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.