Most music genre classification techniques employ pattern recognition algorithms to classify feature vectors extracted from recordings into genres. An automatic music genre classification system using tensor representations is proposed, where each recording is represented by a feature matrix over time. Thus, a feature tensor is created by concatenating the feature matrices associated to the recordings. A novel algorithm for non-negative tensor factorization (NTF), which employs the Frobenius norm between an n-dimensional raw feature tensor and its decomposition into a sum of elementary rank-1 tensors, is developed. Moreover, a supervised NTF classifier is proposed. A variety of sound description features are extracted from recordings from the GTZAN dataset, covering 10 genre classes. NTF classifier performance is compared against multilayer perceptrons, support vector machines, and non-negative matrix factorization classifiers. On average, genre classification accuracy equal to 75% with a standard deviation of 1% is achieved. It is demonstrated that NTF classifiers outperform matrix-based ones.