It is very reasonable assume that since all known human cultures, no matter how remote, embrace music as a central cultural component, that music existed before modern humans existed. It is something bred into us. This would probably be a better question for someone that studies the prehistory or evolution of mankind.

See The Origins of Music, editors Wallen, Merker, and Brown (2010: MIT Press), which happens to be on goggle books.

FYI, this question on early humans would be worth x-posting to /r/AskAnthropology

"Music' as in any kind of organized sound exists from time immemorial. However, the music that we as Western folks would appreciate as such is reliant on some general principles of diatonicism and modality, which later evolved into tonality. The very earliest precursors of this may be dated as early as the Roman imperial era, with e.g. Seikilos' Epitaph, or the preserved songs of Mesomedes (a courtier to Hadrian). Even earlier Greek music was in fact quite different; in fact much of it was clearly non-diatonic. The available evidence suggests that this modal/tonal foundation was then developed further in the secular music of the Middle Ages (of which comparatively little was preserved - whereas religious music was then based on plainchant and developed along largely separate lines) and then became finally established in the Renaissance period, which is when it firmly enters the written tradition of classical music.