how to program a synth

specifically?

There are tons of examples on the net about digital signal processing for sample generation. This is essentially what you want to do.

Start with a basic program that generates a constant tone (a simple sinuosoid will do) and plays it back. Then have the amplitude of the wave decay exponentially in time. Then start adding frequency oscillation and then it starts becoming more of a black art than a science.

After you have a good sounding piano note (and this can be extremely tough unfortunately) all you need to do is generate it at different frequencies and lengths to get your different piano notes. Then you need some sort of music description to play back. Look into the MIDI format, it is very easy to understand (the basic format that is) and is quite compact as well.

Good luck.

This might be of use to you.

http://www.musicdsp.org/archive.php