I think musica mundana has been misunderstood both in antiquity and today. The point is not, I think, to redefine our conception of the heavens into our predefined concept of music, but the other way around, at least, or maybe a redefinition of both. Mostly our conception of music needs redefining, though. It’s interesting to note that Boethius in De Musica registered himself as something of a skeptic of the idea of musica mundana and made fun of people who thought the music was actual music, music that you could hear.
Clearly we can’t hear it, and the usual explanation is that the music of the spheres is something so fundamental to our surroundings that we have ceased to notice it for its ubiquity, but were it to stop, we would immediately notice its absence. I’m not sure where this idea has its origins, but it may be clouding us as to exactly what the ancients thought about the subject.
This is just guesswork, but I think the idea comes from a few observations about music and about astronomy:
(a) Motion through air creates sound.
(b) That sound can be manipulated into music by combining two motions whose movements span certain ratios (2:1 being an octave, 3:2 a fifth, and so on).
(c) The heavens also move.
(d) They can be explained in terms of similar ratios.
(e) Consequently, the heavens are engaged in the same sort of activity.
Music, then, is a genus—the genus of objects in motion according to whole-number ratios—which subsumes the species (a) heavenly bodies (musica mundana) and indeed also (b) what we now call music (musica instrumentalis).