Skip to main content
Springer Nature Link
Log in

Quantum-Musical Explorations on \(\mathbb {Z}_n\)

  • Conference paper
  • First Online:
Mathematics and Computation in Music (MCM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13267))

Included in the following conference series:

  • 869 Accesses

Abstract

Motivated through recent applications of quantum theory to the music-theoretical conceptualisation of tonal attraction, the paper recapitulates basic facts about quantum wave functions over the finite configuration space \(\mathbb {Z}_n\), and proposes a particular musical application.

After an introduction of position and momentum operators, the Fourier transform as well as the translation and ondulation operators, particular attention is plaid to the Quantum Harmonic Oscillator via its Hamilton operator and its eigenstates. In this setup the time development of chosen wave functions is applied to the control of moving sound sources in a Spatialisation scenario.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    https://forum.ircam.fr/projects/detail/spat/.

  2. 2.

    https://www.wolfram.com/mathematica/.

References

  1. Amiot, E.: Music Through Fourier Space. Discrete Fourier Transform in Music Theory, Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45581-5

    Book  Google Scholar 

  2. Blutner, R., beim Graben, P.: Gauge models of musical forces. J. Math. Music 15(1), 17–36 (2021). https://doi.org/10.1080/17459737.2020.1716404

    Article  Google Scholar 

  3. Clampitt, D., Noll, T.: Modes, the height-width duality, and Handschin’s tone character. Music Theor. Online, 17(1) (2011). http://www.mtosmt.org/issues/mto.11.17.1/mto.11.17.1.clampitt_and_noll.html

  4. De La Torre, A.C., Goyeneche, D.: Quantum mechanics in finite-dimensional Hilbert space. Am. J. Phys. 71(1), 49–54 (2003)

    Article  Google Scholar 

  5. Ebeling, M.: Tonhöhe: Physikalisch - Musikalisch - Psychologisch - Mathematisch. Peter Lang, Frankfurt a.M (1999)

    Google Scholar 

  6. Feichtinger, H..G., Hazewinkel, M., Kaiblinger, N., Matusiak, E., Neuhauser, M.: Metaplectic operators on \(\mathbb{C}^n\). Q. J. Math. 59(1), 15–28 (2007). https://doi.org/10.1093/qmath/ham023

    Article  Google Scholar 

  7. beim Graben, P.: Musical pitch quantization as an eigenvalue problem. J. Math. Music 14(3), 329–346 (2020). https://doi.org/10.1080/17459737.2020.1763488

    Article  Google Scholar 

  8. Hall, Brian C..: Quantum Theory for Mathematicians. GTM, vol. 267. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-7116-5

    Book  Google Scholar 

  9. Beim Graben, P., Blutner, R.: Quantum approaches to music cognition. Quantum Theory Mathematicians 91, 38–50 (2019)

    Google Scholar 

  10. Vourdas, A.: Quantum systems with finite Hilbert space. Rep. Prog. Phys. 67(3), 267–320 (2004). https://doi.org/10.1088/0034-4885/67/3/r03

    Article  Google Scholar 

  11. Vourdas, A., Banderier, C.: Symplectic transformations and quantum tomography in finite quantum systems. J. Phys. Math. Theor. 43(4), 042001 (2010). https://doi.org/10.1088/1751-8113/43/4/042001

    Article  Google Scholar 

  12. Yust, Jason: Applications of DFT to the theory of twentieth-century harmony. In: Collins, Tom, Meredith, David, Volk, Anja (eds.) MCM 2015. LNCS (LNAI), vol. 9110, pp. 207–218. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20603-5_22

    Chapter  Google Scholar 

  13. Yust, J.: Schubert’s harmonic language and Fourier phase space. J. Music Theory 59(1), 121–181 (2015). https://doi.org/10.1215/00222909-2863409

    Article  Google Scholar 

  14. Yust, J.: Harmonic qualities in Debussy’s Les sons et les parfums tournent dans l’air du soir. J. Math. Music 11(2–3), 155–173 (2017). https://doi.org/10.1080/17459737.2018.1450457

    Article  Google Scholar 

  15. Yust, Jason: Probing questions about keys: tonal distributions through the DFT. In: Agustín-Aquino, Octavio A.., Lluis-Puebla, Emilio, Montiel, Mariana (eds.) MCM 2017. LNCS (LNAI), vol. 10527, pp. 167–179. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71827-9_13

    Chapter  Google Scholar 

  16. Yust, J.: Geometric Generalizations of the Tonnetz and Their Relation to Fourier Phases Spaces, Chapter 13, pp. 253–277. World Scientific (2018). https://doi.org/10.1142/9789813235311_0013. https://www.worldscientific.com

Download references

Author information

Authors and Affiliations

  1. Departament de Creació i Teoria Musical, Escola Superior de Música de Catalunya, Barcelona, Spain

    Thomas Noll

  2. Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany

    Peter Beim Graben

Authors
  1. Thomas Noll
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Beim Graben
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Noll .

Editor information

Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Mariana Montiel

  2. Technological University of the Mixteca, Huajuapan de León, Mexico

    Octavio A. Agustín-Aquino

  3. Universidad Politécnica de Madrid, Madrid, Spain

    Francisco Gómez

  4. Life University, Marietta, GA, USA

    Jeremy Kastine

  5. National Autonomous University of Mexico, Mexico City, Distrito Federal, Mexico

    Emilio Lluis-Puebla

  6. Georgia State University, Atlanta, GA, USA

    Brent Milam

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Noll, T., Graben, P.B. (2022). Quantum-Musical Explorations on \(\mathbb {Z}_n\). In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-07015-0_32

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-07014-3

  • Online ISBN: 978-3-031-07015-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics