As one of the most intuitive interfaces known to humans, natural language has the potential to mediate many tasks that involve human-computer interaction, especially in application-focused fields like Music Information Retrieval. In this work, we explore cross-modal learning in an attempt to bridge audio and language in the music domain. To this end, we propose MusCALL, a framework for Music Contrastive Audio-Language Learning. Our approach consists of a dual-encoder architecture that learns the alignment between pairs of music audio and descriptive sentences, producing multimodal embeddings that can be used for text-to-audio and audio-to-text retrieval out-of-the-box. Thanks to this property, MusCALL can be transferred to virtually any task that can be cast as text-based retrieval. Our experiments show that our method performs significantly better than the baselines at retrieving audio that matches a textual description and, conversely, text that matches an audio query. We also demonstrate that the multimodal alignment capability of our model can be successfully extended to the zero-shot transfer scenario for genre classification and auto-tagging on two public datasets.