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

1. As in the problem, the zeros in $(0,1)$ are the fractions $\frac pq$ in lowest terms with $2\le q\le6$: $1+2+2+4+2=11$ zeros, cutting $(0,1)$ into $12$ intervals. The product changes sign at $\frac pq$ iff $\lfloor6/q\rfloor$ is odd, i.e. for $q=2,4,5,6$ but not for $q=3$.
2. In increasing order: $\frac16$F, $\frac15$F, $\frac14$F, $\frac13$N, $\frac25$F, $\frac12$F, $\frac35$F, $\frac23$N, $\frac34$F, $\frac45$F, $\frac56$F. Starting with $+$ near $0$, the signs of the $12$ intervals are $+,-,+,-,-,+,-,+,+,-,+,-$.
3. There are $\boxed{6}$ positive intervals.