"Today, I ask for a clear mind, not just a quiet one."

Ganapati names him lord of the ganas — the hosts, the categories, the many. Tradition reads this first at the literal level — lord of Shiva's attendant hosts — and, through the Atharvashirsha's own language ('you are of the nature of knowledge, of discriminative realization'), as lord over the mind's own power to order and distinguish. A prayer to Ganapati, in this reading, is a request for buddhi — the faculty that sorts — and not only for the feeling of calm.