Gammic family: Difference between revisions

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 {{monzo| -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 = {{monzo| 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 family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of {{multival| 1 -8 -15 }} is plainly much less complex than gammic with wedgie {{multival| 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 {{multival| 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.

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -29 -11 20 }}

{{Mapping|legend=1| 1 1 2 | 0 20 11 }}

 

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

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}

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

Mapping: {{mapping| 1 1 2 0 2 | 0 20 11 96 50 }}

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

Mapping: {{mapping| 1 1 2 0 2 3 | 0 20 11 96 50 24 }}

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

Mapping: {{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

[[Subgroup]]: 2.3.5.7

[[Mapping]]: {{mapping| 1 1 2 1 | 0 20 11 62 }}

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

Mapping: {{mapping| 1 1 2 1 2 | 0 20 11 62 50 }}

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

Mapping: {{mapping| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}

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

Mapping: {{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

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 {{multival| 40 22 21 -3 … }} or 68 & 103, and 171 can still be used as a tuning, with [[val]] {{val| 171 271 397 480 591 }}.

[[Subgroup]]: 2.3.5.7

{{Mapping|legend=1| 1 21 13 13 | 0 -40 -22 -21 }}

: mapping generators: 2, ~7/5

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

Mapping: {{mapping| 1 21 13 13 2 | 0 -40 -22 -21 3 }}

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

Mapping: {{mapping| 1 21 13 13 2 27 | 0 -40 -22 -21 3 -48 }}

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

Mapping: {{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

Mapping: {{mapping| 1 21 13 13 52 | 0 -40 -22 -21 -100 }}

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

Mapping: {{mapping| 1 21 13 13 52 27 | 0 -40 -22 -21 -100 -48 }}

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

Mapping: {{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

Mapping: {{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