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It’s about the circle of fifths, really. …Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A#… Notice that all the notes without accidentals (the white keys) are contiguous (in a row). We defined them that way, of course. Also A#=Bb, it repeats, it’s a circle after all.
The circle of fifths itself exists because of mathematics, and arguably it’s not just fifths (a 3:2 ratio) but also thirds (5:4) that are relevant, and the reasons why are complicated. We don’t like dissonance which is highest when note frequencies are near unison but not at unison (think of a graph like x*exp(-x)), including when that happens between overtones/harmonics, and that naturally leads to preference for simple ratios, the simpler the better. 12 equally spaced notes is an optimum that’s much better than 11 or 13, and that’s not a simple fact to derive, but it is “natural” and requires no assumptions about culture. It’s not a coincidence that 2^(19)≈3^(12) or 2^(7)≈5^(3) – those are, effectively, reasons why 12 is an optimum. But that’s all overcomplicating the issue, I think.
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