A dedicated algorithm for sparse spectral representation of music sound is presented. The goal is to enable the representation of a piece of music signal, as a linear superposition of as few spectral components as possible. A representation of this nature is said to be sparse. In the present context sparsity is accomplished by greedy selection of the spectral components, from an overcomplete set called a dictionary. The proposed algorithm is tailored to be applied with trigonometric dictionaries. Its distinctive feature being that it avoids the need for the actual construction of the whole dictionary, by implementing the required operations via the Fast Fourier Transform. The achieved sparsity is theoretically equivalent to that rendered by the Orthogonal Matching Pursuit method. The contribution of the proposed dedicated implementation is to extend the applicability of the standard Orthogonal Matching Pursuit algorithm, by reducing its storage and computational demands. The suitability of the approach for producing sparse spectral models is illustrated by comparison with the traditional method, in the line of the Short Time Fourier Transform, involving only the corresponding orthonormal trigonometric basis.