Given a strictly increasing sequence of abcissae \(\{x_ k\}^ n_{k=1}\) and a sequence of nonnegative numbers \(\{\sigma_ k\}^{n- 1}_{k=1}\) (called tension factors), we say that a real function s defines a generalized tension spline if \(s\in C^ 1[x_ 1,x_ n]\) and \(s^{(4)}-(\sigma_ k/h_ k)^ 2s^{(2)}=0\) in every interval \([x_ k,x_{k+1}]\) with \(h_ k=x_{k+1}-x_ k\) [cf. D. G. Schweikert, J. Math. Phys. 45, 312-317 (1966; Zbl 0146.141); H. Späth, Computing 4, 225-233 (1969; Zbl 0184.198); S. Pruess, J. Approximation Theory 17, 86-96 (1976; Zbl 0327.41009)]. A convenient set of basis functions for the interpolant s taking prescribed values at the nodes has been given in terms of the functions sin h(z)-z and cos h(z)-1. The utility of tension factors in achieving the interpolant with desired properties such as monotonicity and convexity is highlighted in the paper. A local derivative-estimation procedure is given to obtain a \(C^ 1\) interpolant satisfying the desired constraints with minimum tension. An iterative procedure can be used to obtain a \(C^ 2\) spline fit which satisfies the constraints. It is also shown that both the methods produce visually pleasing interpolants.