Abstract: We show that the compactly generated t-structures over any commutative ring are in a 1-1 correspondence with infinite filtrations of the Zariski spectrum by Thomason sets, establishing a common generalization of the analogous result for commutative noetherian rings due to Alonso-Jeremias-Saorin, and the recent classification of tilting classes. Our method can also be used to show that, over a commutative noetherian ring, the homotopically smashing t-structures (as introduced in the recent paper by Saorin-Stovicek-Virili) coincide with the compactly generated ones.