Looking for new perspectives to analyze non-linear dynamics of speech, this paper presents a novel approach based on a microcanonical multiscale formulation which allows the geometric and statistical description of multiscale properties of the complex dynamics. Speech is a complex system whose dynamics can be, to some extent, geometrically and statistically accessed by the computation of Local Predictability Exponents (LPEs) unlocking the determination of the most informative subset (Most Singular Manifold or MSM), leading to associated compact representation and reconstruction. But the complex intertwining of different dynamics in speech (added to purely turbulent descriptions) suggests the definition of appropriate multiscale functionals that might influence the evaluation of LPEs, hence leading more compact MSM. Consequently, by using the classical and generic Sauer/Allebach algorithm for signal reconstruction from irregularly spaced samples, we show that speech reconstruction of high quality can be achieved using MSM of low cardinality. Moreover, in order to further show the potential of the new methodology, we develop a simple and efficient waveform coder which achieves almost the same level of perceptual quality as a standard coder, while having a lower bit-rate.