To what extent do the vibrations of a mechanical system reveal its composition? Despite innumerable applications and mathematical elegance, this question often slips through those cracks that separate courses in mechanics, differential equations, and linear algebra. We address this omission by detailing a classical nite dimensional example: the use of frequencies of vibration to recover positions and masses of beads vibrating on a string. First we derive the equations of motion, then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory's monochord. More challenging is the recovery of masses and positions of the beads from spectral data, a problem elegantly solved, through application of continued fractions, by Mark Krein. After presenting Krein's algorithm in a manner suitable for advanced undergraduates, we confirm its effcacy through physical experiment. We encourage readers to conduct their own explorations using data sets we provide on the web.