A categorical generalization of Klumpenhouwer networks. (English) Zbl 1321.00088

Collins, Tom (ed.) et al., Mathematics and computation in music. 5th international conference, MCM 2015, London, UK, June 22–25, 2015, Proceedings. Cham: Springer (ISBN 978-3-319-20602-8/pbk; 978-3-319-20603-5/ebook). Lecture Notes in Computer Science 9110. Lecture Notes in Artificial Intelligence, 303-314 (2015).
Summary: 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.
For the entire collection see [Zbl 1315.00045].


00A65 Mathematics and music
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18B25 Topoi
Full Text: DOI


[1] Lewin, D.: Transformational techniques in atonal and other music theories. Perspect. New Music 21(1–2), 312–371 (1982)
[2] Lewin, D.: Generalized Music Intervals and Transformations. Yale University Press, New Haven (1987)
[3] Mazzola, G.: Gruppen und Kategorien in der Musik: Entwurf einer mathematischen Musiktheorie. Heldermann, Lemgo (1985) · Zbl 0574.00016
[4] Mazzola, G.: Geometrie der Töne. Birkhäuser, Basel (1990) · Zbl 0729.00008
[5] Mazzola, G.: The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Birkhäuser, Basel (2002) · Zbl 1104.00003
[6] 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) · Zbl 1335.00121
[7] Popoff, A.: Towards A Categorical Approach of Transformational Music Theory. Submitted
[8] Kolman, O.: Transfer principles for generalized interval systems. Perspect. New Music 42(1), 150–189 (2004)
[9] Fiore, T.M., Noll, T., Satyendra, R.: Morphisms of generalized interval systems and PR-groups. J. Math. Music 7(1), 3–27 (2013) · Zbl 1327.00019
[10] Nolan, C.: Thoughts on Klumpenhouwer networks and mathematical models: the synergy of sets and graphs. Music Theory Online 13(3), 1–6 (2007)
[11] Lewin, D.: Klumpenhouwer networks and some isographies that involve them. Music Theory Spectr. 12(1), 83–120 (1990)
[12] Klumpenhouwer, H.: A generalized model of voice-leading for atonal music. Ph.D. Dissertation, Harvard University (1991)
[13] Klumpenhouwer, H.: The inner and outer automorphisms of pitch-class inversion and transposition. Intégral 12, 25–52 (1998)
[14] Mazzola, G., Andreatta, M.: From a categorical point of view: K-nets as limit denotators. Perspect. New Music 44(2), 88–113 (2006)
[15] Ehresmann, C.: Gattungen von lokalen Strukturen. Jahresber. Dtsch. Math. Ver. 60, 49–77 (1957) · Zbl 0097.37803
[16] Vuza, D.: Some mathematical aspects of David Lewin’s book generalized musical intervals and transformations. Perspect. New Music 26(1), 258–287 (1988)
[17] Kan, D.M.: Adjoint functors. Trans. Am. Math. Soc. 87, 294–329 (1958) · Zbl 0090.38906
[18] Agon, C., Assayag, G., Bresson, J.: The OM Composer’s Book. Collection ”Musique/Sciences”. IRCAM-Delatour France, Sampzon (2006)
[19] 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) · Zbl 1270.00024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.