“When I master my projects why does it sound better to compress three times at a 4:1 ratio than one time at 12:1?”
Well, first, this is a matter of opinion, and probably does depend a lot on the material and the equipment you are using. If you were only considering the ratio settings you’d think the two processes would be equivalent. But these don’t take into account things like the attack and release of the compressor. While there are other variables, this is likely where the difference is most noticeable to you. Different compressors have (sometimes very) different attack and release characteristics from one another, and often different depending upon how hard they are being hit. If your compressor’s release works like an optical compressor, for example, you’ll find that (as noted in the Tech Tip from 03/20/2002) the first 50% of the release happens very rapidly, whereas the latter 50% is very slow, and has a sort of non-linear characteristic in that its rate changes over time. Further, the release speed is often somewhat dependent upon how high the input signal has been. Clearly one can see how this would manifest itself differently under the two different scenarios you suggest. Attack characteristics are also different between different units and settings. Furthermore, if you are using over easy compression you have yet another variable to consider in terms of the amount of time the compressor is likely spending in that transition area between normal gain and the onset of compression.
Depending upon the compressor(s) you use it may be possible to get close to that 3×4:1 sound with your 12:1 ratio through experimentation. On the other hand, as illustrated with the optical compressor example above, you may not. Some compressors are much, much more versatile than others, which is usually considered a desirable attribute in mastering work.