A method for calculating unsaturated slow-passage N.M.R. spectra is presented which differs from conventional procedures in that no energy levels and no wavefunctions have to be computed and that no explicit line-shape function need be specified. It is shown, by developing the theory in the Liouville representation of quantum mechanics rather than in the Hilbert space representation, that it is possible to compress the total information content of an N.M.R. spectrum into two complex vectors, a radiofrequency-independent ‘shape vector’ and a ‘spectral vector’ which is a trivially simple function of the radiofrequency. The absorption spectrum is obtained as the negative real part of the scalar product of these two vectors. The method may be regarded as an extension of the super-operator formalism of Banwell and Primas.