How do I analyze a music file ( only contains 5 seconds of note A) using the FFT plot to determine the note being played? To perform this task why can't I just import the file, and directly perform the fft and use the stem plot to determine the note? From the index k of the peak I can do w = 2pi*k/N where N is total number of points. 

Fs, x = wavfile.read('piano.wav')
X = np.fft.fft(x)
plt.plot(abs(X))
plt.show()


However I'm not sure about the correct way to map the fourier coefficients from the plot to the frequency to determine the note.

Please advise.
Thanks.
An FFT magnitude peak frequency (derived from a single FFT result index) is not reliably the same thing as the music pitch frequency or the musical note.  Note pitch is a psychoacoustic phenomena, and more reliably related to periodicity than the (sinusoidal) spectral peak frequency (e.g. a periodic/repeating waveform shape does not need to look like a sine wave).  See FFT Pitch Detection methods: Autocorrelation or other? for a list of some better methods for estimating note pitch frequency or the musical note.