In the world of music science, there is a lot of interest in "animal music", or, perhaps more precisely, in those animal sounds that are plausibly analogous in some way to human music.
The Origins of Music (MIT Press 2000), edited by Wallin, Merker and Brown, contains a section "Vocal Communication in Animals" with eight articles about different animal calls and their possible relationship to human music. Groups of animals considered to be of interest include primates, birds and whales.
The very existence of terms like "birdsong" gives testament to our tendency to perceive the sounds that other animals make as forms of music. There is also an apparent level of creativity in some forms of animal song; in particular whales and some birds are found to invent new "songs".
But we must be very cautious about over-interpreting our own human reactions to these non-human compositions and repertoires. Music is a very subjective phenomenon, and subjective phenomena are exactly the one category of phenomena that we can't study in animals (at least not until we have a full understanding of the relationship between subjective phenomena and physiological phenomena in the human brain, which we could then apply retrospectively to our observations of animals – but this would imply that we had already solved the problems that we were studying).
The more mundane explanation of most animal calls is that they are forms of communication. The frustrating thing about trying to explain music is precisely that it is not a form of communication. The only thing we "learn" from hearing an item of music played is that the particular item has certain subjective effects on us. If the composition of music is a "communication" of some important information, one would have to wonder why the ability to communicate this information is restricted to that tiny percentage of the human race that actually knows how to compose new music.
For those animals whose "songs" have an apparent degree of creativity, the most straightforward explanation is that the creativity is done for the benefit of a listener who appreciates creativity.
Sexual selection can explain arbitrary features of animal structures and behaviours, and such features are in effect super-stimuli for themselves. So if the male of some species of bird sings a certain mating call, and the female bird responds to "creative" variations within that mating call, then there will be a selective pressure for the male bird to evolve a tendency to be as creative as possible. In "Birdsong Repertoires" (Chapter 4, The Origins of Music) Peter Slater cites C.K. Catchpole ("Temporal and sequential organisation of song in the sedge warbler" (Acrocephalus schoenobaenus), Behaviour 59:226-246) and M. Eens, R. Pinxten and R.F. Verheyen ("Male song as a cue for mate choice in the European starling" Behaviour 116:210-238) as giving evidence that female birds of some species choose sexual partners based on size of song repertoire.
My theory of music is also a theory about a super-stimulus, but it is a different kind of super-stimulus, and in particular I do not suppose that music is a super-stimulus for itself. Rather I suppose that music is a super-stimulus for the perception of "musicality", where musicality is a perceived attribute of normal speech. Thus the theory relates music to speech, and speech is obviously communication, but the theory denies that music is a form of communication.
So what does this tell us about animal music? Applying the full logic of my theory to non-human animals, the following can be concluded:
A final question is whether the perception of "musicality" is something that non-human animals would want to or be able to do. My best guess as to what musicality represents is that it is a perception of the level of conscious arousal in the speaker. It seems reasonable to suppose that other animals do have some form of consciousness similar to human consciousness. We cannot know whether animals subjectively experience consciousness, but we can make judgements about what sort of problems consciousness is used to solve, and observe that other animals have some degree of success in solving those types of problem.
Whether or not other animals need to be able to perceive each other's consciousness is another issue. An ability to perceive the current state of consciousness in other individuals would seem most relevant to those animals that were both intelligent and had complex social lives. Candidates for such a requirement would include (non-human) primates, especially the great apes, and perhaps other intelligent social mammals such as elephants, dolphins and whales.