Deep generative neural networks have been successful in tasks such as composing novel music and rendering expressive performance. Controllability is essential for building creative tools from such models. Recent work in this area has focused on disentangled latent space representations, but this is only part of the solution. Efficient control of semantic attributes must handle non-linearities and holes that occur in latent spaces, whilst minimising unwanted changes to other attributes. This paper introduces SeNT-Gen, a neural traversal algorithm that uses a secondary neural network to model the complex relationships between latent codes and musical attributes. This enables precise editing of semantic attributes that adapts to context. We demonstrate the method using the dMelodies dataset, and show strong performance for several VAE models.