You're right - music as a logical system is pretty much a mess. But that reflects both its long history and lack of understanding that went along with it for a very long time. A great series to watch if you're interested in that is
Howard Goodall's "Big Bangs" which is very interesting and approachable even for non-musicians.
The idea of musical notes being a straightforward geometric series took an awful long time to be realised, because of the idea (set in train by Pythagoras) that musical intervals HAD TO BE simple integer ratios. Just like Pythag's blind-spot about irrational numbers such as root 2, he assumed that nature abhorred nasty fractions. That thinking influenced music for millennia, and people still think that equal temperament is a "compromise" because it's slightly off from whole number ratios - but who says it has to be? That's just trying to force it to fit an idealised model of nature that is an entirely human idea.
Musical notation is also barking mad in the light of a 12-tone scale. It would make much more sense if a stave had six lines, so every note in the scale had its own fixed place, instead of wandering all over the place because 12 doesn't divide into 5. It would do away with clefs, the need to notate keys by use of special sharps and flats, and generally make it much easier to learn and use. However, it'll never happen because those who read the existing notation think it's fine, and a change would be too disruptive.