Counting words satisfying the rhythmic oddity property
Franck Jedrzejewski
TL;DR
This paper provides a method to enumerate and count words with the rhythmic oddity property, revealing their relationship with necklaces and offering a formula for their enumeration.
Contribution
It introduces a bijection between necklaces and rop-words, enabling precise counting of these words based on their length.
Findings
Established a bijection between necklaces and rop-words
Derived a counting formula for rop-words of a given length
Connected the rhythmic oddity property to combinatoric structures in words
Abstract
This paper describes an enumeration of all words having a combinatoric property called "rhythmic oddity property"named \emph{rop-words}.\ This property was introduced by Simha Aron in the 1990s. The set of rop-words is not a subset of the set of Lyndon words, but is very closed. We show that there is a bijection between some necklaces and rop-words. This leads to a formula for counting the rop-words of a given length. \textsc{Keywords:} Combinatoric on words. Lyndon words. Rhythmic oddity. Music formalization
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Taxonomy
TopicsMusicology and Musical Analysis · Music and Audio Processing · Music Technology and Sound Studies