[philosophical throat-clearing]

We can get the finite number of possible songs down quite a bit more by considering the identity conditions covering songs. The number of songs is going to be much smaller than the number of five-minute noise-chunks that count as music. Why? Well, that set of noise-chunks contains, for example, every actual and possible performance of "Summertime" -- Coltrane's, Nina Simone's, mine in the shower this morning, etc. Unless you want to adopt an implausibly strict notion of identity for songs, where any difference in sound --> different song, it's clear that these are all versions of the same song.

So now consider that all the possible performances of "Summertime" include performances by every distinctive voice that's ever existed or will, and on every instrument; and for each voice and instrument and combination, the possible performances include every performance by that voice etc that differs slightly from another in pitch, timbre, tempo, etc etc. So in the set of musical five-minute noise-chunks, there's going to be billions (zillions? more?) of performances of "Summertime". But they're all versions of the same song.

My maths isn't good enough to see exactly what we can do with that, but if the set of musical noise-chunks is guaranteed to contain zillions of versions of every song, perhaps we can safely divide the number of noise-chunks in that set by zillions and achieve a more credible (though still incredibly large) result.

Things get tricky when we try to specify exactly what the identity conditions are for songs. How far do we have to deviate from the template before we don't count as playing the same song any more? Until you resolve that, you're never going to be able to give a precise answer to the question of how many possible songs there are (similar to the point you make that which five-minute chunks count as music is partly a matter of taste).