I'm looking for some metric that can score fractions or decimals by how "clean" or "round" they are. Put another way, I want to measure inverse "dissonance". I want to give a high score to numbers like 1, 1/2, 2/3, 1/4, etc., and low scores to 3/13, or $\sqrt{2}$.
One idea I had was to just add the numerator and the denominator of the most simplified form, but I wanted to see if there were other ideas out there. I also looked briefly at entropy in cryptography and data compression.
Huge bonus points if the method could handle numbers close to round fractions. So that for example $2^\frac{7}{12} \approx 3/2$ would get a good score even though it's irrational.
Any thoughts? thanks!