Nonlinear silicon photonics offers unique abilities to generate, manipulate and detect optical signals in nano-devices, with applications based on field localization and large third order nonlinearity. However, at the nanoscale, inefficient nonlinear processes, absorption, and the lack of realistic models limit the nano-engineering of silicon. Here we report measurements of second and third harmonic generation from undoped silicon membranes. Using experimental results and simulations we identify the effective mass of valence electrons, which determines second harmonic generation efficiency, and oscillator parameters that control third order processes. We can then accurately predict the nonlinear optical properties of complex structures, without introducing and artificially separating the effective χ into surface and volume contributions, and by simultaneously including effects of linear and nonlinear dispersions. Our results suggest that judicious exploitation of the nonlinear dispersion of ordinary semiconductors can provide reasonable nonlinear efficiencies and transformational device physics well into the UV range.