Notes and Note-Pairs in Noergaard's Infinity Series
Combinatorics
2014-02-14 v1 Discrete Mathematics
Abstract
The Danish composer Per Noergaard 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.
Cite
@article{arxiv.1402.3091,
title = {Notes and Note-Pairs in Noergaard's Infinity Series},
author = {Christopher Drexler-Lemire and Jeffrey Shallit},
journal= {arXiv preprint arXiv:1402.3091},
year = {2014}
}