How many DFT bins must I use to get best precision when analyzing melody scale I need to analyze some signals containing chords. 

How big DFT spectrum is needed to identify each possible musical note in such a signal sampled at 44100 Hz? I can see that 8192 bins gives some frequencies close to some notes like C3 A4 etc., but how big should it be to contain the whole scale, let's say to C5 and with good precision, like for a very good tuner?
You can always improve the accuracy of frequency estimates by quadratically interpolating the peaks in the DFT. This page should explain how to do this: https://ccrma.stanford.edu/~jos/parshl/Peak_Detection_Steps_3.html

You still end up with the issue of converting peaks in the spectrum to pitch, i.e. which peaks correspond to the fundamentals. The fundamental may not even be present as well!