Normalizing a spectrogram or a pitch class profile
Audio volume is usually measured in decibels (a logarithmic scale) precisely because there's such a wide range of input energies. But in your case, it's unlikely that you need volume at all. What you need is relative volume: what fraction of the total energy in the signal is in each FFT bin? Since the energy in each FFT bin is positive, it follows that the relative contribution of each FFT bin is strictly in the range [0,1]. Multiply by 2, subtract 1, and you're in the correct range. And as a bonus, the extremes of the range are quite unlikely in practice.