Gammic family: Difference between revisions
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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. | 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. | ||
= Gammic = | == Gammic == | ||
[[Comma]]: {{monzo| -29 -11 20 }} | [[Comma]]: {{monzo| -29 -11 20 }} | ||
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[[Badness]]: 0.087752 | [[Badness]]: 0.087752 | ||
== 7-limit == | === 7-limit === | ||
[[Comma list]]: 4375/4374, 6591796875/6576668672 | [[Comma list]]: 4375/4374, 6591796875/6576668672 | ||
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[[Badness]]: 0.047362 | [[Badness]]: 0.047362 | ||
= Neptune = | == 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 {{multival|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]]. | 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 {{multival|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]]. | ||
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[[Badness]]: 0.023427 | [[Badness]]: 0.023427 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 385/384, 1375/1372, 2465529759/2441406250 | Comma list: 385/384, 1375/1372, 2465529759/2441406250 | ||