Ranking Intervals

I've been going through Rayleigh's The Theory of Sound and have been trying to make sense of the intervals refered as thirds, fourths, fifths, etc. and tried ranking them logrithmicly which gives the correct order for them in a list. Trial and error led to a ranking function which seems to correlate well with the numbers,

rank(r) = 7·[log₂(9/8) + log₂(r)]



This is just an empirical formula that works for the intervals listed. The 9/8th term suggests that the ranking is shifted by a tone and the factor 7 suggests 7 steps to an octave. Using 8 steps and no offset doesn't work for the thirds. I have to admit that I am not well informed on the history of the nomenclature of these intervals.