Is it possible to generate a tone using an LRC circuit?
You would be quite limited using just an LRC circuit- the tone would quickly die down. To get more sustained tones you probably need to add some gain (transistors or op-amps). Usually you want some kind of a controlled attack and a slower decay, maybe with some noise added in. 

This was often done in the days before digital by using pot core inductors and film capacitors to resonate at audio frequencies, along with a bit of active circuitry.  

It's quite possible to do this- you might want to look at an old project- a modular synthesizer that was published in Practical Electronics back in 1973 (it's spread over 13 months of issues).