Notes and note pairs in Nørgård's infinity series. Issue 1 (2nd January 2017) - Hin (Gary) Au, Yu; Drexler-Lemire, Christopher; Shallit, Jeffrey
Abstract : The Danish composer Per Nørgård (born July 13, 1932) defined the "infinity series"s = ( s ( n ) ) n ≥ 0 by the ruless ( 0 ) = 0, s ( 2 n ) = − s ( n ) forn ≥ 1, ands ( 2 n + 1 ) = s ( n ) + 1 forn ≥ 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.