# Synthesis and ortogonal functions

2005-01-25

Several synthesis techniques are linked to ortogonal functions in one way or another: Fourier analysis FM synthesis (amplitudes of sidebands can be calculated using Bessel functions) and waveshaping (controlling partials using lookup tables calculated from Chebyshev polynomials).

I wouldn’t be surprised if wavelet analysis also depends on ortogonal functions but I don’t know the theory behind wavelet analysis well enough.

One of the challenges of additive synthesis is how to handle the massive amount of control data required. I’ve been thinking for a while that if combinations of amplitudes of partials could be described as a set of finite (to avoid aliasing) series it should be possible to reduce the amount of control data required significantly and still maintain the ability to create rich sonic events. This could be done by describing a set of weight functions for each of the series and have the weight function varying over time. Maybe it could be raised one level further and describe the weight function themselves according to ortogonal functions. If the zeroth order was a constant the zero order by itself would produce a static spectrum as in organ tones.

**Idea for further investigation:** If a harmonic sound is made up of a maximum of N partials the spectral development of the sound can be thought of as a function F(t) in a space of N dimentions one dimension for each of the partials. Will it then be possible to approach this function by linear approximation using the method of least squares?

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