Summary: The Danish composer Per Nørgård (born July 13, 1932) defined the “infinity series” \(\mathbf s=(s(n))_{n\geq 0}\) by the rules \(s(0)=0\), \(s(2n)=-s(n)\) for \(n \geq 1\) and \(s(2n+1)=s(n)+1\) for \(n\geq 0\); it figures prominently in many of his compositions. Here we give several new results about this sequence: first, the set of binary representations of the positions of each note forms a context-free language that is not regular; second, a complete characterization of exactly which note pairs appear; third, consecutive occurrences of identical phrases are widely separated. We also consider to what extent the infinity series is unique.