Algebraic K-theory and topological cyclic homology are invariants defined very generally (e.g., for stable \infty-categories). When applied to a commutative ring, they inherit an additional important (and slightly mysterious) structure: a motivic filtration. For topological cyclic homology, the motivic filtration has been constructed in the recent work of Bhatt--Morrow--Scholze using a very direct approach; the graded pieces are a type of filtered Frobenius eigenspace on prismatic cohomology. I will describe the BMS construction and some new structural features of the motivic filtration on TC, in particular, an identification of the graded pieces, either in low weights or with denominators, with syntomic cohomology -- a construction which relies only on more classical techniques, and which is easier to compute in practice. Joint work with Benjamin Antieau, Matthew Morrow, and Thomas Nikolaus.