Two different approaches are used for evaluating two-time correlations of Fourier velocity modes in isotropic, and in rotating, homogeneous incompressible turbulence. A synthetic model is proposed and compared with the results of direct numerical simulations. This kinematic model presents the advantage to incorporate different sources of unsteadiness, from arbitrary timescales to linear dynamics of rotating flows. The two main timescales characterizing the various processes involved in the dynamics of isotropic turbulence can be included: the sweeping effect, in which small scales of the flow are advected by the large scale motion, and the straining hypothesis–also called Kolmogorov or eddy turnover timescale. In the rotating case, the dispersion relation of inertial waves can also be included. Using Lighthill's theory, the effect of these different timescales on the sound emitted by such a model is also examined, and compared with direct numerical simulations.