In this paper we investigate the relation between geometrical clocks usually used in the canonical formulation of general relativity and edge modes mostly used in the canonical phase space formulation. We show that with a suitable extension of the ADM phase space different from the one in the existing literature a relation can be established in the context of linearised gravity. Furthermore, we explore how the Barbero-Immirzi parameter is involved in the generators for angle-dependent asymptotic symmetries and the associated super-translation charges.