Fourier Transform of an audio signal
How can we extract the frequency content of an audio signal given the
  histogram of beat energy versus beats per minutes?


You can't. The histograms you have shown are meaningless in the context of the fourier analysis of the full musical signal. In fact the histograms you have shown don't actually represent any part of the frequency spectrum of an audio signal. 

To extract beat information from an audio signal requires demodulation of that signal via a frequency selective envelope detector - this largely loses any other spectral content that you might be wishing to extract.