I saw Bernoulli's principal mentioned. Although not a mathematician not do I play one on tv, there is a formula for octave and pitch. A local man who is a mathematician had us build an instrument based on a Bernoulli involute. I cut it on the water jet and the lads machined it on the bridge mill. That was a few years ago. Yesterday we were making a different design for the same guy but based on Bernoulli's math. Except instead of aluminum, this one was highly valuable.... Yup. Black walnut. 
. This is a link to the first one.
. This is a link to the first one.
