Gammic family: Difference between revisions
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Comma: {{monzo| -29 -11 20 }} | Comma: {{monzo| -29 -11 20 }} | ||
[[POTE generator]]: 35. | [[POTE generator]]: ~1990656/1953125 = 35.0964 | ||
Map: [<1 1 2|, <0 20 11|] | Map: [<1 1 2|, <0 20 11|] | ||
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{{EDOs|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }} | {{EDOs|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }} | ||
= 7-limit = | == 7-limit == | ||
Commas: 4375/4374, 6591796875/6576668672 | Commas: 4375/4374, 6591796875/6576668672 | ||
[[POTE generator]]: 35. | [[POTE generator]]: ~234375/229376 = 35.0904 | ||
Map: [<1 1 2 0|, <0 20 11 96|] | Map: [<1 1 2 0|, <0 20 11 96|] | ||
{{EDOs|legend=1| 171, 1402, 1573, 1744, 1915 }} | {{EDOs|legend=1| 34d, 171, 205, 1402, 1573, 1744, 1915 }} | ||
= 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 <<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]]. | ||
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Commas: 2401/2400, 48828125/48771072 | Commas: 2401/2400, 48828125/48771072 | ||
[[POTE generator]]: 582.452 | [[POTE generator]]: ~7/5 = 582.452 | ||
Map: [<1 21 13 13|, <0 -40 -22 -21|] | Map: [<1 21 13 13|, <0 -40 -22 -21|] | ||
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Commas: 385/384, 1375/1372, 2465529759/2441406250 | Commas: 385/384, 1375/1372, 2465529759/2441406250 | ||
[[POTE generator]]: 582.475 | [[POTE generator]]: ~7/5 = 582.475 | ||
Map: [1 21 13 13 2|, <0 -40 -22 -21 3|] | Map: [1 21 13 13 2|, <0 -40 -22 -21 3|] | ||