The authors present new analytic formulae for the phonon density of states and inverse decay length of a one-dimensional granular structure and show that for large grains with no disorder these are universal functions of a dimensionless frequency Omega and a velocity of sound ratio, C. The density of states at frequencies Omega <C is shown to exhibit a granularity-induced enhancement, which is unaffected by the presence of disorder. For the first time a one-dimensional analogue of a network glass below a smeared rigidity percolation transition is analysed. It is shown that this system also exhibits a low-frequency density-of-states enhancement, whose magnitude is determined by the velocity of sound ratio C alone.