My children asked me why the piano has seven white keys and five black. I played around with graph paper and discovered in 1992 that you could use 19 keys to the octave. Further research, using logarithms, showed that you could also divide the octave into 31 or 53 equal steps. These systems had been known for centuries, but I thought I had discovered a new one: 34 gives a better all-round performance than either 19 or 31 (let alone 12, which is very rough). Eventually I found that 34 too had been discovered by Dirk de Klerk in 1979. Indeed, Cyriac Schneegass had advocateded the 34 system in 1590.
All that remained was to explain this strange series of numbers - 12 - 19 - 31 - 34 and 53. I thought I had solved the problem and in 1996 I wrote an article to say so, but I did not find anyone to publish it. Perhaps it was too philosophical; I try to relate the solution to the Platonic idea of the 'music of the spheres'. I also try to be practical, suggesting a keyboard that could be used for 19 or 31 keys as well as 34. In the end though the piano keyboard has conquered the world, so the alternatives are not going to find a use on this planet. Anyway, here is my article as an attachment.