A new method for the analysis of dielectric or viscoelastic data, the calculation of the fine structure in relaxation parameters, is described. The dielectric case is worked out in detail. Use is made of the partial derivatives of the dielectric constant with respect to temperature and frequency. It is shown how the fine structure in activation energy and frequency factor (Arrhenius) or, equivalently, the fine structure in activation enthalpy and activation entropy (Eyring) can be extracted from the data using these derivatives. Results of simulations using a Cole-Cole distribution in frequency factors and a Gaussian distribution in activation energies are discussed. The values of both relaxation parameters are accurately deconvoluted with the fine-structure method. Finally, some experimental results for rigid poly(vinyl chloride) are presented. The typical behavior at the alpha- and beta-transitions is discussed. An additional effect is observed at temperatures close to the alpha-transition or the structural transition from the glassy to the rubbery state. This effect offers an interesting tool for a nearly frequency-independent assessment of the glass-rubber transition.