Also, the white keys are far from being an arbitrary set of pitches. Very roughly, they're chosen so that as many combinations of them as possible sound reasonably harmonious together when played on an instrument whose sound has a harmonic spectrum (which applies to most of the tuned instruments used in Western music). I don't mean that someone deliberately sat down and solved the optimization problem, of course, but it turns out that the Western "diatonic scale" (= the white notes) does rather well by that metric. So it's not like we'd particularly want to change the scale for the sake of making either the mathematics or the music sound better.