Wanna see where you made an ass out of u and me? With a string you've got one-dimensional oscillation modes. With a plate you've got two-dimensional oscillation modes with all the misadventures that entails; with a surface you've got three-dimensional oscillation modes and they're all interdependent. Also keep in mind that try as you might, the material properties of that bell aren't necessarily homogeneous across the bell and non-homogeniety is going to change your interdependencies away from what you know to what you don't. "Non-homogeniety" in this case being "damping one part but not all parts of the bell" because you suddenly changed the damping coefficient for part of the bell but not all of it. I accidentally made a bell in beat-on-shit-with-a-hammer class. It was dope. But I had two half-spheres of nickel silver, same treatment, same size, and one of them was a semitone more flat than the other. Why? Because bells are fucking complicated. I dug into it just enough to go "well damn, they were grinding the things in like seven different places to try and get the first five harmonics to line up to their temperament by ear" and recognized that me and my silver hammer should be happy with a semitone.oscillation mode