We present a new technique, based on the Direct Solution Method (DSM) [Geller et al., 1990; Hara et al., 1991; Geller and Ohminato, 1994>, for calculating SH (toroidal) synthetic seismograms for spherically symmetric, isotropic, media. No asymptotic approximations are used and the synthetics, which fully include both body and surface waves, can be computed for a broad range of frequencies. Our algorithm accurately handles sources that require discontinuities in the vertically dependent part of the displacement. We use matrix operators that minimize the numerical error of the solutions. An example synthetic profile is presented for a 600 km deep source in the IASP91 model for the period range 4--5000 s. ¿ American Geophysical Union 1994