The matrix pencil spectrum analysis method provides an interesting alternative to classical Fourier analysis. The method is based on the assumption that the signal can be represented as a sum of exponentially damped sinusoids whose parameters (frequency, damping factor, amplitude and phase) are estimated by a matrix-based algorithm. In spite of its computational cost, the matrix pencil method achieves extremely good results in the case of highly damped signals, short data records, or close frequencies. The matrix pencil method is first briefly described and compared to other spectrum analysis techniques (Fourier transform, Prony method). Several implementations are proposed and their computational costs evaluated. The good performance of the method is illustrated by analysis examples involving low-frequency beats of piano tones. These beats are decomposed as sums of closely spaced damped sinusoids, which makes it possible to verify some of results on the coupling between orthogonal vibrating modes of strings [G. Weinreich, ‘‘Coupled piano strings,’’ J. Acoust. Soc. Am. 62, 1474 (1977)]. The matrix pencil method is also applied to the analysis of the bridge admittance of a guitar and is used to model the admittance as a set of simple mechanical systems in parallel.