The intellectual capacities of infants learning language have always amazed those studying it. The ingenuity of researchers is on display as they figure out how to interrogate a non-verbal infant. In this case it’s figuring just what to study.
One feature of language is the use of the same sound to mean different things, particularly emotional things. Think of how many emotional states there are in cool — neutral as a description, positive, negative as an emotional descriptor. I’m sure you can think of better examples. The fancy word for this is functional flexibility — e.g. when a single vocal category expresses a variety of emotional states on different occasions.
So when do infants develop this? Surprisingly early according to [ Proc. Natl. Acad. Sci. vol. 110 pp. 6318 – 3623 ’13 ] The authors measured functional flexibility by looking at the infants faces as they made different sounds. It begins very early (3 – 4 months). Functional flexibility is found with squeals, vowel-like sounds and growls (but not by laughing or crying which are always taken to the same emotional state — I’m not sure how you’d take them to mean something else.). It appears far earlier than syntax, or word learning.
Prosody is the melody of speech. When we talk the rhythm of our words changes along with their pitch. It’s why I love to hear people from the (Caribbean) islands talk. Their speech is incredibly musical. It is invariably lost after severe head injury, and one good prognostic sign is when it starts to come back.
I don’t know if anyone has studied when infant babbling acquires prosody, but it’s present long before speech.
Chemiotics II 
Comments
We report on the emergence of functional flexibility in vocalizations of human infants. This vastly underappreciated capability becomes apparent when prelinguistic vocalizations express a full range of emotional content—positive, neutral, and negative. The data show that at least three types of infant vocalizations (squeals, vowel-like sounds, and growls) occur with this full range of expression by 3–4 mo of age. In contrast, infant cry and laughter, which are species-specific signals apparently homologous to vocal calls in other primates, show functional stability, with cry overwhelmingly expressing negative and laughter positive emotional states. Functional flexibility is a sine qua non in spoken language, because all words or sentences can be produced as expressions of varying emotional states and because learning conventional “meanings” requires the ability to produce sounds that are free of any predetermined function. Functional flexibility is a defining characteristic of language, and empirically it appears before syntax, word learning, and even earlier-developing features presumed to be critical to language (e.g., joint attention, syllable imitation, and canonical babbling). The appearance of functional flexibility early in the first year of human life is a critical step in the development of vocal language and may have been a critical step in the evolution of human language, preceding protosyntax and even primitive single words. Such flexible affect expression of vocalizations has not yet been reported for any nonhuman primate but if found to occur would suggest deep roots for functional flexibility of vocalization in our primate heritage.