We introduce tangential singular beams with non-integer or fractional topological charge (TC), which have not yet been studied. These beams are constructed by modulating the spatial phase by a piecewise tangent function. The TC can be controlled by the periodicity parameter and the parameter that limits the range of the function allowing to achieve non-integer phase jumps. We study the effect on their amplitude profiles and TC using numerical simulations. Our results are supported by an experimental implementation using the collinear phase-shifting digital holography technique. By means of experimental phase reconstruction, we calculate the TC of these new beams in the near field.