  We describe an analytic procedure whereby scattering amplitudes are
bootstrapped directly from an input mass spectrum and a handful of physical
constraints: crossing symmetry, boundedness at high energies, and finiteness of
exchanged spins. For an integer spectrum, this procedure gives a first
principles derivation of a new infinite parameter generalization of the
Veneziano amplitude that is unitary while exhibiting dual resonance and
consistent high-energy behavior. Lifting to a $q$-deformed integer spectrum, we
derive the Coon amplitude and its analogous generalizations. Finally, we apply
this logic to derive an infinite class of deformed Virasoro-Shapiro amplitudes.
