I've made some recent progress on a math problem related to waveshapers...specifically, the simplest one, a cubic polynomial.
All good waveshaper functions should have a odd polynomial expansion (for symmetry about x=0)
Suppose our transfer function looks like o(x)= x - .001x^3 we have a dominant linear term and a small (eps=.001) multiplied by x^3, making it non-linear. This transfer function reveals amplitude dependent additions of odd harmonic spectra, just as we would expect. When we look at spectral representations of x^3, we have by the relationship between product/convolution using Fourier transforms, g(t)=u(t)^3: G(f) the FT of g, and U(f) the FT of u
G(f) = ( U * U * U )(f) where * represents convolution in the Fourier domain
With a little exerimentation with the spectrum of u(t), we can see that iff U(f) is an odd harmonic spectrum, G(f) also is an odd harmonic spectrum (this is merely conjecture, as I have not proved it yet). We can examine eigenfunctions of a spectral representation by looking at a non-linear differential equation,
u' ' (t) + eig*u(t) - eps*u(t)^3 = 0 (if we add a forcing term on the RHS, we get the Duffing equation)
if we expand u(t) = u0(t) + eps*u1(t) + eps^2*u2(t) + ... (series expansion of u with respect to eps) and we put appropriate periodic B.C.'s on some interval [0,T], with a little work, we can show the eigenvalues remain exactly the same, and we can derive a series expansion of u, which shows the odd harmonic spectrum. I'm looking for a good source on this equation in order to verify my series expansion terms, before writing them all out.
the series goes (as harmonic numbers) 1, 3, 5, 7, 9, etc...
now if we change x^3 to x^5, we get 1, 5, 9, 13, 17 etc...
which is a bit different spectrum
so, the idea here is that if we have a function such as arctan (x) as our waveshaper, we can expand arctan(x) as a Taylor series to know the coeff. of x^3, x^5 or x^7 or for instance, we could work with x-.001*x^5 as our waveshaper, and use harmonic intervals of 9/5, 13/5, and so on
I'm currently working on a new tuning for 19-tet guitar based on odd harmonics. When I turn on the distortion, the open tuning is very consonant, as one would expect from the mathematical side of things above. However, it's hard to find some chords that really fit in with a uniquely odd harmonic spectrum (changes the system of harmony drastically)
Chuck