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 list: [-29 -11 20⟩

Mapping: [⟨1 1 2], ⟨0 20 11]]

mapping generators: ~2, ~1990656/1953125

Optimal tuning (POTE): ~2 = 1\1, ~1990656/1953125 = 35.0964

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~234375/229376 = 35.0904

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.089

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.091

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.090

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~1990656/1953125 = 34.984

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.985

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.988

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.997

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.452

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.475

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.480

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.475

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.478

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.477

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.475

Optimal ET 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

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 17.545

Optimal ET sequence: 68, 206b, 274, 342

Badness: 0.041727