I gotta say this:
The interval = tune idea is not adequate in my opinion.
The intervals sound different (they have a different quality) based on where in the chord they are. So a fourth is not just a fourth.
Take a major chord:
There is a difference between the intervals Fifth up to Tonic and Tonic up to Fourth - both are 5 semitones intervals, but they do not sound the same at all
'When I fall in Love' starts this way:Fifth-up to-Tonic-up to-Fourth. (The fourth is actually the #5 of the next chord if you wanna be pedantic, but the argument still stands: A fourth and a fourth is not necessarily the same thing)
So memorizing a fourth by remembering "Prince of Denmark" (starts with fifth up to tonic) wont help you if you're hovering on the tonic and you want the sound the fourth to be your next color.
There is a cool application here:
The Home Page of Alain Benbassat that helps you recognize notes based on their place in the scale. This is much more worth the effort in my opinion.
Just my 2 kopeks