Skip to main content

Sturmian Canons

  • Conference paper
Mathematics and Computation in Music (MCM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7937))

Included in the following conference series:


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

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adler, C.: Signals Intelligence, for percussion solo or ensemble. Liber Pulveris Press (2002)

    Google Scholar 

  2. Amiot, E.: Auto Similar Melodies. Journal of Mathematics and Music 3(1), 1–26 (2009)

    Article  MathSciNet  Google Scholar 

  3. Amiot, E., Rahn, J.: Tiling Rhythmic Canons (Special Issue). Perspectives of New Music 49, 2 (2011)

    Google Scholar 

  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. 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),

  6. Canright, D.: Fibonacci Gamelan Rhythms. 1/1: The Journal of the Just Intonation Network 6(4) (1990)

    Google Scholar 

  7. Carey, N., Clampitt, D.: Self-Similar Pitch Structures, Their Duals, and Rhytymic Analogues. Perspectives of New Music 34, 62–87 (1996)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  9. Clough, J., Douthett, J.: Maximally Even Sets. Journal of Music Theory 35, 93–173 (1991)

    Article  Google Scholar 

  10. Hall, R., Klinsberg, P.: Asymmetric Rhythms and Tiling Canons. The Mathematical Association of America 113, 887–896 (2006)

    MATH  Google Scholar 

  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. Shallit, J.: Mathematics of Per Nøgård’s Rhythmic Infinity System. The Fibonacci Quarterly 27(2), 262–268 (2005)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints 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.

Download citation

  • DOI:

  • 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)

Publish with us

Policies and ethics