  We analyze so-called generalized Veneziano and generalized Virasoro
amplitudes. Under some physical assumptions, we find that their spectra must
satisfy an over-determined set of non-linear recursion relations. The recursion
relation for the generalized Veneziano amplitudes can be solved analytically
and yields a two-parameter family which includes the Veneziano amplitude, the
one-parameter family of Coon amplitudes, and a larger two-parameter family of
amplitudes with an infinite tower of spins at each mass level. In the
generalized Virasoro case, the only consistent solution is the string spectrum.
