A way of measuring beauty? I think an answer to that might be more profound than how the universe started :) It's probably best to try explaining an area of aesthetics such as music. And then split this up further to explain one aspect of music; what makes a good chord?
Question: Why is the chord C#, F#, B, Eb, F#, Bb, much, much better than C#, F#, C, Eb, G, Bb ?
Here are some more bad chords: C#, F, C, Eb, F#, Bb, ...or this one... D, F#, B, Eb, F, C
Here are some more good chords: C (in the bass followed by) Bb C D F G and A ... or: C Bb C# E G and A
All you have to do is work out exactly why the good chords are better. Once you've done that, try the countless two chord combinations, and then the multitudes of three chord combinations etc. etc. Find a logical pattern to distinguish them scientifically, work out a multi-dimensional rating system, and you're half way there! Presumably, there would be chords and chord sequences which are excellent (even perfect), very poor/boring, and everything in between.
Yes, an element of subjectivity would come into it, but as long as practically everyone agrees on whether a chord is good/bad (and in the examples I gave above, I believe they would), then we can try to find out the pattern behind it. That's a start.
The same principle could be applied to pictures too. Imagine a long one-dimensional 100-pixel black and white 'picture'. Hundreds of people could rate thousands of these randomly generated 'pictures'. Because each picture is relatively abstract, there would be little room for cultural bias. On the average, I believe we could establish a general consensus on the 'scores' for such pictures.