Gammic family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The gammic family of temperaments tempers out the gammic comma (monzo: [-29 -11 20⟩), a 5-limit comma of about 4.77 cents in size.

Gammic

The Carlos Gamma rank-1 temperament divides a ~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⟩, the gammic comma. 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, which makes for an ideal gammic tuning.

As a 5-limit temperament supported by 171edo, the schismic temperament makes for a natural comparison. Schismic, tempering out the schisma ([-15 8 1⟩), is plainly much less complex than gammic, 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 34edo, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.

Subgroup: 2.3.5

Comma list: [-29 -11 20⟩

Mapping: [⟨1 1 2], ⟨0 20 11]]

mapping generators: ~2, ~1990656/1953125

Optimal tunings:

• WE: ~2 = 1200.0419 ¢, ~1990656/1953125 = 35.0977 ¢

error map: ⟨+0.042 +0.399 -0.156]

• CWE: ~2 = 1200.0000 ¢, ~1990656/1953125 = 35.0981 ¢

error map: ⟨0.000 +0.008 -0.234]

Optimal ET sequence: 34, 103, 137, 171, 547, 718, 889, 1607

Badness (Sintel): 2.06

Overview to extensions

7-limit extensions

Because 171 is such a strong 7-limit system, it is well motivated to extend gammic to the 7-limit. This we may do by adding 4375/4374 to the comma list. 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 generator chain is possible.

Subgroup extensions

Gammic also naturally extends with the 17th harmonic, as is given in #Subgroup extensions.

Septimal gammic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 6591796875/6576668672

Mapping: [⟨1 1 2 0], ⟨0 20 11 96]]

Optimal tunings:

• WE: ~2 = 1200.0712 ¢, ~234375/229376 = 35.0924 ¢

error map: ⟨+0.071 -0.035 -0.154 +0.049]

• CWE: ~2 = 1200.0000 ¢, ~234375/229376 = 35.0913 ¢

error map: ⟨0.000 -0.130 -0.310 -0.065]

Optimal ET sequence: 34d, …, 137d, 171, 1402, 1573, 1744, 1915, 2086c, …, 2599c, 5369bccd

Badness (Sintel): 1.20

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 tunings:

• WE: ~2 = 1199.8949 ¢, ~45/44 = 35.0855 ¢
• CWE: ~2 = 1200.0000 ¢, ~45/44 = 35.0872 ¢

Optimal ET sequence: 34d, …, 137d, 171

Badness (Sintel): 3.21

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 tunings:

• WE: ~2 = 1199.8098 ¢, ~45/44 = 35.0855 ¢
• CWE: ~2 = 1200.0000 ¢, ~45/44 = 35.0888 ¢

Optimal ET sequence: 34d, 137d, 171

Badness (Sintel): 1.98

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 tunings:

• WE: ~2 = 1199.8393 ¢, ~45/44 = 35.0851 ¢
• CWE: ~2 = 1200.0000 ¢, ~45/44 = 35.0882 ¢

Optimal ET sequence: 34d, 137d, 171

Badness (Sintel): 1.60

Gammy

Subgroup: 2.3.5.7

Comma list: 225/224, 94143178827/91913281250

Mapping: [⟨1 1 2 1], ⟨0 20 11 62]]

Optimal tunings:

• WE: ~2 = 1200.5055 ¢, ~1990656/1953125 = 34.9984 ¢

error map: ⟨+0.506 -1.482 -0.321 +1.577]

• CWE: ~2 = 1200.0000 ¢, ~1990656/1953125 = 34.9947 ¢

error map: ⟨0.000 -2.060 -1.372 +0.848]

Optimal ET sequence: 34d, 69d, 103, 240, 343b

Badness (Sintel): 5.84

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 tunings:

• WE: ~2 = 1200.5129 ¢, ~45/44 = 34.9999 ¢
• CWE: ~2 = 1200.0000 ¢, ~45/44 = 34.9967 ¢

Optimal ET sequence: 34d, 69de, 103, 240, 343be

Badness (Sintel): 2.16

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 tunings:

• WE: ~2 = 1200.4356 ¢, ~45/44 = 35.0008 ¢
• CWE: ~2 = 1200.000 ¢, ~45/44 = 34.9975 ¢

Optimal ET sequence: 34d, 69de, 103, 240, 343be

Badness (Sintel): 1.38

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 tunings:

• WE: ~2 = 1200.2936 ¢, ~45/44 = 35.0057 ¢
• CWE: ~2 = 1200.0000 ¢, ~45/44 = 35.0021 ¢

Optimal ET sequence: 34d, 69de, 103, 137, 240

Badness (Sintel): 1.28

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 7-odd-limit 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.

Gene Ward Smith once described neptune as an analog of miracle.

7-limit

Subgroup: 2.3.5.7

Comma list: 2401/2400, 48828125/48771072

Mapping: [⟨1 -19 -9 -8], ⟨0 40 22 21]]

mapping generators: 2, ~10/7

Optimal tunings:

• WE: ~2 = 1200.0660 ¢, ~10/7 = 617.5815 ¢

error map: ⟨+0.066 +0.053 -0.114 -0.141]

• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5489 ¢

error map: ⟨0.000 +0.000 -0.238 -0.299]

Optimal ET sequence: 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778, 1949d, 3727cdd, 5676ccddd

Badness (Sintel): 0.593

2.3.5.7.17 subgroup

Extending 2.3.5.17 gammic via neptune, we find that both 2401/2400 (S49) and 2500/2499 (S50) are tempered out; their product, 1225/1224 (S35) is therefore also tempered out.

Subgroup: 2.3.5.7.17

Comma list: 1225/1224, 2401/2400, 24576/24565

Subgroup-val mapping: [⟨1 -19 -9 -8 1], ⟨0 40 22 21 6]]

Optimal tunings:

• WE: ~2 = 1200.0136 ¢, ~10/7 = 617.5572 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5504 ¢

Optimal ET sequence: 35, 68, 103, 171, 581, 752, 923, 1094

Badness (Sintel): 0.404

2.3.5.7.17.31 subgroup

Since neptune splits the interval of 5/3 into two, we can accurately map each part to 40/31~31/24 by tempering out 961/960 (S31). This is especially natural, as combined with tempering out 1225/1224 (S35) and 24576/24565 (S16/S17), we can map (17/16)² (6 gammic generators) to 35/31. This also gives us its complement with respect to 5/4, the interval of 5 gammic generators representing a quarter of a perfect fifth, as 31/28.

Subgroup: 2.3.5.7.17.31

Comma list: 868/867, 961/960, 1225/1224, 2401/2400

Subgroup-val mapping: [⟨1 -19 -9 -8 1 -11], ⟨0 40 22 21 6 31]]

Optimal tunings:

• WE: ~2 = 1200.0519 ¢, ~10/7 = 617.5760 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5501 ¢

Optimal ET sequence: 35, 68, 103, 171, 752k, 923k

Badness (Sintel): 0.393

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 78408/78125

Mapping: [⟨1 -19 -9 -8 5], ⟨0 40 22 21 -3]]

Optimal tunings:

• WE: ~2 = 1200.4655 ¢, ~10/7 = 617.7648 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5317 ¢

Optimal ET sequence: 35, 68, 103, 171e, 274e, 445ee

Badness (Sintel): 2.10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 1188/1183, 1375/1372

Mapping: [⟨1 -19 -9 -8 5 -21], ⟨0 40 22 21 -3 48]]

Optimal tunings:

• WE: ~2 = 1200.4067 ¢, ~10/7 = 617.7290 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5257 ¢

Optimal ET sequence: 35f, 68, 103, 171e, 274e

Badness (Sintel): 1.54

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 385/384, 561/560, 625/624, 715/714, 1188/1183

Mapping: [⟨1 -19 -9 -8 5 -21 1], ⟨0 40 22 21 -3 48 6]]

Optimal tunings:

• WE: ~2 = 1200.2971 ¢, ~10/7 = 617.6784 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5291 ¢

Optimal ET sequence: 35f, 68, 103, 171e, 274e

Badness (Sintel): 1.32

Salacia

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 9765625/9732096

Mapping: [⟨1 -19 -9 -8 -48], ⟨0 40 22 21 100]]

Optimal tunings:

• WE: ~2 = 1200.2180 ¢, ~10/7 = 617.6341 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5253 ¢

Optimal ET sequence: 68e, 103, 171, 274

Badness (Sintel): 2.30

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 625/624, 2200/2197

Mapping: [⟨1 -19 -9 -8 -48 -21], ⟨0 40 22 21 100 48]]

Optimal tunings:

• WE: ~2 = 1200.1492 ¢, ~10/7 = 617.5993 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5249 ¢

Optimal ET sequence: 68e, 103, 171, 274

Badness (Sintel): 1.45

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 375/374, 441/440, 625/624, 2200/2197

Mapping: [⟨1 -19 -9 -8 -48 -21 1], ⟨0 40 22 21 100 48 6]]

Optimal tunings:

• WE: ~2 = 1200.0872 ¢, ~10/7 = 617.5702 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 617.5264 ¢

Optimal ET sequence: 68e, 103, 171, 274, 445e

Badness (Sintel): 1.25

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 tunings:

• WE: ~99/70 = 600.0509 ¢, ~99/98 = 17.5466 ¢
• CWE: ~99/70 = 600.0000 ¢, ~99/98 = 17.5458 ¢

Optimal ET sequence: 68, 206b, 274, 342, 2804cdee, 3146cdee, …, 5198bccdddeeee

Badness (Sintel): 1.38

Subgroup extensions

Gammic (2.3.5.17)

The interval of 3 generators represents 1/3 of 6/5, which is very close to 17/16, with the comma between 6/5 and (17/16)³ being 24576/24565 (S16/S17). This then naturally interprets the generator as 51/50 with two generators representing 25/24, tempering out 15625/15606 (S49⋅S50²).

Subgroup: 2.3.5.17

Comma list: 15625/15606, 24576/24565

Subgroup-val mapping: [⟨1 1 2 4], ⟨0 20 11 3]]

mapping generators: ~2, ~51/50

Optimal tunings:

• WE: ~2 = 1199.9899 ¢, ~51/50 = 35.1008 ¢
• CWE: ~2 = 1200.0000 ¢, ~51/50 = 35.1008 ¢

Optimal ET sequence: 34, 103, 137, 171, 376, 547

Badness (Sintel): 0.320