Escapade family

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.

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

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.

5-limit escapade

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. 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

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 (16/15)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

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: 2.3.5.7.11

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 ~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.

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: 2.3.5.7.11

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: 2.3.5.7.11

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