Presents an analytic ideal MHD normal-mode analysis of the axisymmetric, m=1, free-boundary mode. The ordering Btheta /Bp approximately O(1) is considered and an incompressible perturbation is used which mainly has implications for the horizontal mode. The analysis is based on an expansion in the inverse aspect ratio and on a shaping parameter, characterizing the effect of a weakly non-circular (up-down symmetric) cross-section. A closed analytical expression, giving the leading order contributions to omega 2 due to toroidicity, non-circularity and the coupling between toroidicity and non-circularity, is obtained in terms of a number of simple parameters, describing the effects of non-circularity and diffuse current and field profiles. When the inverse aspect ratio and the non-circularity are approximately of the same order, the toroidicity can give a contribution to omega 2 which is comparable in magnitude with the contribution due to non-circularity present in a straight pinch. In particular, the relative effect of toroidicity increases with peaking of the current profile.