# List of Fourier coefficientes to simulate musical instruments

I'm trying to find a list of Fourier Coefficients (frequencies/amplitudes, ie, spectral data) to simulate different musical instruments using additive synthesis. Are these data published somewhere? This is for a Math class: I just would like to add more frequencies and listen how the sound generated compares with the real sound.

Thanks!

• do you have MATLAB (like the student version that's s'pose to be cheap) or something similar? and do you have .wav files of some notes from musical instruments? if you wanted to demo this, i would think you would need both. – robert bristow-johnson May 12 '15 at 20:24

## 1 Answer

look up Additive Synthesis somewhere. those Fourier coefficients are not constants. the musical note is not normally a perfectly periodic function in time. it is what i like to call "quasi-periodic". so this is not exactly true (given some period $P$):

$$x(t) = x(t+P) \quad \text{for all } t$$

but in the neighborhood of some given time $t_0$ this is approximately true

$$x(t) \approx x(t+P(t_0)) \quad \text{for } t \approx t_0$$

so the exact Fourier series for $x(t)$

$$x(t) = \sum\limits_{n=-\infty}^{+\infty} c_n \ e^{j 2 \pi n f_0 t} \quad f_0 \triangleq \frac{1}{P}$$

is not precisely true, but more like

$$x(t) = \sum\limits_{n=-\infty}^{+\infty} c_n(t) \ e^{j \theta_n(t)}$$

where

$$\theta_n(t) = \int\limits_{0}^{t} 2 \pi f_n(u) du + \theta_n(0) \quad f_n(t) \triangleq \frac{n}{P(t)}$$

so what you're looking for is a collection of envelopes, rather than a collection of constant coefficients, for some musical instrument.

in a previous life, living on an older computer (that might not power up now) and maybe even on some 3.5 inch floppy disks, i have some of that additive synth data. actually, i am not sure where to get it all digested. within the context of Wavetable Synthesis, i know how i would go about extracting it from "samples" of notes (like what you would get from a sample library like from Synthogy ).