Consider the set of equidistant nodes in [0, 2π), θ k:= k· 2 π n, k= 0,⋯, n− 1. For an arbitrary 2π–periodic function f (θ), the barycentric formula for the corresponding trigonometric interpolant between the θ k’s is T [f](θ)=∑ k= 0 n− 1 (− 1) k cst (θ− θ k 2) f (θ k)∑ k= 0 n− 1 (− 1) k cst (θ− θ k 2), where cst (·):= ctg (·) if the number of nodes n is even, and cst (·):= csc (·) if n is odd. Baltensperger [3] has shown that the corresponding barycentric rational trigonometric interpolant given by the right-hand side of the above equation for arbitrary nodes introduced in [9] converges exponentially toward f when the nodes are the images of the θ k’s under a periodic conformal map. In the present work, we introduce a simple periodic conformal map which accumulates nodes in the neighborhood of an arbitrarily located front, as well as its extension to several fronts. Despite its simplicity, this map allows for a very accurate approximation of smooth periodic functions with steep gradients.