Lie algebra cochains have a differential d where d^2 =0 because of the Jacobi identity, which can be written in the cyclic form or the Leibniz form. L_\infty algebra cochains have a differential d where d^2 =0 because of higher Jacobi identities written in the unshuffle generalization of Leibniz. Is there some kind of algebra for which cochains have a differential d where d^2 =0 because of the higher cyclic generaalization of the Jacobi identity?