I originally started using MAP at the suggestion of the gunsmith who made my first flintlock... but then thought about it and decided that the soap and peroxide weren't likely to add much to the mixture, and both had the potential to cause damage (soap residues and oxidizing effects of peroxide). So now I just use denatured alcohol, which seems to work great. Alcohol dissolves both oils and water soluble residues, so seems to me like the perfect compound, in addition to the fact that it dries quickly.
I wonder -- was MAP thought up by someone who had no knowledge of chemistry, or does the peroxide or soap offer anything (solvent wise) that I'm missing? Soap is just a lipid (i.e. fat/oil) chemically joined with a base (lye), such that either end of the molecule is still soluble in lipids and water. Seems to offer nothing over alcohol in that regard. I can't think of what the peroxide would offer to the mixture, and wonder if it was just a case of someone throwing the kitchen sink at the issue and everyone else just copying them because it works.