Math formulae for organs?

Question

I already know that an organ can be synthesized by using sine waves at specific frequencies.

But if you generate sine waves like that, the timbre of the note is very very plain (it sounds just like a plain tone)

Is there anywhere I can find the math formulae for blending sine waves of different shapes (or triangle waves even) to get more pleasant sounding generated tones?

Answer 1

Additive synthesis can be more than just stacking a bunch of sine waves to emulate an instrument. For instance are the sine waves following the harmonic series or some other series of harmonics? are the sine waves at a constant amplitude? are the sine waves in the same phase?

The key thing to remember is that almost all natural sounds will have dynamic events that are in constant change so only stacking waves in a static manner will not render a complex tone.

One might start with making an analysis of what tone or sound you are interested in replicating. Here is the 101 on this:

TAKING THE WAVEFORM APART

http://artsites.ucsc.edu/ems/music/tech_background/TE-04/teces_04.html

and THE MATHEMATICS OF ELECTRONIC MUSIC

http://artsites.ucsc.edu/ems/music/tech_background/TE-11/teces_11.html

and Frequency modulation synthesis

http://en.wikipedia.org/wiki/Frequency_modulation_synthesis

and a tutorial:

http://www.sfu.ca/~truax/fmtut.html

This is not trivial work.

Answer 2

adding sin waves together of different amplitude,phase,frequency will get you to ALL the tones that are periodic. But periodic tones are pretty boring to the ear by themselves. So you wrap a volume/filtercutoff controlling envelope or lfo around em and they start getting interesting. then throw them thru some fx filters like chorus and reverb and they get MORE interesting. An organ is basically the first form of software synthesizer that we came up with. But, mmm, I'm no expert in these rather deep matters.