Skip to main content

The Structure of Z-Related Sets

  • 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:

Abstract

The paper presents some new results on Z-related sets obtained by computational methods. We give a complete enumeration of all Z-related sets in ℤ N for small N. Furthermore, we establish that there is a reasonable permutation group action representing the Z-relation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Forte, A.: The Structure of Atonal Music, 2nd edn. Yale University Press (1977)

    Google Scholar 

  2. Patterson, A.L.: Ambiguities in the X-ray analysis of crystal structures. Physical Review 65(5-6), 195–201 (1944)

    Article  Google Scholar 

  3. Bullough, R.K.: On homometric sets I: Some general theorems. Acta Crystallographica 14, 257–268 (1961)

    Article  MathSciNet  Google Scholar 

  4. Rosenblatt, J.: Phase retrieval. Communications in Mathematical Physics 95, 317–343 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Soderberg, S.: Z-related sets as dual inversions. Journal of Music Theory 39(1), 77–100 (1995)

    Article  Google Scholar 

  6. Goyette, J.S.: The Z-Relation in Theory and Practice. PhD thesis, University of Rochester, NY (2012)

    Google Scholar 

  7. Mandereau, J., Ghisi, D., Amiot, E., Andreatta, M., Agon, C.: Z-relation and homometry in musical distributions. Journal of Mathematics and Music 5(2), 83–98 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mandereau, J., Ghisi, D., Amiot, E., Andreatta, M., Agon, C.: Discrete phase retrieval in musical structures. Journal of Mathematics and Music 5(2), 99–116 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press (1987)

    Google Scholar 

  10. O’Rourke, J., Taslakian, P., Toussaint, G.: A pumping lemma for homometric rhythms. In: Proceedings of the 20th Canadian Conference on Computational Geometry, pp. 121–123 (2008)

    Google Scholar 

  11. Althuis, T.A., Göbel, F.: Z-related pairs in microtonal systems. Memorandum 1524, University of Twente, The Netherlands (2000)

    Google Scholar 

  12. Callender, C., Hall, R.: Crystallography and the structure of Z-related sets, Paper given at the annual meeting of the Society for Music Theory in Nashville, TN. Handout accessed (2008), http://myweb.fsu.edu/ccallender/z-relationhandout.pdf.

  13. Buerger, M.J.: Exploration of cyclotomic point sets for tautoeikonic complementary pairs. Zeitschrift für Kristallographie 145, 377–411 (1977)

    Google Scholar 

  14. Chieh, C.: Analysis of cyclotomic sets. Zeitschrift für Kristallographie 150, 261–277 (1979)

    Article  MathSciNet  Google Scholar 

  15. Lewin, D.: On extended Z-triples. Theory and Practice 7, 38–39 (1981)

    Google Scholar 

  16. Wild, J.: Enumerating set-classes and Z-related tuplets in equal temperaments of up to thirty-one notes per octave. Unpublished Graduate Seminar Paper. McGill University (1996)

    Google Scholar 

  17. Johnson, T., Jedrzejewski, F.: Looking at Numbers. Birkhauser (forthcoming)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jedrzejewski, F., Johnson, T. (2013). The Structure of Z-Related Sets. 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_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39357-0_10

  • 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