||Recently there has been a great deal of interest in the use of "tension" parameters to augment control mesh vertices as design handles for piecewise polynomials. A particular local cubic basis called p-splines, which has been termed a "generalization of B-splines", has been proposed as an appropriate basis. These functions are defined only for floating knot sequences. This paper uses the known property of B-splines that with appropriate knot vectors they span what are called here spaces of tensioned splines, and that particular combinations of them, called LT-splines, form bases for the spaces of tensioned splines. In addition, this paper shows that these new proposed bases have the variation diminishing property, the convex hull property, and straightforward knot insertion algorithms, and that both curves and individual basis functions can be easily computed. Sometimes it is desirable to interpolate points and also use these tension parameters so interpolation methods using the LT-spline bases are presented. Finally, the above properties are established for uniform and nonuniform knot vectors, open and floating end conditions, and homogeneous and nonhomogeneous tension parameter pairs.