Gammic family: Difference between revisions

The [[Carlos_Gamma|Carlos Gamma]] rank one temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank two microtemperament tempering out |-29 -11 20>. This temperament, gammic, takes 11 [[generator|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|171edo]], [[Schismatic_family|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 1L33s refinement of the [[34edo|34edo]] tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of [[Carlos_Gamma|Carlos Gamma]] if used for it.

Because 171 is such a strong [[7-limit|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.

[[POTE_tuning|POTE generator]]: 35.096

[[POTE_tuning|POTE generator]]: 35.090

===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, with wedgie <<40 22 21 -58 -79 -13||. 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|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|Carlos Gamma]].  

Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit|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|val]] <171 271 397 480 591|.

An article on Neptune as an analog of miracle can be found [http://tech.groups.yahoo.com/group/tuning-math/message/6001 here].

[[POTE_tuning|POTE generator]]: 582.452

[[POTE_tuning|POTE generator]]: 582.475

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