Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 39, 21 June 2024
Open
Access | Article
Dancing with math: Using Klein’s quartic for music generation
* Author to whom correspondence should be addressed.
Abstract
Studying music from a mathematical perspective is often based on the notions of symmetry and topological space. A group G acting on some set S of musical objects in a meaningful way is seen as a symmetry group in music. This can be used in analysing motif development and chord progressions. In neo-Riemannian analysis, one has three principal chord transformations that generate a group G≅D_12, acting on the set S of 24 major and minor triads. Moreover, this theory is visualized through a simplicial complex whose underlying space is a topological torus. In this paper, we first introduce various symmetry groups and graphic presentations in music theory. We then propose a way of doing neo-Riemannian analysis on Klein’s quartic, which is a genus 3 surface instead of a torus, realizing an idea of John Baez. Finally, we use our theory to perform a harmonic analysis of The Imperial March from “Star Wars”.
Keywords
Mathematical music theory, symmetry group, topology
References
1. M.A. Roig-Francolí. Harmony in Context. McGraw-Hill Education, 2019.
2. John Baez. This week’s finds in mathematical physics. https://math.ucr.edu/home/baez/TWF. html.
3. E.M. Stein and R. Shakarchi. Fourier Analysis: An Introduction. Princeton University Press, 2003.
4. Dave Benson. Music: A Mathematical Offering. Cambridge University Press, 2006.
5. Thomas M Fiore. Music and mathematics. University of Michigan, 2004.
6. W. Berry. Structural Functions in Music. Prentice-Hall, 1976.
7. Tony Phillips. Bach and the musical mobius strip. Plus Magazine, 2016.
8. J.J. Rotman. An Introduction to the Theory of Groups. Graduate Texts in Mathematics. Springer New York, 2012.
9. S. Levy. The Eightfold Way: The Beauty of Klein’s Quartic Curve. Mathematical Sciences Research Institute Publications. Cambridge University Press, 2001.
10. H.S.M. Coxeter and W.O.J. Moser. Generators and Relations for Discrete Groups. Ergebnisse der Mathe matik und ihrer Grenzgebiete. 2. Folge. Springer Berlin Heidelberg, 2013.
11. G. James and M. Liebeck. Representations and Characters of Groups. Cambridge University Press, 2001.
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. Authors who publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open Access Instruction).
- Volume Title
- Proceedings of the 2nd International Conference on Mathematical Physics and Computational Simulation
- ISBN (Print)
- 978-1-83558-463-7
- ISBN (Online)
- 978-1-83558-464-4
- Published Date
- 21 June 2024
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/39/20240565
- Copyright
- 21 June 2024
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited