I'm working on the mathematically modeled piano synthesizer. Right now I'm using modal based approach, where the sound is formed by a sum of exponentially decaying sinusoids (harmonics).
Right now I can calculate frequencies of sinusoids (taking inharmonicity into account, of course). For initial amplitude values I'm using data obtained from a real piano samples, since it is just easier right now. I don't care about phases just jet.
My main concern is decay rate. To simulate beating and two stage decay I use 2 or 3 exponents of about the same frequency (depending on the number of strings for a particular note).
For a case of 3 strings theory suggests that one of the exponents will decay faster and two will decay at about the same rate, but due to slight difference in frequencies they will produce beating.
This method is described in the literature, but no algorithm is given to obtain actual values.
At the moment I use simple formulas that I've created myself and they are not based on any physics at all, but produce decent sound after a lot of trial and error.
My question is: is there a method to calculate those decay rates based on a more physical approach?
Key moment - method should produce values for the decay rates before synthesis itself occurs, so methods based on building system with feedback that reproduce this behavior won't work for me.
If you synth your string partials with multiple overlapped modes (with slight detuning), and you start your partials with the same phase, you'll naturally get 2 stage decay - loud at first as your partials are in sync, then a quick decay caused by having your partials go out of phase.