Abstract
This article introduces a method for building and studying various harmonic structures in the actual conceptual framework of graph theory. Tone-networks and chord-networks are therefore introduced in a generalized form, focusing on Hamiltonian graphs, iterated line graphs and triangles graphs and on their musical meaning. Reference examples as well as notable music-related Hamiltonian graphs are then presented underlining their relevance for composers.
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Notes
- 1.
The well-known term lattice has been deliberately avoided in favor of the more abstract term tone-network to define our general note-based graphs. In fact, all lattices are tone-networks as we defined them (in particular, the vertex-transitive ones) but not all tone-networks are, or can be seen as, lattices. The term chord-network followed accordingly.
- 2.
A definition of the Generalized Interval System equivalent to the one given in [15].
- 3.
Although a GIS in its original formulation admits more general musical elements in its set, a tone-network defined as such admits only Generalized Interval Systems so that P is a set of pitches, pitch classes or similar one-note musical elements (such as for example scale degrees). This allows us to build a framework in which certain graphs obtained from tone-networks always represent chords or general n-note musical elements (such as for example a collection of scale degrees).
- 4.
Proper directed Cayley graphs are also known as Cayley graphs. The definition we present of undirected Cayley graphs is the one offered in [7].
- 5.
Note that the complement of a \((P,I,\varphi )\)-proper tone-network is not always \((P,I,\varphi )\)-unproper!.
- 6.
This paper is limited to n-iterated line graphs with \(n<3\), but it is possible to extend the results and study the cases for \(n\ge 3\).
- 7.
A chord can be redundantly presented in case it is a limited transposition one.
- 8.
\(L^2(T(Q,H))\) represents paths of length two on T(Q, H), while \(\varDelta (T(Q,H))\) represents 3-cycles on T(Q, H). It means that the former admits all the trichords in all the possible inversions that can be built on T(Q, H). In fact, it could be that some inversions are not possible because of missing edges/intervals. The latter consider only the trichords that admit all the inversions. As a matter of fact a vertex in \(\varDelta (T(Q,H))\) represents all of them.
- 9.
A representation introduced by Douthett and Steinbach in [6].
- 10.
Cf. [1].
- 11.
Note that Dirac’s theorem is a corollary of Ore’s. In 1976 Bondy and Chvátal’s proved a more general result of which Dirac’s and Ore’s theorem are both corollaries.
- 12.
Lewin did not require the group of a GIS to be abelian [15], but we think that commutativity is a strong requirement for the intuitiveness and consistency of a group of interval. Nevertheless, the result of Proposition 10 seems to apply also to the general case: excluding \(K_2\) all the known non-Hamiltonian vertex transitive graphs are not Cayley graphs and this leads to the conjecture that all Cayley graphs are Hamiltonian [12].
- 13.
This can be obviously done also in an arbitrary n-tone system.
- 14.
The software employed by the Authors is Groups and Graph version 3.6 by William Kocay.
References
Albini, G., Antonini, S.: Hamiltonian cycles in the topological dual of the Tonnetz. In: Chew, E., Childs, A., Chuan, C.H. (eds.) MCM 2009. Communications in Computer and Information Science, vol. 38, pp. 1–10. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02394-1_1
Baroin, G.: The planet-4D model: an original hypersymmetric music space based on graph theory. In: Agon, C., Andreatta, M., Assayag, G., Amiot, E., Bresson, J., Mandereau, J. (eds.) MCM 2011. LNCS (LNAI), vol. 6726, pp. 326–329. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21590-2_25
Bigo, L., Giavitto, J.-L., Spicher, A.: Building topological spaces for musical objects. In: Agon, C., Andreatta, M., Assayag, G., Amiot, E., Bresson, J., Mandereau, J. (eds.) MCM 2011. LNCS (LNAI), vol. 6726, pp. 13–28. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21590-2_2
Bigo, L., Andreatta, M., Giavitto, J.-L., Michel, O., Spicher, A.: Computation and visualization of musical structures in chord-based simplicial complexes. In: Yust, J., Wild, J., Burgoyne, J.A. (eds.) MCM 2013. LNCS (LNAI), vol. 7937, pp. 38–51. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39357-0_3
Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)
Douthett, J., Steinbach, P.: Parsimonious graphs: a study in parsimony, contextual transformations, and modes of limited transposition. J. Music Theor. 42(2), 241–63 (1998)
Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001). https://doi.org/10.1007/978-1-4613-0163-9
Gould, R.J.: Updating the Hamiltonian problem - a survey. J. Graph Theor. 15(2), 121–157 (1991)
Gould, R.J.: Advances on the Hamiltonian Problem - A Survey. Emory University, Atlanta (2002)
Gould, R.J.: Recent advances on the Hamiltonian problem: survey III. Graphs Comb. 30(1), 1–46 (2014)
Harary, F.: Graph Theory. Addison-Wesley Publishing Company, Reading (1969)
Heus, A.: A study of necessary and sufficient conditions for vertex transitive graphs to be Hamiltonian. Master’s thesis, University of Amsterdam (2008)
Hiller, P.: Arvo Pärt. Clarendon Press, Oxford (1997)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)
Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)
Ore, O.: A note on Hamiltonian circuits. Am. Math. Mon. 67, 55 (1960)
Tymoczko, D.: A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford University Press, Oxford (2011)
Witte, D.S., Gallian, J.A.: A survey: Hamiltonian cycles in Cayley graphs. Discrete Math. 51, 293–304 (1984)
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Albini, G., Bernardi, M.P. (2017). Hamiltonian Graphs as Harmonic Tools. In: Agustín-Aquino, O., Lluis-Puebla, E., Montiel, M. (eds) Mathematics and Computation in Music. MCM 2017. Lecture Notes in Computer Science(), vol 10527. Springer, Cham. https://doi.org/10.1007/978-3-319-71827-9_16
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