Abstract
This paper presents a computational approach to a particular theory in the work of Julian Hook—Uniform Triadic Transformations (UTTs). A UTT defines a function for transforming one chord into another, and is useful for explaining triadic transitions that circumvent traditional harmonic theory. By combining two UTTs and extrapolating, it is possible to create a two-dimensional chord graph. Meanwhile, graph theory has long been studied in the field of Computer Science. This work describes a software tool which can compute the shortest path between two points in a two-dimensional transformational chord space. Utilizing computational techniques, it is then possible to find the optimal chord space for a given musical piece. The musical work of Michael Nyman is analyzed computationally, and the implications of a weighted chord graph are explored.
R. Groves—Many thanks to Professor Robert Hasegawa for much encouragement and guidance during the process of this research.
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Notes
- 1.
The code for the generation of UTT spaces can be found at Ryan Groves’ github landing page at http://github.com/bigpianist/UTTSpaces.
References
Callender, C.: Interactions of the lamento motif and jazz harmonies in György Ligeti’s arc en ciel. Intégral 21, 41–77 (2007)
Cohn, R.: Introduction to Neo-Riemannian theory: a survey and historical perspective. J. Music Theor. 42(2), 167–180 (1998)
Cook, S.A.: Moving through triadic space: an examination of Bryars’s seemingly haphazard chord progressions, March 2009. http://www.mtosmt.org/issues/mto.09.15.1/mto.09.15.1.cook.html
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)
Gollin, E.: Some unusual transformations in Bartóks minor seconds, major sevenths. Intégral 12(2), 25–51 (1998)
Hook, J.: Uniform triadic transformations. J. Music Theor. 46(1/2), 57–126 (2002)
Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)
Lewin, D.: Generalized interval systems for Babbitt’s lists, and for Schoenberg’s string trio. Music Theor. Spectr. 17(1), 81–118 (1995)
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© 2015 Springer International Publishing Switzerland
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Groves, R. (2015). Finding Optimal Triadic Transformational Spaces with Dijkstra’s Shortest Path Algorithm. In: Collins, T., Meredith, D., Volk, A. (eds) Mathematics and Computation in Music. MCM 2015. Lecture Notes in Computer Science(), vol 9110. Springer, Cham. https://doi.org/10.1007/978-3-319-20603-5_12
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DOI: https://doi.org/10.1007/978-3-319-20603-5_12
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