In the previous post I raised the question of what might be the mathematical relationship between length and pitch in kalimba tines and similar vibrating bodies. (I had to add a bunch of specific limitations to make the situation mathematically manageable; you can see the details in the original post preceding this one.) Based on empirical observation, I speculated parenthetically that there might be an inverse-squared relationship (e.g., halving the length would quadruple the frequency). Phill Styles, of the physics department at North Carolina State University, sent this comment:
The square root of the length applies when and only when the bar/rod is so rigid clamped that there is no motion at the point and no slope/bending. In a kalimba, the bar/rod rests on a stop, where most people measure from, and then continues to a poor clamp. Because of this the conditions on the bar are not uniform as you mentioned so it doesn’t come out just right. Not bad tho in some cases. Easy to understand in the treatment given in “The Physics of Musical Instruments,” second edition pg.64 case b. The book is written by Fletcher and Rossing., USBN 0-387-98374-0. A great book.
… And this fits well with my observations, in which the relevent factor appeared to be just a bit little less than what the inverse-squared factor would have been. A less-than-perfect immobility/rigidity at the mounting nicely accounts for this “just a bit less.” Thanks to Phill for this, as well as to Tom Rossing and Neville Fletcher.