Exploring the stochastic gravitational wave background (SGWB) could unlock invaluable information about astrophysical source populations and processes that remain out of reach through any other means. Gravitational waves can, for example, give us information all the way back to the onset of inflation. But to learn from this complex data, we must have precise predictions for the observables commonly used to characterize it, such as the frequency dependence of the fractional energy density spectrum. Generically, UV-complete inflationary models predict an abundance of light scalars, so any inflationary stochastic background may well be formed in a model with several interacting degrees of freedom. In some regions of parameter space, multi-field inflation models predict an SGWB visible in LISA. In this talk, I will explore how generic the frequency profile is as a function of the number of fields.