How many DFT bins must I use to get best precision when analyzing melody scale
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!