A more detailed explanation by Kevin Houston on this topic for the British Science Festival, "The Beatles' Magical Mystery Chord" (8m49s) ~ http://www.youtube.com/watch?v=xKx0DG-5Qyg&feature=playe...
If you really want to kill it, you could always come in with a learning model which accounts for the characteristics of the instruments and the way in which the sound evolves over time. An FFT is effectively looking for the integrated power in a frequency band. This is often more than enough to do the job, but it misses potentially useful information found in the time domain.