From generators to vals

I've remarked how easy it is to move from the notation defined by two
vals to an octave-generator system of notation; it is worth remarking
that we can go in the opposite direction easily also, and in a
canonical way.

If g/n is the generator in lowest terms, we have two adjacent
fractions in the Farey sequence Fn next to it, p1/q1 and p2/q2, such
that p1/q1 < g/n < p2/q2 and (p1+p2)/(q1+q2) = g/n; since p1/q1 is
adjacent to p2/q2 in the Farey sequences before we reach Fn, we have
q1p2 - q2p1 = 1, so that the matrix

[q1 q2]
[p1 p2]

has determinant 1, and hence is invertible. This is the matrix which
transforms from the n,g system of coordinates to one based on two
vals; we can find the vals by transforming the vectors for 2, 3, etc
using the above matrix.

For instance, given the 4/19 generator, we have 1/5 < 4/19 < 3/14,
and this generator is the same as the 5-14 system. If we write 3 in
the octave-generator form, it is [2,-2], since 2 + (-2)*(4/19) =
30/19. Then

[14 5]
[2 -2] [ 3 1] = [22 8],

so that if g5 is the 5 val and g14 is the 14 val, we get g5(3) = 8
and g14(3) = 22. In this way we find g14(5) = 32, g4(5) = 12 and
g14(7) = 39, g4(7) = 14; hence g5 = h5, but since h14(5) = 33 we
don't have g14 = h14. The real point, of course, is that g14+g5 = h19.

In the same way, we can find that the minor third, or 5/19 system, is
the 4-15 system, that the 2/19 generator is the 9-10 system, and that
the meantone, as we might have expected, works out to be 7-12. We may
also do this when the interval of repetition is a fraction of an
octave, so that from 1/4<2/7<1/3 we get that the 8/28 generator is
the 12-16 system.

We may also express the same system in terms of the comma group dual
to the val group, so that the 5/19 system is the 4-15 system is the
system of 49/48 and 126/125. From there we can pick an appropriate
chroma, such as 25/24 or 28/27 (which is almost exactly a 19-tone
step) and get a block which the system approximates.