The quest for comprehensive generative models of intonation that link linguistic and paralinguistic functions to prosodic forms has been a longstanding challenge of speech communication research. Traditional intonation models have given way to the overwhelming performance of deep learning (DL) techniques for training general purpose end-to-end mappings using millions of tunable parameters. The shift towards black box machine learning models has nonetheless posed the reverse problem -- a compelling need to discover knowledge, to explain, visualise and interpret. Our work bridges between a comprehensive generative model of intonation and state-of-the-art DL techniques. We build upon the modelling paradigm of the Superposition of Functional Contours (SFC) model and propose a Variational Prosody Model (VPM) that uses a network of variational contour generators to capture the context-sensitive variation of the constituent elementary prosodic contours. We show that the VPM can give insight into the intrinsic variability of these prosodic prototypes through learning a meaningful prosodic latent space representation structure. We also show that the VPM is able to capture prosodic phenomena that have multiple dimensions of context based variability. Since it is based on the principle of superposition, the VPM does not necessitate the use of specially crafted corpora for the analysis, opening up the possibilities of using big data for prosody analysis. In a speech synthesis scenario, the model can be used to generate a dynamic and natural prosody contour that is devoid of averaging effects.