An application of multiple-stage non-negative matrix factorization to music analysis

Abstract

Non-negative matrix factorization (NMF) is an analysis technique for matrix with non-negative elements. NMF decomposes a sound spectrogram X obtained by a short-time Fourier transform into the product of two non-negative matrices H and U. The matrix H represents the bases of X and the matrix U represents the activation gains of H. As an extension of NMF, multiple-stage NMF has been proposed in order to represent the hierarchy of data. This research focuses on the hierarchy of music data, and attempts to apply multiple-stage NMF to music analysis. For simplicity, sounds treated in this research are assumed as follows: single tone consists of fundamental tone and harmonic overtones and a chord consists of some single tones. According to this assumption, observed sound spectrogram can be decomposed into three stages. In the first stage, the basis matrix is approximated by a Gaussian distribution in order to represent a spectrum of fundamental tone and harmonic overtones. In the second stage, the basis matrix consists of the set of single tone with fundamental tone and harmonic overtones. And in the third stage, the basis matrix consists of the set of chord with some single tones. Evaluation was carried out by MIDI data.