  In this work we construct an Hermite interpolant starting from basis
functions that satisfy a Lagrange property. In fact, we extend and generalise
an iterative approach, introduced by Cirillo and Hormann (2018) for the
Floater-Hormann family of interpolants. Secondly, we apply this scheme to
produce an effective barycentric rational trigonometric Hermite interpolant at
general ordered nodes using as basis functions the ones of the trigonometric
interpolant introduced by Berrut (1988). For an easy computational
construction, we calculate analytically the differentation matrix. Finally, we
conclude with various examples and a numerical study of the rate of convergence
at equidistant nodes and conformally mapped nodes.
