What we were looking for in the problem Euclid's Algorithm and Musical Intervals was, if you like, a 'ratio of ratios' but we were not able to find that exactly. In that problem you are asked to find rational approximations for the 'ratio of ratios' using Euclid's algorithm. If the process terminates then you will have found an exact 'ratio of ratios' but generally the process does not terminate.

Here you are asked to prove that 'ratios of ratios' in this sense are (nearly) always irrational.