Theories of music origins and the role of musical emotions in the mind are reviewed. Most existing theories contradict each other, and cannot explain mechanisms or roles of musical emotions in workings of the mind, nor evolutionary reasons for music origins. Music seems to be an enigma. Nevertheless, a synthesis of cognitive science and mathematical models of the mind has been proposed describing a fundamental role of music in the functioning and evolution of the mind, consciousness, and cultures. The review considers ancient theories of music as well as contemporary theories advanced by leading authors in this field. It addresses one hypothesis that promises to unify the field and proposes a theory of musical origin based on a fundamental role of music in cognition and evolution of consciousness and culture. We consider a split in the vocalizations of proto-humans into two types: one less emotional and more concretely-semantic, evolving into language, and the other preserving emotional connections along with semantic ambiguity, evolving into music. The proposed hypothesis departs from other theories in considering specific mechanisms of the mind-brain, which required the evolution of music parallel with the evolution of cultures and languages. Arguments are reviewed that the evolution of language toward becoming the semantically powerful tool of today required emancipation from emotional encumbrances. The opposite, no less powerful mechanisms required a compensatory evolution of music toward more differentiated and refined emotionality. The need for refined music in the process of cultural evolution is grounded in fundamental mechanisms of the mind. This is why today's human mind and cultures cannot exist without today's music. The reviewed hypothesis gives a basis for future analysis of why different evolutionary paths of languages were paralleled by different evolutionary paths of music. Approaches toward experimental verification of this hypothesis in psychological and neuroimaging research are reviewed.