Both the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups and the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups are examples of the brackets arising from the general noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an e_3-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads. Joint work with Niels Kowalzig (arXiv:1712.09717)