  K. Kato has recently defined and studied heights of mixed motives and
proposed some interesting questions. In this paper, we relate the study of
heights to the study of Tamagawa numbers of motives. We also partially answer
one of Kato's questions about the number of mixed motives of bounded heights in
the case of mixed Tate motives with two graded quotients. Finally, we provide a
concrete computation with the number of mixed Tate motives with three graded
quotients.
