One of the most challenging aspects of sound analysis and representation is the definition of a good model for the noisy part of sounds. We need a good representation of those components of sound whose spectra lie outside the frequencies of the partials. In this article, we subdivide a sound into its deterministic and stochastic components. The deterministic part of sounds provides the pitch and the global timbre of a sound; it is, in a sense, the fundamental structure of the sound. The stochastic part contains the ”life” of a sound—all the ”microfluctuations” with respect to a non-evolving sound plus the noise due to the physical excitation system. The main idea behind our method is that these ”microfluctuations” with respect to a pure harmonic behavior can be reconstructed from the power spectrum. In this article, we describe a model for the particular case of voiced sounds, namely, sounds with a harmonic spectrum. We define a well-suited analysis and resynthesis method for our model based on the Harmonic-Band Wavelet Transform (HBWT). Owing to the mathematical properties of the HBWT, the synthesis of signals with pseudoperiodic 1/f -like power spectra is straightforward. These spectra are very good approximations of those of real-life voiced sounds. In a first approximation, the only thing we need to do is control the energies of white noise coefficients according to very few parameters derived from the analysis of real sounds. As a further insight, the HWBT analysis reveals the existence of a small but non-zero correlation between the coefficients. An autoregressive (AR) analysis and resynthesis model employing white noise as excitation for AR filters, in conjunction with the above-mentioned loose correlation, can substitute for the trivial white noise coefficient model. A refinement of the technique also takes into account scale-dependent time evolution of the resynthesis parameters. The model does not apply to the harmonic components. We preserve the restricted set of analysis wavelet coefficients corresponding to the narrow bands of the harmonics in order to have perfect reconstruction data for the deterministic part, that is, the harmonic amplitudes and their time envelopes. The attack transients are preserved as well. This method can be seen both as a musical tool for sound synthesis able to provide synthetic sounds with a natural timbre dynamic and as a compression technique. In the first part of this article we review ordinary wavelets, Harmonic-Band Wavelets, and the pseudo-periodic 1/f-like model. In the second part we present new developments of the synthesis method from methodological and experimental points of view.