Abstract
Drawing inspiration from both the classical Guerino Mazzola’s symmetry-based model for first-species counterpoint (one note against one note) and Johann Joseph Fux’s Gradus ad Parnassum, we propose an extension for second-species (two notes against one note).
This work was partially supported by a grant from the Niels Hendrik Abel Board.
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Notes
- 1.
The discantus can be understood in the sweeping (\(x+y\)) or the hanging (\(x-y\)) orientations, but we will only use the sweeping orientation from this point on.
- 2.
The original inspiration for using dual numbers in counterpoint was the Zariski tangent space, thus the definition of the tangent space of a morphism of schemes can be seen as a cue to use this kind of algebraic structure for second-species. See [7] for details.
- 3.
For the converse swap the standard rules of counterpoint suffice: we can arbitrarily define the third component of the 2-interval. This is coherent with the local application of counterpoint rules in Fux’s theory, and also with the particular idea of projection that stems from the fact that, in order to analyze a fragment, we “disregard” notes on the upbeat [3, pp. 41–43].
- 4.
The first example is the student’s attempt to write a second-species discantus by himself, but he makes two mistakes near the end of the exercise, namely the steps from the sequence \(7+\epsilon _{1}.7+\epsilon _{2}.4\), \(5+\epsilon _{1}.7+\epsilon _{2}.4\), \(4+\epsilon _{1}.7+\epsilon _{2}.9\). They are also forbidden steps in the projection model!.
- 5.
References
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Agustín-Aquino, O.A., Junod, J., Mazzola, G.: Computational Counterpoint Worlds. Springer, Heidelberg (2015)
Mann, A.: The Study of Counterpoint. W. W. Norton & Company (1965)
Mazzola, G.: The Topos of Music, vol. I, 2nd edn. Springer, Heidelberg (2017)
Nieto, A.: Una aplicación del teorema de contrapunto. B.Sc. thesis (2010)
Sachs, K.J.: Der Contrapunctus im 14. und 15. Jahrhundert, Beihefte zum Archiv fü Musikwissenschaft, vol. 13. Franz Steiner Verlag (1974)
The Stacks project authors: The Stacks project, Section 0B28 (2022). https://stacks.math.columbia.edu/tag/0B28
Acknowledgements
We thank the anonymous reviewers whose suggestions significantly improved the exposition and clarity of this paper.
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Agustín-Aquino, O.A., Mazzola, G. (2022). A Projection-Oriented Mathematical Model for Second-Species Counterpoint. 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_7
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