It has long been known that two scaling factors and three spectra, corresponding to three different end conditions, are required to determine the masses, lengths and stiffnesses in a discrete model of a beam in flexural vibration. What had not been known was how to find a family of beams that all have the same spectrum under one set of end conditions, say those corresponding to a cantilever. This paper presents two procedures for finding families of such isospectral beams. The first uses shifted QR factorization and yields a four–parameter family. The second uses Toda flow to find another, more restricted family.