Gammic family: Difference between revisions
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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]]. | 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)<sup>3</sup> 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 [[ | Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> 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]. | An article on Neptune as an analog of miracle can be found [http://tech.groups.yahoo.com/group/tuning-math/message/6001 here]. | ||
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[[Category:Gammic]] | [[Category:Gammic]] | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
{{todo|review}} | |||
Revision as of 12:32, 11 January 2021
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 [-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 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: [-29 -11 20⟩
POTE generator: 35.096
Map: [<1 1 2|, <0 20 11|]
EDOs: 34, 103, 137, 171, 547, 718, 889, 1607
7-limit
Commas: 4375/4374, 6591796875/6576668672
POTE generator: 35.090
Map: [<1 1 2 0|, <0 20 11 96|]
EDOs: 171, 1402, 1573, 1744, 1915
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 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].
An article on Neptune as an analog of miracle can be found here.
Commas: 2401/2400, 48828125/48771072
POTE generator: 582.452
Map: [<1 21 13 13|, <0 -40 -22 -21|]
Generators: 2, 7/5
EDOs: 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778
11-limit
Commas: 385/384, 1375/1372, 2465529759/2441406250
POTE generator: 582.475
Map: [1 21 13 13 2|, <0 -40 -22 -21 3|]
Generators: 2, 7/5