Abstract
This article proposes a functorial framework for generalizing some constructions of transformational theory. We focus on Klumpenhouwer Networks for which we propose a categorical generalization via the concept of set-valued poly-K-nets (henceforth PK-nets). After explaining why K-nets are special cases of these category-based transformational networks, we provide several examples of the musical relevance of PK-nets as well as morphisms between them. We also show how to construct new PK-nets by using some topos-theoretical constructions.
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References
Lewin, D.: Transformational techniques in atonal and other music theories. Perspect. New Music 21(1–2), 312–371 (1982)
Lewin, D.: Generalized Music Intervals and Transformations. Yale University Press, New Haven (1987)
Mazzola, G.: Gruppen und Kategorien in der Musik: Entwurf einer mathematischen Musiktheorie. Heldermann, Lemgo (1985)
Mazzola, G.: Geometrie der Töne. Birkhäuser, Basel (1990)
Mazzola, G.: The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, Basel (2002)
Fiore, T.M., Noll, T.: Commuting groups and the topos of triads. In: Agon, C., Andreatta, M., Assayag, G., Amiot, E., Bresson, J., Mandereau, J. (eds.) MCM 2011. LNCS, vol. 6726, pp. 69–83. Springer, Heidelberg (2011)
Popoff, A.: Towards A Categorical Approach of Transformational Music Theory. Submitted
Kolman, O.: Transfer principles for generalized interval systems. Perspect. New Music 42(1), 150–189 (2004)
Fiore, T.M., Noll, T., Satyendra, R.: Morphisms of generalized interval systems and PR-groups. J. Math. Music 7(1), 3–27 (2013)
Nolan, C.: Thoughts on Klumpenhouwer networks and mathematical models: the synergy of sets and graphs. Music Theory Online 13(3), 1–6 (2007)
Lewin, D.: Klumpenhouwer networks and some isographies that involve them. Music Theory Spectr. 12(1), 83–120 (1990)
Klumpenhouwer, H.: A generalized model of voice-leading for atonal music. Ph.D. Dissertation, Harvard University (1991)
Klumpenhouwer, H.: The inner and outer automorphisms of pitch-class inversion and transposition. Intégral 12, 25–52 (1998)
Mazzola, G., Andreatta, M.: From a categorical point of view: K-nets as limit denotators. Perspect. New Music 44(2), 88–113 (2006)
Ehresmann, C.: Gattungen von lokalen Strukturen. Jahresber. Dtsch. Math. Ver. 60, 49–77 (1957)
Vuza, D.: Some mathematical aspects of David Lewin’s book generalized musical intervals and transformations. Perspect. New Music 26(1), 258–287 (1988)
Kan, D.M.: Adjoint functors. Trans. Am. Math. Soc. 87, 294–329 (1958)
Agon, C., Assayag, G., Bresson, J.: The OM Composer’s Book. Collection “Musique/Sciences”. IRCAM-Delatour France, Sampzon (2006)
Andreatta, M., Ehresmann, A., Guitart, R., Mazzola, G.: Towards a categorical theory of creativity for music, discourse, and cognition. In: Yust, J., Wild, J., Burgoyne, J.A. (eds.) MCM 2013. LNCS, vol. 7937, pp. 19–37. Springer, Heidelberg (2013)
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Popoff, A., Andreatta, M., Ehresmann, A. (2015). A Categorical Generalization of Klumpenhouwer Networks. 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_31
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DOI: https://doi.org/10.1007/978-3-319-20603-5_31
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