Defines the “even filtration” on an arbitrary commutative ring spectrum; uses this to recharacterize Bhatt–Morrow–Scholze’s motivic filtrations on THH, TC–, TP, TC of quasisyntomic commutative rings (and hence their syntomic cohomology, i.e. the associated graded of the filtration on TC), and then extend these to “chromatically quasisyntomic” commutative ring spectra, such as S, MU, ku (and its Adams summand at any prime), ko, and tmf; finally, calculates the mod (p,v1) syntomic cohomology of the Adams summand at p, realizing part of a picture conjectured by Rognes.