Abstract
Musical chords and chord relations can be described through mathematics. Abstract permutations can be visualized through the Rubik’s cube, born as a pedagogical device [7, 21]. Permutations of notes can also be heard through the CubeHarmonic, a novel musical instrument. Here, we summarize the basic ideas and the state of the art of the physical implementation of CubeHarmonic, discussing its conceptual lifting up to the fourth dimension, with the HyperCubeHarmonic (HCH). We present the basics of the hypercube theory and of the 4-dimensional Rubik’s cube, investigating its potential for musical applications. To gain intuition about HCH complexity, we present two practical implementations of HCH based on the three-dimensional development of the hypercube. The first requires a laptop and no other devices; the second involves a physical Rubik’s cube enhanced through augmented and virtual reality and a specifically-designed mobile app. HCH, as an augmented musical instrument, opens new scenarios for STEAM teaching and performing, allowing us to hear the “sound of multiple dimensions.”
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
In music theory, slot-machine transformations are permutations. If we have three discs with three notes in each of them, they give a sequence of 3-note chords. Rotating the discs, the chord changes. For example, the vertical sequence \(0 - 1 - 2\) becomes \(1 - 2 - 0\) after a rotation of one of the discs [1].
- 3.
Video: https://tinyurl.com/3j5csh36.
- 4.
- 5.
It had been called ars magna by Ramon Llull [4].
- 6.
- 7.
- 8.
- 9.
- 10.
- 11.
- 12.
Night Forest: https://www.youtube.com/watch?v=1OClm1pn7-g&t=1s, Blues and Grays: https://www.youtube.com/watch?v=oMldYLbfIRE, Female Laptop Orchestra (FLO).
References
Alegant, B., Mead, A.: Having the last word: Schoenberg and the ultimate cadenza. Music Theor. Spectr. 34(2), 107–136 (2012)
Amiot, E., Baroin, G.: Old and new isometries between pc sets in the planet-4D model. Music Theor. Online 21(3) (2015)
Andreotti, E., Frans, R.: The connection between physics, engineering and music as an example of STEAM education. Phys. Educ. 54, 045016 (2019)
Ars combinatoria. Treccani Enciclopedia (in Italian). https://www.treccani.it/enciclopedia/ars-combinatoria/
Baroin, G.: The planet-4D model: an original hypersymmetric music space based on graph theory. In: Agon, C., Andreatta, M., Assayag, G., Amiot, E., Bresson, J., Mandereau, J. (eds.) Mathematics and Computation in Music, MCM 2011. LNCS (LNAI), vol. 6726, pp. 326–329. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21590-2_25
D’amore, B.: Salvador Dalì e la geometria a quattro dimensioni (2015). http://www.saperescienza.it/biologia/salvador-dali-e-la-geometria-a-quattro-dimensioni-18-2-15/
Hofstadter, D.: Metamagical Themas, March 1981. The Magic Cube’s cubies are twiddled by cubists and solved by cubemeisters. Scientific American (1981). https://www.scientificamerican.com/article/metamagical-themas-1981-03/
Huang, J., Sugawara, R., Chu, K., Komura, T., Kitamura, Y.: Reconstruction of dexterous 3D motion data from a flexible magnetic sensor with deep learning and structure-aware filtering. IEEE Trans. Vis. Comput. Graph. (2020). https://doi.org/10.1109/TVCG.2020.3031632
Mannone, M.: Have fun with math and music! In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds.) Mathematics and Computation in Music, MCM 2019. LNCS (LNAI), vol. 11502, pp. 379–382. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-21392-3_33
Mannone, M., Kitamura, E., Huang, J., Sugawara, R., Kitamura, Y.: Musical combinatorics, tonnetz, and the cubeharmonic. Collect. Pap. Acad. Arts Novi Sad. 6, 137–153 (2018). Zbornik Radova Akademije Umetnosti
Mannone, M., Kitamura, E., Huang, J., Sugawara, R., Chiu, P., Kitamura, Y.: CubeHarmonic: a new musical instrument based on Rubik’s cube with embedded motion sensor. ACM SIGGRAPH Posters, Article no. 53 (2019). https://doi.org/10.1145/3306214.3338572
Mazzola, G., Mannone, M., Pang, Y.: Cool Math for Hot Music. CMS. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-42937-3
Miyazaki, K., Ishii, M., Yamaguchi, S.: Science of Higher-Dimensional Shape and Symmetry. Kyoto University Press, Kyoto (2005). (in Japanese)
Montiel, M., Gómez, F.: Music in the pedagogy of mathematics. J. Math. Music 8(2), 151–166 (2014)
Montiel, M., Gómez, F. (eds.): Theoretical and Practical Pedagogy of Mathematical Music Theory: Music for Mathematics and Mathematics for Music. World Scientific, From School to Postgraduate Levels, Singapore (2018)
Polfreman, R., Oliver, B.: Rubik’s cube, music’s cube. In: NIME Conference Proceedings (2017)
Riepel, J.: Grundregeln zur Tonordnung Insgemein. Nabu Press, Charleston (1755, 2014)
Rotman, J.: An Introduction to the Theory of Groups. Springer, New York (1991). https://doi.org/10.1007/978-1-4612-4176-8
Staff, M.: How to play a Rubik’s cube like a piano (2015). https://ed.ted.com/lessons/group-theory-101-how-to-play-a-rubik-s-cube-like-a-piano-michael-staff
Tymoczko, D.: The geometry of musical chords. Science 313, 72–74 (2006)
Undark, H.P.: A brief history of the Rubik’s cube. Smithsonian Mag. (2020) https://www.smithsonianmag.com/innovation/brief-history-rubiks-cube-180975911/
Webb, R.: Image and stella software. http://www.software3d.com/Stella.php
Weisstein, E.W.: Hypercube. From MathWorld-A Wolfram Web Resource (2014). https://mathworld.wolfram.com/Hypercube.html
Williams, R.: The Geometric Foundation of Natural Structure. Dover, New York (1979)
Yoshino, T.: Rubik’s 4-cube. Math. J. (2017). https://www.mathematica-journal.com/2017/12/31/rubiks-4-cube/
Zassenhaus, H.: Rubik’s cube: a toy, a galois tool, group theory for everybody. Phys. A 114, 629–637 (1982)
Zweig, J.: Ars Combinatoria. Art J. 56(3), 20–29 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Mannone, M., Yoshino, T., Chiu, P., Kitamura, Y. (2022). Hypercube + Rubik’s Cube + Music = HyperCubeHarmonic. 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_20
Download citation
DOI: https://doi.org/10.1007/978-3-031-07015-0_20
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)