I've never really been much of a audio creation buff, but when I heard that Propellerhead were releasing their ReBirth audio program for free, I quickly downloaded it because I've always been curious as to what all the fuss was about. ReBirth is apparently an emulator of the TB-303, which is a piece of techno hardware from the 80s that could be used to generate weird instrument sounds and to sequence those sounds into some simple tracks. It also emulates some other pieces of hardware (namely some rhythm boxes). I actually found ReBirth to be really difficult to use and completely confusing, so I uninstalled it.
Fortunately, I also downloaded the free Rubberduck program from d-lusion, and it was quite fun and easy to use (not that I have sufficient talent to get the program to do anything that sounds good, but at least I can make those crappy sounds easily as opposed to with ReBirth). After playing with it for a while, I noticed that if I turned off all the filtering effects, I could still make some nice instrument sounds just by combining sine waves and square waves.
This was significant to me because I occasionally create little Java games, but it's hard to get any music into those games. Java includes support for MIDI output, and the full Java VM even comes with Roland sound patches for its instruments, but the existence of these patches isn't guaranteed, so you can't rely on them for game music. Instead, you have to supply game music via sound files, but then you need to generate these sound files somehow. I purchased some cheap music composition software, but it creates sound files by playing the music out through the computer sound card and recording it. Since my sound card is really cheap, the resulting sound files end up containing much too much noise to be useful. There are open-source programs that can take MIDI songs and output sound files directly, meaning you don't get any noise, but these programs all use sound patch sets of questionable legality, so I didn't want to use them. But if I could just make my own instrument sounds by combining some sine waves and square waves, then I could just create a program for converting MIDI songs into sound files myself without the problems of noise or legal encumbrances.
Unfortunately, once I wrote a program for combining sine waves and square waves in various ways, I realized that the sounds that my program was generating were puny and weak. Even the simple sine wave sound generated by d-lusion's Rubberduck was much fuller and colourful than the sine wave generated by my program.
The documentation for d-lusion's Rubberduck said that they pass their sound through a low-pass filter, but I thought I had turned all the filtering parameters off. Besides, a low-pass filter shouldn't really affect a sine wave. In any case, I'm really weak with signal processing, so I couldn't really design such a filter anyways.
Well, anyways, I was looking over the "Cookbook formulae for audio EQ biquad filter coefficients" document by Robert Bristow-Johnson which is, essentially, the dummies guide to creating various filters using digital feedback mechanisms. When I looked at what Rubberduck did to square waves (it rounded out the corners), I suspected that maybe I could just sort of randomly feedback things into itself and maybe get a similar effect.
So that's what I did:
y[n] = 0.8 * y[n-1] + 0.2 * x[n]
where x[n] is a sine or square wave and y[n] is the final sound output, and the result is exactly what I wanted. Whereas a simple sine wave is teeny and annoying, when you pass it through this filtering, you get a richer, rounder, more bloated sound. Of course, when you look at the raw wave form, it still looks like a sine wave, but I suspect that it's slightly reshaped so that you get some interesting side frequencies going on. I tried doing a spectral analysis, but I couldn't really interpret it (it looked consistent with a sine wave to me), but it sounds a lot different, so there must be some sort of subtle colouring there.
Of course, to get really interesting sounds, you need adjustable filters that allow you to get various cool effects, but I think I have pieces of code for making blip and bop noises for my Java games.