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TES: The largest network of teachers in the world

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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 <<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 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.

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

EDOs: 35, 68, 103, 171, 274, 445