Notes and note pairs in Nørgård's infinity series. Issue 1 (2nd January 2017)

• Record Type: Journal Article
• Authors: Hin (Gary) Au, Yu; Drexler-Lemire, Christopher; Shallit, Jeffrey
• Is Part Of: Journal of mathematics and music. Volume 11:Issue 1(2017)
• Page Start: 1
• Page End: 19
• Publication Date: 2017-01-02
• Subjects: Nørgård's infinity series -- hierarchical music -- note pair -- Pascal's triangle -- context-free language -- binomial coefficient -- running sum -- interval -- repetition -- self-similarity; 00A65 -- 11B85 -- 05A10 -- 05A15 -- 11B37 -- 11B65; Music theory -- Periodicals; Music -- Mathematical models -- Periodicals; Music -- Mathematics -- Periodicals; 781.2
• DOI: 10.1080/17459737.2017.1299807
• Languages: English
• ISSNs: 1745-9737

Abstract

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.