Gammic family: Difference between revisions

The [[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 {{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 <<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]] 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 =

Comma: {{monzo| -29 -11 20 }}

[[POTE generator]]: 35.096

Map: [<1 1 2|, <0 20 11|]

{{EDOs|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }}

= 7-limit =

Commas: 4375/4374, 6591796875/6576668672

Map: [<1 1 2 0|, <0 20 11 96|]

{{EDOs|legend=1| 171, 1402, 1573, 1744, 1915 }}

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]] 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|val]] <171 271 397 480 591|.

Commas: 2401/2400, 48828125/48771072

[[POTE generator]]: 582.452

Map: [<1 21 13 13|, <0 -40 -22 -21|]

Generators: 2, 7/5

{{EDOs|legend=1| 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778 }}

Commas: 385/384, 1375/1372, 2465529759/2441406250

Map: [1 21 13 13 2|, <0 -40 -22 -21 3|]

Generators: 2, 7/5

[[Category:Theory]]

[[Category:Temperament family]]

[[Category:Gammic]]