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 generalization of the Jacobi identity?