Formal structures of a harmony in the parabola. (English) Zbl 1495.00017

Delgado Vega, Edgar Armando

Montiel, Mariana (ed.) et al., Mathematics and computation in music. 8th international conference, MCM 2022, Atlanta, GA, USA, June 21–24, 2022. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 13267, 356-362 (2022).

Summary: We develop a geometric analog of musical harmony from the group law of the affine parabola. First, we associate musical notes and intervals with points of a parabola. Immediately, we can define the usual affine and linear transformations for musical chords in module theory. Subsequently, we show that the actions of the groups \(T/I\) in PK-nets, PLR, UTTs, and JQZ behave identically to the circle space. Then, we propose to recreate the Planet-4D model, the study of musical distance and the DFT for subsets of points on the parabola. We believe that we have an innovative and motivational perspective to approach the parabola in a musical meaning.
For the entire collection see [Zbl 1492.00048].

MSC:

00A65

Mathematics and music

Keywords:

parabola; group law; pitch-class set theory; affine transformations; neo-Riemannian theory; music Fourier space

References:

[1]	Amiot, E., Music Through Fourier Space: Discrete Fourier Transform in Music Theory (2016), Cham: Springer, Cham · Zbl 1356.00006
[2]	Amiot, E., Baroin, G.: Old and new isometries between pc-sets in the Planet-4D model. Music Theor. Online, 21(3), 2 (2015)
[3]	Arias-Valero, J.S., Agustín-Aquino, O.A., Lluis-Puebla, E.: On first-species counterpoint theory (2020). arXiv:2004.07983
[4]	Baroin, G.; Agon, C.; Andreatta, M.; Assayag, G.; Amiot, E.; Bresson, J.; Mandereau, J., The Planet-4D model: an original hypersymmetric music space based on graph theory, Mathematics and Computation in Music, 326-329 (2011), Heidelberg: Springer, Heidelberg · Zbl 1321.00079
[5]	Bigo, L.; Ghisi, D.; Spicher, A.; Andreatta, M., Representation of musical structures and processes in simplicial chord spaces, Comput. Music J., 39, 3, 9-24 (2015)
[6]	Cohn, R., Neo-Riemannian operations, parsimonious trichords, and their “Tonnetz” representations., J. Music Theor., 41, 1, 1-66 (1997)
[7]	Crans, AS; Fiore, TM; Satyendra, R., Musical actions of dihedral groups, Am. Math. Mon., 116, 6, 479-495 (2009) · Zbl 1229.00013
[8]	Douthett, J.; Steinbach, P., Parsimonious graphs: a study in parsimony, contextual transformations, and modes of limited transposition, J. Music Theor., 42, 2, 241-263 (1998)
[9]	Fiore, TM; Noll, T., Voicing transformations of triads, SIAM J. Appl. Algebra Geom., 2, 2, 281-313 (2018) · Zbl 1391.00035
[10]	Forte, A., The Structure of Atonal Music (1973), New Haven: Yale University Press, New Haven
[11]	Genuys, G., Pseudo-distances between chords of different cardinality on generalized voice-leading spaces, J. Math. Music, 13, 3, 193-206 (2019) · Zbl 1429.00015
[12]	Gollin, E., Some aspects of three-dimensional “Tonnetze”, J. Music Theor., 42, 2, 195-206 (1998)
[13]	Hook, J., Uniform triadic transformations, J. Music Theor., 46, 1-2, 57-126 (2002)
[14]	Jedrzejewski, F.; Montiel, M.; Gomez-Martin, F.; Agustín-Aquino, OA, Non-contextual JQZ transformations, Mathematics and Computation in Music, 149-160 (2019), Cham: Springer, Cham · Zbl 1456.00076
[15]	Lemmermeyer, F.: Pell Conics, An Alternative Approach to Elementary Number Theory (2012). https://www.mathi.uni-heidelberg.de/ flemmermeyer/pell/bfc02.pdf. Accessed 11 Jan 2022
[16]	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. In: ACM SIGGRAPH 2019 Posters. SIGGRAPH 2019, Association for Computing Machinery, New York (2019)
[17]	Popoff, A.; Andreatta, M.; Ehresmann, A.; Montiel, M.; Gomez-Martin, F.; Agustín-Aquino, OA, Groupoids and wreath products of musical transformations: a categorical approach from poly-klumpenhouwer networks, Mathematics and Computation in Music, 33-45 (2019), Cham: Springer, Cham · Zbl 1456.00095
[18]	Shirali, SA, Groups associated with conics, Math. Gaz., 93, 526, 27-41 (2009)
[19]	Tymoczko, D.; Chew, E.; Childs, A.; Chuan, C-H, Three conceptions of musical distance, Math. Comput. Music, 258-272 (2009), Heidelberg: Springer, Heidelberg · Zbl 1247.00029
[20]	Tymoczko, D.; Yust, J.; Montiel, M.; Gomez-Martin, F.; Agustín-Aquino, OA, Fourier phase and pitch-class sum, Mathematics and Computation in Music, 46-58 (2019), Cham: Springer, Cham · Zbl 1456.00099
[21]	Yust, J.; Collins, T.; Meredith, D.; Volk, A., Applications of DFT to the theory of twentieth-century harmony, Mathematics and Computation in Music, 207-218 (2015), Cham: Springer, Cham · Zbl 1320.00036
[22]	Yust, J., Generalized Tonnetze and Zeitnetze, and the topology of music concepts, J. Math. Music, 14, 2, 170-203 (2020) · Zbl 1466.00023