Is there a way to calculate the required diameter of a cylindrical metal (let's say aluminum) pin of a fixed length so that it has a specific resonant frequency? Let's say I have an aluminum pin that is 4 inches long, and I want it to have a resonant frequency of 256Hz (middle C), how would I go about calculating its diameter?
Edit:
Thanks to Farcher for this link, I think this is the equation I am looking for, however not being a physicist I'm not too sure what each term represents, or the units used. Can someone help me break this down:
$$f = \frac{v}{2L} = \sqrt{\frac{E}{4L^2\rho}}\\ E = \rho\; \left(\;2\;L\;f\;\right)^2$$
I get that L is the length, and F the frequency, but what are p and E? Also I do not see where the diameter or any dimension other than the length comes into effect... Am I just mistaken, and the resonant frequency is only dependent on the material and length, no matter the other dimensions? Somehow that doesn't seem right to me...