Frequency Ratio Circuit
Since you only want an integer multiple, as far as I can tell ("\$12f_\text{ref} \le f_\text{sig}&lt;13f_\text{ref}\,\rightarrow 12\$"), what about simply using a cheap synthesizer IC that takes a reference oscillation, and produces multiples of that?

Use a microcontroller to program your synthesizer to generate 10, 11, 12, …, 19,20 × reference frequency. 

Use that local oscillation to mix down your \$f_\text{sig}\$, and low-pass filter with a filter bandwidth of \$\frac{f_\text{ref}}2\$. If something passes through that filter, you know you've hit "close" to that local oscillator.

You can of course increase resolution by using smaller steps (synthesizers can also do things like 10.25545×\$f_\text{ref}\$), and narrower low-pass filtering.

By the way, what you're building with this is practically a spectrum analyzer :)