Notes and Note-Pairs in Noergaard's Infinity Series

The Danish composer Per Nørg̊ard defined the “infinity series” s = (s(n))n≥0 by the rules s(0) = 0, s(2n) = −s(n) for n ≥ 1, and s(2n + 1) = s(n) + 1 for n ≥ 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, that consecutive occurrences of identical phrases are widely separated. We also consider to what extent the infinity series is unique.