Tunings that contain their own combinational tones

I've recently become fascinated with the idea of tunings/scales whose
harmonies produce combination tones that are other notes from the
scale. So far I've only heard of two scales that can do this: the
harmonic series and the 1:phi pythagorean scale. I think I remember
reading somewhere that 20-EDO is close to having this property as well.

Are there any other scales/tunings that would have this property? And
please, be gentle with the explanations! Thank you.

-Igliashon Jones

--- In tuning-math@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:

> Are there any other scales/tunings that would have this property? And
> please, be gentle with the explanations! Thank you.

For one example consider the Fibonacci sequence, defined by
F_1 = F_2 = 1, F_n = F_(n-1) + F_(n-2). Then the difference tone
F_n - F_(n-1) is F_(n-2), and the sum F_(n-1)+F_(n-2) is F_n.
Other recurrence relationships work also. If you take Siegel's
constant, the real root of x^3-x-1=0, it is analogous to phi, and
recurrence relations can be defined with this as the characteristic
polynomial, and so on and so forth, etc.

> For one example consider the Fibonacci sequence, defined by
> F_1 = F_2 = 1, F_n = F_(n-1) + F_(n-2). Then the difference tone
> F_n - F_(n-1) is F_(n-2), and the sum F_(n-1)+F_(n-2) is F_n.
> Other recurrence relationships work also. If you take Siegel's
> constant, the real root of x^3-x-1=0, it is analogous to phi, and
> recurrence relations can be defined with this as the characteristic
> polynomial, and so on and so forth, etc.

That works for quadratic combinational tones, but what about cubic?
Those are much more audible, if I'm correctly remembering what Paul
wrote me. Also, aren't fibonacci scales only recursive with combi
tones produced by adjacent notes? I had different harmony relations in
mind, say perhaps with notes two to four scale steps from each other.
Are there any scales that would work with these criteria?

Much obliged,

-Igs

--- In tuning-math@yahoogroups.com, "Igliashon Jones" <igliashon@s...>
wrote:

> That works for quadratic combinational tones, but what about cubic?
> Those are much more audible, if I'm correctly remembering what Paul
> wrote me. Also, aren't fibonacci scales only recursive with combi
> tones produced by adjacent notes? I had different harmony relations in
> mind, say perhaps with notes two to four scale steps from each other.
> Are there any scales that would work with these criteria?

There's more going on than that. Successive terms of the Fibonacci
sequnce are F_n, F_(n+1), F_n + F_(n+1), 2F_(n+1) + F_n, so F_(n+3) is
2F_(n+1) + F_n, and therefore F_(n+3) - 2F_(n+1) = F_n, for example.