There's a nice high-level (human readable) technical overview here if you'd like to learn more: http://arstechnica.com/features/2007/10/the-audiofile-unders...
To help squeeze every last drop of perceived dynamic range out of a track you can use a technique known as sidechain compression (http://samplesfrommars.com/blogs/tips-tricks/18999227-how-to...) to dynamically compress other components of the mix in response to (usually) a kick drum.
"Sideband compression" sounds like an interesting dsp project! But I think the term is "sidechain compression", right?
Re-arranging is definitely one 'solution' and the only real approach when you're talking music played live on acoustic instruments. It's one of the elements of good orchestral music I find most fascinating - that many sources of noise, each playing their part and either perfectly blending or completely juxtaposing one another.
I'd argue that using all the various forms of DSP at your disposal when working with sound in the recorded / amplified sense (and especially electronic music) is an equally valid aesthetic choice. Different limitations, different capabilities and difference aesthetic possibilities for different mediums.
There's also simultaneous or frequency masking. If you've ever wondered why loud noises prevent you from hearing something that would otherwise be perfectly audible (e.g. a loud motorcycle driving by your apartment can temporarily drown out a noisy air conditioner), that's why.
Both of these psychoacoustic concepts are used in perceptual audio encoders like MP3 - the idea being that certain bits can be discarded because the listener wouldn't be able to hear them anyway.