In recent years, the adoption of deep learning techniques has allowed to obtain major breakthroughs in the automatic music generation research field, sparking a renewed interest in generative music. A great deal of work has focused on the possibility of conditioning the generation process in order to be able to create music according to human-understandable parameters. In this paper, we propose a technique for generating chord progressions conditioned on harmonic complexity, as grounded in the Western music theory. More specifically, we consider a pre-existing dataset annotated with the related complexity values and we train two variations of Variational Autoencoders (VAE), namely a Conditional-VAE (CVAE) and a Regressor-based VAE (RVAE), in order to condition the latent space depending on the complexity. Through a listening test, we analyze the effectiveness of the proposed techniques.