I agree, the white keys on a piano represent a diatonic scale, but because today’s pianos are rarely tuned to anything other than 12TET, there are few interesting mathematical relationships between notes in practice (and pianos are normally tuned with high notes sharp and low notes flat because that’s how piano strings tend to produce their partials anyway).

Also worth noting the black keys represent a major pentatonic scale and the major pentatonic scale is how many of the earliest bone flutes are tuned.