Let $f(x)={\log }_{10}(\sin (\pi x)\cdot \sin (2\pi x)\cdot \sin (3\pi x)\cdot \sin (4\pi x)\cdot \sin (5\pi x)\cdot \sin (6\pi x))$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n$?