Improving controllability or the ability to manipulate one or more attributes of the generated data has become a topic of interest in the context of deep generative models of music. Recent attempts in this direction have relied on learning disentangled representations from data such that the underlying factors of variation are well separated. In this paper, we focus on the relationship between disentanglement and controllability by conducting a systematic study using different supervised disentanglement learning algorithms based on the Variational Auto-Encoder (VAE) architecture. Our experiments show that a high degree of disentanglement can be achieved by using different forms of supervision to train a strong discriminative encoder. However, in the absence of a strong generative decoder, disentanglement does not necessarily imply controllability. The structure of the latent space with respect to the VAE-decoder plays an important role in boosting the ability of a generative model to manipulate different attributes. To this end, we also propose methods and metrics to help evaluate the quality of a latent space with respect to the afforded degree of controllability.