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Thesis Info
- LABS ID
- 00553
- Thesis Title
- Rhythmic Canons and Modular Tiling
- Author
- Hélianthe Caure
- E-mail
- helianthe.caure AT ircam.fr
- 2nd Author
- 3rd Author
- Degree
- Ph.D.
- Year
- 2016
- Number of Pages
- 158
- University
- Université Pierre et Marie Curie
- Thesis Supervisor
- Moreno Andreatta
- Supervisor e-mail
- Moreno.andreatta AT ircam.fr
- Other Supervisor(s)
- Jean-Paul Allouche
- Language(s) of Thesis
- French
- Department / Discipline
- Mathématics
- Copyright Ownership
- Languages Familiar to Author
- English, French
- URL where full thesis can be found
- tel.archives-ouvertes.fr/tel-01338353
- Keywords
- Vuza canons, modular canons
- Abstract: 200-500 words
- This thesis is a contribution to the study of modular tiling. A rhythmic tiling canon can be defined by two rhythmic patterns. The first is a finite rhythm and the second represents the beats where the first rhythm starts. Those rhythms are such that at every beat of the time, one can hear one and exactly one onset of the first rhythm. Such canon can also be understood as a discrete tiling of the line by translation. Modulo p canons follow the same definition, but instead of one onset per beat, we want to hear one onset modulo p. Hence, modular canons can be defined as a controlled cover of the time, or again as a modular tiling.
Many mathematical and computational tools were used for the study of rhythmic tiling canons for the past 50 years. Recent research has mainly focused in finding tiles without inner periodicity called Vuza canons, which are exponentially long to obtain despite a few recent enhancements. Musically, rhythmic tiling canons have been used a lot by composers (like Lanza, Tangian, Bloch, Levy, Johnson, or very recently Roux). Thanks to finite field theory we can prove that any rhythmic pattern can tile a finite interval modulo 2. Moreover, we present a greedy algorithm to produce such a tile in linear time. Because we show that the algorithm is optimal, we can define the tiles it returns as modular Vuza canons.
We show combinatorial results on the second rhythmic pattern of modulo 2 canons. For some families of tile, we can prove recursive properties that permit a construction of a tile in logarithmic time, and enumerations results, and other number theory theorems.
All the results of the thesis are implemented in OpenMusic, a visual programming language developed at IRCAM.