Although one might argue that little wisdom can be conveyed in messages of 140 characters or less, this paper sets out to explore whether the aggregation of messages in social awareness streams, such as Twitter, conveys meaningful information about a given domain. As a research community, we know little about the structural and semantic properties of such streams, and how they can be analyzed, characterized and used. This paper introduces a network-theoretic model of social awareness stream, a so-called "tweetonomy", together with a set of stream-based measures that allow researchers to systematically define and compare different stream aggregations. We apply the model and measures to a dataset acquired from Twitter to study emerging semantics in selected streams. The network-theoretic model and the corresponding measures introduced in this paper are relevant for researchers interested in information retrieval and ontology learning from social awareness streams. Our empirical findings demonstrate that different social awareness stream aggregations exhibit interesting differences, making them amenable for different applications.