It isn't circular at all. What Artful Codger is suggesting is a variant of the scheme used by Indian and Middle Eastern tonal systems (and in a simpler way by the chant modes) - look for characteristic cadential formulas, or at the bare minimum note frequencies. Phil Taylor's BarFly ABC processor for the Mac uses a frequency-based approach for its mode-guessing utility - it ignores the order of the notes in the tune entirely. Nonetheless that's often enough to go on. Here's the frequency data it reports for "I Know Where I'm Going":

  F       *************************
  Gb      
  G       *************************
  Ab      
  A       *************************
  Bb      ******
  B       
  C       *******************
  Db      
  D       
  Eb      
  E       

Most of the notes fall into an F major triad. Sing it in a resonant acoustic and you'd hear that as the predominant vertical sonority (the other one being ambiguously Gm or C7).