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.

Keywords

Sturmian word canon self-similarity 

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References

  1. 1.
    Adler, C.: Signals Intelligence, for percussion solo or ensemble. Liber Pulveris Press (2002)Google Scholar
  2. 2.
    Amiot, E.: Auto Similar Melodies. Journal of Mathematics and Music 3(1), 1–26 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Amiot, E., Rahn, J.: Tiling Rhythmic Canons (Special Issue). Perspectives of New Music 49, 2 (2011)Google Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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
  6. 6.
    Canright, D.: Fibonacci Gamelan Rhythms. 1/1: The Journal of the Just Intonation Network 6(4) (1990)Google Scholar
  7. 7.
    Carey, N., Clampitt, D.: Self-Similar Pitch Structures, Their Duals, and Rhytymic Analogues. Perspectives of New Music 34, 62–87 (1996)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Clough, J., Douthett, J.: Maximally Even Sets. Journal of Music Theory 35, 93–173 (1991)CrossRefGoogle Scholar
  10. 10.
    Hall, R., Klinsberg, P.: Asymmetric Rhythms and Tiling Canons. The Mathematical Association of America 113, 887–896 (2006)MATHGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    Shallit, J.: Mathematics of Per Nøgård’s Rhythmic Infinity System. The Fibonacci Quarterly 27(2), 262–268 (2005)MathSciNetGoogle Scholar
  13. 13.
    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)MATHCrossRefGoogle Scholar
  14. 14.
    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) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Clifton Callender
    • 1
  1. 1.Florida State UniversityUSA

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