Modulus p Vuza canons: generalities and resolution of the case \lbrace 0,1,2^k\rbrace with p =2

2015/05/26 by Helianthe Caure, Caure, Helianthe
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1505.06930

arxiv created 2015/08/17 · arxiv updated 2015/08/18

Abstract

Non-periodic tilings of ℤ_N are a difficult to obtain key to a wide range of complex mathematical issues. In a musician's eyes, they are called rhythmic Vuza canons. After a short summary of the main properties of rhythmic tiling canons and after stressing the importance of Vuza canons, this article presents a new approach: modulo p Vuza canons. Working modulo p (especially with p=2) allows Vuza canons to be computed very quickly. But going back to traditional Vuza canons requires understanding of when and how the tiling is a covering. The article's main result is a complete case construction of a modulo 2 tiling. This first solution of a modulo 2 tiling is proved in a constructive way, which, it is hoped, is expandable to every modulo p tiling. Following this path may lead to a solution to the problem of building modulo p tilings and thus a solution to the problem of easily finding traditional Vuza canons.

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