Tiling the (Musical) Line with polynomials: some Theoretical and implementational Aspects

This paper aims at discussing the polynomial approach to the problem of tiling the (musical) time axis with translates of one tile. This mathematical construction naturally leads to a new family of rhythmic tiling canons having the property of being generated by cyclotomic polynomials (tiling cyclotomic canons).

[1]  H. E. Slaught.  THE CARUS MATHEMATICAL MONOGRAPHS , 1923 .

[2]  I. Laba.  The spectral set conjecture and multiplicative properties of roots of polynomials , 2000, math/0010169.

[3]  Aaron D. Meyerowitz,et al.  Tiling the Integers with Translates of One Finite Set , 1998 .

[4]  M. Andreatta.  On group-theoretical methods applied to music : some compositional and implementational aspects , 2022 .

[5]  E. Amiot.  Rhythmic canons and Galois theory , 2005 .

[6]  L. Rédei.  Zwei Lückensätze Über Polynome in Endlichen Primkörpern mit Anwendung auf die Endlichen Abelschen Gruppen und die Gaussischen Summen , 1947 .

[7]  Emmanuel Amiot,et al.  Tiling problems in music composition: Theory and Implementation , 2002, ICMC.

[8]  Moreno Andreatta.  Méthodes algébriques dans la musique et la musicologie du XXème siècle : aspects théoriques, analytiques et compositionnels , 2003 .

[9]  Georg Hajós,et al.  Über einfache und mehrfache Bedeckung desn-dimensionalen Raumes mit einem Würfelgitter , 1942 .

[10]  I. Łaba.  The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials , 2002 .

[11]  Richard Cohn.  Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective , 1998 .

[12]  Bent Fuglede,et al.  Commuting self-adjoint partial differential operators and a group theoretic problem , 1974 .

[13]  E. Amiot.  Why rhythmic canons are interesting , 2003 .

[14]  H. Fripertinger.  Tiling problems in music theory , 2003 .

[15]  Camilo Rueda,et al.  Computer-Assisted Composition at IRCAM: From PatchWork to OpenMusic , 1999, Computer Music Journal.

[16]  Dan Tudor Vuza,et al.  Supplementary Sets and Regular Complementary Unending Canons (Part Four) , 1992 .

[17]  H. Minkowski,et al.  Diophantische Approximationen : eine Einführung in die Zahlentheorie , 1907 .

[18]  O. Keller,et al.  Über die lückenlose Erfüllung des Raumes mit Würfeln. , 1930 .