32 -19 29 ]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 55.327

Optimal ET sequence: 21, 22, 43, 65d

Badness: 0.077950

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

Optimal ET sequence: 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

Mapping: [⟨1 2 2 3 3 3], ⟨0 -9 7 -4 10 15]]

Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 55.550

Optimal ET sequence: 21, 22, 43

Badness: 0.035261

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).

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]]

Wedgie: ⟨⟨ 9 -7 26 -32 16 80 ]]

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 55.122

Optimal ET sequence: 22, 65, 87, 196, 283

Badness: 0.088746

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

Optimal ET sequence: 22, 65, 87, 196, 283

Badness: 0.035844

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

Optimal ET sequence: 22, 65, 87, 283

Badness: 0.031366

Alphaquarter

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

Comma list: 5120/5103, 29360128/29296875

Mapping: [⟨1 2 2 0], ⟨0 -9 7 61]]

Wedgie: ⟨⟨ 9 -7 -61 -32 -122 -122 ]]

Optimal tuning (POTE): ~2 = 1\1, ~16128/15625 = 55.243

Optimal ET sequence: 65d, 87, 152, 239, 391

Badness: 0.116594

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

Optimal ET sequence: 65d, 87, 152, 239, 391

Badness: 0.029638

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

Optimal ET sequence: 65d, 87, 152f, 239f

Badness: 0.025344

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, ~16875/16384

Wedgie: ⟨⟨ 18 -14 30 -64 -3 109 ]]

Optimal tuning (POTE): ~343/243 = 1\2, ~16875/16384 = 55.320

Optimal ET sequence: 22, 86, 108, 130, 152, 282

Badness: 0.059241

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

Optimal ET sequence: 22, 86, 108, 130, 152, 282

Badness: 0.024619

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

Optimal ET sequence: 22f, 108f, 130, 282

Badness: 0.022887

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

Optimal ET sequence: 22, 108, 130

Badness: 0.033038

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

Wedgie: ⟨⟨ 18 -14 -35 -64 -106 -42 ]]

Optimal tuning (POTE): ~2 = 1\1, ~64/63 = 27.668

Optimal ET sequence: 43, 87, 130, 217, 347, 824c, 1171c, 1518cd

Badness: 0.094345

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

Optimal ET sequence: 43, 87, 130, 217, 347e, 911cde

Badness: 0.036541

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

Optimal ET sequence: 43, 87, 130, 217, 347e, 564e

Badness: 0.019504

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