An entirely new way of analyzing linear and nonlinear oscillatory material responses is presented. The new quantitative sequence of physical processes (SPP) method views generic oscillatory responses within the Frenet-Serret frame as sequences of planar 2D curves embedded in the 3D space defined by the strain, strain rate, and stress axes. Associated with the curve within each so-called "osculating" plane is a binormal vector that wholly determines the orientation of the plane. Physically meaningful information is obtained by calculating the angles between a modified form of the binormal vector and two reference vectors from local information at any point in a cycle. Information from a full period of oscillation is not a requirement of this technique, so that moduli can be calculated from partial or incomplete oscillations. Time-dependent phase and complex modulus information, or dynamic moduli information are obtainable throughout a period for arbitrary oscillatory responses. The SPP method is applied to the Bingham plastic model, power-law fluids, the Herschel-Bulkley model, and the nonlinear Giesekus model to accustom readers to its function. Application of the SPP method to these nonlinear models necessitates the refinement of some common language, as well as changes to the way results from strain amplitude sweeps are displayed and discussed. (C) 2012 The Society of Rheology. [http://dx.doi.org/10.1122/1.4726083]