posted 20 October 2000 02:17            

He gives the hint that I needed... I have long wondered why CSound incorporated all those Tchebyshev polynomial tables. I knew the definition of the Tn(x) polynomials (I have known this for years) but I needed the kick in the pants to see the use for them in sound synthesis...

Since Kyma is basically a wavetable synth in its oscillators, and the Wing modulator is a kind of waveshaper, apparently using a Sin(x)/x shaper, we can use the Tchebyshev polynomials in several ways...

The Tn(x) = cos(n arccos(x)) so if x = cos(2 pi f t) then each Tn(x) will generate the n'th harmonic. Adding several of these together in different amplitudes creates a composite polynomial with a defined spectrum according to the selected Tn(x) polynomials.

When used as a waveshaper, a la Wing, a full amplitude cosine produces the harmonic spectrum, but a sinusoid of lesser amplitude will generate a richer spectrum containing subharmonics.

When the polynomial sum is used as a wavetable for an oscillator you can generate the pure harmonic spectrum regardless of amplitudes imposed by envelopes or other modulators.

Computing the sums of the Tn(x) polynomials should be pretty easy in the wave editor of Kyma, so this opens up some rich territory for exploration!

Cheers,

posted 20 October 2000 16:22            

tchebylab.kym tcheby9.aif

This Sound demonstrates a very desirable property - namely that the loudest oscillator inputs generate the richest harmonic spectra, and the quietest ones generate few harmonics. This is similar to real instruments.