12. The smallest positive period of the function $y=|\sin x|+|\sin 2 x|+|\sin 3 x|$ is
12. $2 \pi$. Let $|\sin (x+T)|+|\sin 2(x+T)|+|\sin 3(x+T)|=|\sin x|+|\sin 2 x|+|\sin 3 x|$. Set $x=0$. We get $|\sin T|+|\sin 2 T|+|\sin 3 T|=0$. Thus, $\sin T=\sin 2 T=\sin 3 T=0$, which gives $T=2 k \pi, k \in \mathbf{Z}$. It is easy to see that $2 \pi$ is the period of the given function, and it is also the smallest positive period.