Gammic family
The Carlos Gamma rank-1 temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank-2 microtemperament tempering out [-29 -11 20⟩. This temperament, gammic, takes 11 generator steps to reach 5/4, and 20 to reach 3/2. The generator in question is 1990656/1953125 = [13 5 -9⟩, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by 171EDO, schismatic temperament makes for a natural comparison. Schismatic, with a wedgie of ⟨⟨1 -8 -15]] is plainly much less complex than gammic with wedgie ⟨⟨20 11 -29]], but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L 33s refinement of the 34EDO tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.
Because 171 is such a strong 7-limit system, it is natural to extend gammic to the 7-limit. This we may do by adding 4375/4374 to the comma list, giving a wedgie of ⟨⟨20 11 96 -29 96 192]]. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note MOS is possible.
Gammic
Subgroup: 2.3.5
Comma: [-29 -11 20⟩
Mapping: [⟨1 1 2], ⟨0 20 11]]
POTE generator: ~1990656/1953125 = 35.0964
Optimal GPV sequence: 34, 103, 137, 171, 547, 718, 889, 1607
Badness: 0.087752
Septimal gammic
Subgroup: 2.3.5.7
Comma list: 4375/4374, 6591796875/6576668672
Mapping: [⟨1 1 2 0], ⟨0 20 11 96]]
POTE generator: ~234375/229376 = 35.0904
Optimal GPV sequence: 34d, 171, 205, 1402, 1573, 1744, 1915
Badness: 0.047362
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 4375/4356, 100352/99825
Mapping: [⟨1 1 2 0 2], ⟨0 20 11 96 50]]
POTE generator: ~45/44 = 35.089
Optimal GPV sequence: 34d, 137d, 171
Badness: 0.097061
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 625/624, 2200/2197
Mapping: [⟨1 1 2 0 2 3], ⟨0 20 11 96 50 24]]
POTE generator: ~45/44 = 35.091
Optimal GPV sequence: 34d, 137d, 171
Badness: 0.047822
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 364/363, 375/374, 595/594, 2200/2197
Mapping: [⟨1 1 2 0 2 3 4], ⟨0 20 11 96 50 24 3]]
POTE generator: ~45/44 = 35.090
Optimal GPV sequence: 34d, 137d, 171
Badness: 0.031466
Gammy
Subgroup: 2.3.5.7
Comma list: 225/224, 94143178827/91913281250
Mapping: [⟨1 1 2 1], ⟨0 20 11 62]]
Wedgie: ⟨⟨20 11 62 -29 42 113]]
POTE generator: ~1990656/1953125 = 34.984
Optimal GPV sequence: 34d, 69d, 103, 240, 343b
Badness: 0.230839
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 215622/214375
Mapping: [⟨1 1 2 1 2], ⟨0 20 11 62 50]]
POTE generator: ~45/44 = 34.985
Optimal GPV sequence: 34d, 69de, 103, 240, 343be
Badness: 0.065326
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 351/350, 1188/1183
Mapping: [⟨1 1 2 1 2 3], ⟨0 20 11 62 50 24]]
POTE generator: ~45/44 = 34.988
Optimal GPV sequence: 34d, 69de, 103, 240, 343be
Badness: 0.033418
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 243/242, 351/350, 375/374, 1188/1183
Mapping: [⟨1 1 2 1 2 3 4], ⟨0 20 11 62 50 24 3]]
POTE generator: ~45/44 = 34.997
Optimal GPV sequence: 34d, 69de, 103, 137, 240
Badness: 0.025030
Neptune
A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds 2401/2400 to the gammic comma, and may be described as the 68&171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. 171EDO makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending Carlos Gamma.
Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)3 equates to 11/4. This may be described as ⟨⟨40 22 21 -3 …]] or 68&103, and 171 can still be used as a tuning, with val ⟨171 271 397 480 591].
Gene Ward Smith once described neptune as an analog of miracle.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 48828125/48771072
Mapping: [⟨1 21 13 13], ⟨0 -40 -22 -21]]
Mapping generators: 2, ~7/5
Wedgie: ⟨⟨40 22 21 -58 -79 -13]]
POTE generator: ~7/5 = 582.452
Optimal GPV sequence: 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778
Badness: 0.023427
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 78408/78125
Mapping: [⟨1 21 13 13 2], ⟨0 -40 -22 -21 3]]
POTE generator: ~7/5 = 582.475
Optimal GPV sequence: 35, 68, 103, 171e, 274e, 445ee
Badness: 0.063602
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 385/384, 625/624, 1188/1183, 1375/1372
Mapping: [⟨1 21 13 13 2 27], ⟨0 -40 -22 -21 3 -48]]
POTE generator: ~7/5 = 582.480
Optimal GPV sequence: 35f, 68, 103, 171e, 274e
Badness: 0.037156
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 385/384, 561/560, 625/624, 715/714, 1188/1183
Mapping: [⟨1 21 13 13 2 27 7], ⟨0 -40 -22 -21 3 -48 -6]]
POTE generator: ~7/5 = 582.475
Optimal GPV sequence: 35f, 68, 103, 171e, 274e, 445ee
Badness: 0.025909
Salacia
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 9765625/9732096
Mapping: [⟨1 21 13 13 52], ⟨0 -40 -22 -21 -100]]
POTE generator: ~7/5 = 582.478
Optimal GPV sequence: 68e, 103, 171, 274, 719be, 993bcde, 1267bbcde
Badness: 0.069721
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 441/440, 625/624, 2200/2197
Mapping: [⟨1 21 13 13 52 27], ⟨0 -40 -22 -21 -100 -48]]
POTE generator: ~7/5 = 582.477
Optimal GPV sequence: 68e, 103, 171, 274, 719be, 993bcde
Badness: 0.034977
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 375/374, 441/440, 625/624, 2200/2197
Mapping: [⟨1 21 13 13 52 27 7], ⟨0 -40 -22 -21 -100 -48 -6]]
POTE generator: ~7/5 = 582.475
Optimal GPV sequence: 68e, 103, 171, 274, 445e, 719be, 1164bcdeef
Badness: 0.024577
Poseidon
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 9453125/9437184
Mapping: [⟨2 2 4 5 8], ⟨0 40 22 21 -37]]
Mapping generators: ~99/70, ~99/98
POTE generator: ~99/98 = 17.545
Optimal GPV sequence: 68, 206b, 274, 342
Badness: 0.041727