The recognition of music genre and the discrimination between music and speech are important components of modern digital music systems. Depending on the acquisition conditions, such as background environment, these signals may come from different probability distributions, making the learning problem complicated. In this context, domain adaptation is a key theory to improve performance. Considering data coming from various background conditions, the adaptation scenario is called multi-source. This paper proposes a multi-source domain adaptation algorithm called Wasserstein Barycenter Transport, which transports the source domains to a target domain by creating an intermediate domain using the Wasserstein barycenter. Our method outperforms other state-of-the-art algorithms, and performs better than classifiers trained with target-only data.