As a computer scientist, I think of each element of musical analysis as a computation we have chosen to do on the music for some reason. I'm not saying I understand how humans decide what the meter is, but I know that we hear a piece (our input) and somehow determine the metrical structure (our output). There are possible analyses that we care about, such as the interval between pitches (defined as the result when you subtract one MIDI number from another), and those we don't, such as the product of pitches (defined as the multiplication of two MIDI numbers). At first glance, from a purely theoretical perspective, the two analyses seem equally valid, and yet obviously interval is a much more important feature to specify than product.
Can we think of a way to formally show that intervals are more important than pitch products? One way I can imagine justifying this belief is by proving through experiments that we perceive of two instances of the same interval as very similar, and don't think of two sets of notes with the same product as necessarily similar at all. What other experiments could we run that would provide real reason to believe that we are right in studying intervals and not products?
Or rather, can we think of aspects we can point to of our body of existing music that confirm that we should care about interval structure rather than pitch structure? We can say that, given the corpus of music that exists, we are used to a given interval being predictive of what will happen next - after a large leap up, we expect a small leap down - which is not true of the product of pitches. Are there other statistical or mathematical properties of existing music which "prove" that the idea of an interval is more useful than the idea of the product of a pair of pitches?
Edit: Realized "product of pitches" was unclear, as addition in frequency space is multiplication in pitch space - I meant that, while MIDI notes are a useful concept and multiplication in the frequency space is a useful concept, multiplication of MIDI notes is not.