Abstract
Sturmian words are balanced, almost periodic, self-similar and hierarchical infinite sequences that have been studied in music theory in connection with diatonic scale theory and related subjects. Carey and Camplitt (1996) give a brief but suggestive rhythmic example in which these properties are made manifest in a particularly visceral manner. The present paper expands upon this example, considering the properties of canons based on Sturmian words, or Sturmian canons. In particular, a Sturmian word of irrational slope a with a hierarchical periodicity of p gives rise to p-tuple canons, the voices and relations of which are determined by the terms of the continued fraction expansion of a.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adler, C.: Signals Intelligence, for percussion solo or ensemble. Liber Pulveris Press (2002)
Amiot, E.: Auto Similar Melodies. Journal of Mathematics and Music 3(1), 1–26 (2009)
Amiot, E., Rahn, J.: Tiling Rhythmic Canons (Special Issue). Perspectives of New Music 49, 2 (2011)
Berthé, V., Labbé, S.: Uniformly balanced words with linear complexity and prescribed letter frequencies. In: Electronic Proceedings in Theoretical Computer Science, pp. 44–52 (2011)
Callender, C.: Performing the Irrational: Paul Ushers Arrangement of Nancarrows Study no. 33, Canon \(2:\sqrt{2}\). Online Symposium: Conlon Nancarrow, Life and Music (2012), http://conlonnancarrow.org/symposium/papers/callender/irrational.html
Canright, D.: Fibonacci Gamelan Rhythms. 1/1: The Journal of the Just Intonation Network 6(4) (1990)
Carey, N., Clampitt, D.: Self-Similar Pitch Structures, Their Duals, and Rhytymic Analogues. Perspectives of New Music 34, 62–87 (1996)
Clampitt, D., Domínguez, M., Noll, T.: Plain and Twisted Adjoints of Well-Formed Words. In: Chew, E., Childs, A., Chuan, C.-H. (eds.) MCM 2009. CCIS, vol. 38, pp. 65–80. Springer, Heidelberg (2009)
Clough, J., Douthett, J.: Maximally Even Sets. Journal of Music Theory 35, 93–173 (1991)
Hall, R., Klinsberg, P.: Asymmetric Rhythms and Tiling Canons. The Mathematical Association of America 113, 887–896 (2006)
Noll, T.: Sturmian Sequences and Morphisms: A Music-Theoretical Application. In: Yves André: Mathématique et Musique. Journée Annuelle de la Société Mathématique de France à lInstitut Henri Poincaré (2008)
Shallit, J.: Mathematics of Per Nøgård’s Rhythmic Infinity System. The Fibonacci Quarterly 27(2), 262–268 (2005)
Uscka-Wehlou, H.: Run-hierarchical structure of digital lines with irrational slopes in terms of continued fraction and the Gauss map. Pattern Recognition 42, 2247–2254 (2009)
Vuza, D.: Supplementary Sets and Regular Complementary Unending Canons. Perspectives of New Music 29(2, pt. 1), 22–49, 30(1, pt. 2), 184–207, 30(2, pt. 3), 102–125, 31(1, pt. 4), 207–305 (1991–1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Callender, C. (2013). Sturmian Canons. In: Yust, J., Wild, J., Burgoyne, J.A. (eds) Mathematics and Computation in Music. MCM 2013. Lecture Notes in Computer Science(), vol 7937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39357-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-39357-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39356-3
Online ISBN: 978-3-642-39357-0
eBook Packages: Computer ScienceComputer Science (R0)