What are you trying to do?

Using straight-forward math:

You could get the RMS and peak energy.
You could get pitch, probably using auto-correlation.
You could get spectrum energy density, probably using FFT.

Using more advanced algorithms, and probably more than a second of the music:

You could possibly detect instrumentation.
You could possibly determine the genre of the music.
You could possibly determine the tempo of the music.

Using statistical algorithms:

You could match the sample against a database of known samples for various degrees of match confidence.

You can get a lot of information...
root mean sqare average of time, frequency / frequencies, phase shifts, amplitudes, damping, fourier integral of the waveform ...

Depends on what is interesting for you.
And no, you don't need FFT unless you want to analyse the frequency structure, and even then you can do to by integration (even if FFT is substantially faster, of course)