Length of Hamming Window to Resolve a Harmonic
Around 1000 samples to resolve the psychoacoustic pitch of concert A.  

The fundamental period of 440 Hz is just over 100 samples at a sample rate of 44100.  But humans typically need somewhere around the range 6 to 10 pitch periods to hear a pitched sound instead of noise or clicks.  Since FFT windows attenuate the edges of the waveform, using the larger number of periods (10) is likely more reliable in matching human perception.

So I'd use a 1024 length FFT for initial testing.

To inspect higher harmonics if you already know the exact pitch frequency, you could use a shorter window, at least 2 pitch periods for a Hamming window, and zero pad after windowing to use a much longer DFT that is an exact integer multiple (say 16X) of the pitch period in length.