Many single-channel signal decomposition techniques rely on a low-rank factor-ization of a time-frequency transform. In particular, nonnegative matrix factoriza-tion (NMF) of the spectrogram – the (power) magnitude of the short-time Fourier transform (STFT) – has been considered in many audio applications. In this set-ting, NMF with the Itakura-Saito divergence was shown to underly a generative Gaussian composite model (GCM) of the STFT, a step forward from more empiri-cal approaches based on ad-hoc transform and divergence specifications. Still, the GCM is not yet a generative model of the raw signal itself, but only of its STFT. The work presented in this paper fills in this ultimate gap by proposing a novel signal synthesis model with low-rank time-frequency structure. In particular, our new approach opens doors to multi-resolution representations, that were not pos-sible in the traditional NMF setting. We describe two expectation-maximization algorithms for estimation in the new model and report audio signal processing results with music decomposition and speech enhancement.