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.
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 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 (16/15)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
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