The approach of training sequence models using supervised learning and next-step prediction suffers from known failure modes. For example, it is notoriously difficult to ensure multi-step generated sequences have coherent global structure. We propose a novel sequence-learning approach in which we use a pre-trained Recurrent Neural Network (RNN) to supply part of the reward value in a Reinforcement Learning (RL) model. Thus, we can refine a sequence predictor by optimizing for some imposed reward functions, while maintaining good predictive properties learned from data. We propose efficient ways to solve this by augmenting deep Q-learning with a cross-entropy reward and deriving novel off-policy methods for RNNs from KL control. We explore the usefulness of our approach in the context of music generation. An LSTM is trained on a large corpus of songs to predict the next note in a musical sequence. This Note-RNN is then refined using our method and rules of music theory. We show that by combining maximum likelihood (ML) and RL in this way, we can not only produce more pleasing melodies, but significantly reduce unwanted behaviors and failure modes of the RNN, while maintaining information learned from data.