Representation of Musical Structures and Processes in Simplicial Chord Spaces Louis Bigo, Daniele Ghisi, Antoine Spicher, Moreno Andreatta To cite this version: Louis Bigo, Daniele Ghisi, Antoine Spicher, Moreno Andreatta. Representation of Musical Struc- tures and Processes in Simplicial Chord Spaces. Computer Music Journal, Massachusetts Institute of Technology Press (MIT Press), 2015, 39 (3), pp.9 - 24. 10.1162/COMJ_a_00312. hal-01263299 HAL Id: hal-01263299 https://hal.archives-ouvertes.fr/hal-01263299 Submitted on 29 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. 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Louis Bigo,∗ Daniele Ghisi,† Antoine Spicher,∗∗ Representation of Musical and Moreno Andreatta† ∗University of the Basque Country UPV/EHU Structures and Processes in Department of Computer Science and Artificial Intelligence Simplicial Chord Spaces Paseo de Manuel Lardizabal, 1 San-Sebastian, 20018 Spain [email protected] †Institut de Recherche et Coordination Acoustique/Musique Lab CNRS-IRCAM-UPMC 1 Place Igor-Stravinsky 75004 Paris, France {daniele.ghisi, moreno.andreatta}@ircam.fr ∗∗Universite´ Paris-Est Laboratoire d’Algorithmique, Complexite´ et Logique 61 avenue du Gen´ eral´ de Gaulle 94010 Creteil,´ France [email protected] Abstract: In this article, we present a set of musical transformations based on the representations of chord spaces derived from the Tonnetz. These chord spaces are formalized as simplicial complexes. A musical composition is represented in such a space by a trajectory. Spatial transformations are applied on these trajectories and induce a transformation of the original composition. These concepts are implemented in two applications, the software HexaChord and the Max object bach.tonnetz, dedicated to music analysis and composition, respectively. Spatial structures are commonly used by music to track key boundaries (Chew 2002), although we theorists to represent sets of symbolic objects such as will not delve further into these representations in notes, chords, and rhythms. If geometry and topology this article. play a major role in the representation of the musical Our focus in this article is on the harmonic di- objects, an algebraic formalization is necessary to mension, and we use a set of chord spaces to describe grasp the combinatorial potential of all these and execute certain musical transformations in the structures. The study of the combinatorial properties pitch domain. These spaces, which are generaliza- of these geometrical and topological spaces is also a tions of the Tonnetz, are formalized as simplicial source of inspiration for new approaches in music complexes, a topological concept that has proved to theory, analysis, and composition. Geometrical and be useful in many music-theoretical and analytical topological spaces are conceived as “support spaces” situations (Bigo et al. 2013). In this work, we focus in the representation of given musical objects. on the use of simplicial complexes in the repre- This is the case for the Tonnetz, a support space sentation of musical structures and process, with for representing neo-Riemannian transformations particular emphasis on musical transformations. (Cohn 2012). Other examples of topological spaces After briefly introducing the concepts of simplicial used for musical analysis include work by Dmitri complexes, simplicial collections, and structural Tymoczko (2011) on “orbifolds,” which can be used inclusion in the next section, in the subsequent to describe voice-leading progressions, and Elaine section we discuss the main support space of our Chew’s work with spiral arrays, which can be used approach (the Tonnetz and its generalized versions). Generalized Tonnetze are then described as sim- Computer Music Journal, 39:3, pp. 9–24, Fall 2015 plicial complexes in which trajectories represent doi:10.1162/COMJ a 00312 musical sequences. This is followed by a discussion c 2015 Massachusetts Institute of Technology. of transformation of a sequence in a generic chord Bigo et al. 9 Figure 1. A 0-simplex (a), a 1-simplex (b), a 2-simplex (c), and a 3-simplex (d). space, and we discuss two musical cases correspond- ing to the isomorphism between two support spaces and the automorphism within a given support space. Finally, we present two environments that are based on the geometric approach described in the arti- cle. The first environment, called HexaChord, is experimental software dedicated to computer-aided music analysis. The second one is the Max object bach.tonnetz, which focuses primarily on musi- For the sake of simplicity, we will often consider cal notation and new strategies for computer-aided that the term “simplex” designates the subcomplex music composition (Agostini and Ghisi 2015). containing a simplex and all its incident simplices of lower dimensions. Figure 1 illustrates examples Technical Background of n-simplices for n ∈{0, 1, 2, 3}. A simplicial d-complex is a simplicial complex In this section, we present several spatial tools where the highest dimension of any simplex is d. used in the course of this article. First, we describe Graphs are special cases of simplicial complexes simplicial complexes and collections, which allow where the highest dimension of any simplex is equal us to formalize chord spaces. Then, we describe to 1. Figure 2a illustrates a simplicial 3-complex. For structural inclusions, which provide a formalization any natural integer n,then-skeleton of a simplicial K S K of transformations within chord spaces. complex is defined by the subcomplex n( )of this complex formed by its simplices of dimension n or less. Figure 2a illustrates a complex and its Simplicial Complexes 1-skeleton (see Figure 2b). Given a set V of elements, a simplicial complex K defined on V is a collection of nonempty finite Simplicial Collections subsets of V, called simplices and denoted σ ∈ K, verifying the closure condition: A simplicial collection K is a simplicial complex in which every simplex is labeled by an arbitrary value, for any simplex σ ∈ K, every nonempty subset as illustrated in Figure 2c. The term “collection” σ ⊂ σ is also an element of K, i.e., σ ∈ K. comes from the notion of topological collections The definition of the closure condition is equivalent used in the MGS programming language (Giavitto to saying that there is an incidence relation between and Michel 2001), which has strongly inspired this two simplices σ and σ , which will be written as work. σ ≺ σ . Every simplex σ of K is characterized by More formally, a simplicial collection is a func- its dimension, dim() such that dim(σ ) = card(σ) − 1, tion that associates values from an arbitrary set with where the function card() gives the cardinality of σ . the simplices of a simplicial complex. The notation A simplex of dimension n is called an n-simplex. K(σ) enables one to address the label associated with Vertices can be used to represent 0-simplices, edges the cell σ in the collection K. We denote as |K| the for 1-simplices, triangles for 2-simplices, tetrahedra support of the collection K, which is the simplicial for 3-simplices, etc. complex without labels. A collection K is a subcol- The closure condition implies that an edge is lection of K if |K|⊂|K| and K(σ) = K(σ ) for every incident to two vertices, a triangle is incident σ of K. When there is no danger of ambiguity, the to three edges, etc. In general, every n-simplex notation |·|can be omitted. Similarly, we will often is incident to n + 1(n − 1)-simplices. A proper use the term “complex” to designate a simplicial subset of a simplicial complex K that is also a collection, that is, the simplicial complex and its simplicial complex is called a subcomplex of K. labels. 10 Computer Music Journal Figure 2. A simplicial Figure 3. Two possible complex K (a), the structural inclusions of 1-skeleton S1(K) (b), and a one simplicial complex in simplicial collection another. whose support is a subcomplex of K (c). Figure 2 Structural Inclusions In this work, we will be interested in the ways one complex can be embedded into another (or into itself). In order to deal with this notion, we introduce Figure 3 the concepts of morphism and structural inclusion. Let K and K be two simplicial complexes. A function φ : K → K is a morphism of simplicial into K. A morphism of complexes is injective if complexes if for every cell σ and σ of K: ∀σ , σ ∈ K, φ(σ ) = φ(σ ) ⇒ σ = σ . Injectivity enables K one to distinguish in a subcomplex that has σ ≺ σ ⇒ φ(σ ) ≺ φ(σ ), the same structure as K. We thus say that K is K dimK (φ(σ )) = dimK(σ). structurally included in . Figure 3 illustrates two different ways in which a complex may be These two conditions preserve, respectively, the structurally included in another. neighborhood between simplices and their dimen- Finally, every automorphism in a complex defines sion. In other words, a morphism of complexes a structural inclusion into itself. The set of auto- is a function in which structure is preserved. morphisms of a complex represents its structural Two complexes are said to be isomorphic if symmetries. they have exactly the same structure (i.e., φ is bijective). A morphism between the support complexes of Musical Representations two simplicial collections provides a way to modify values labeling the simplices. Let K and K be two simplicial collections and φ : |K|→|K| a morphism In this section, we present two well-known music- of complexes from the support complex of K into theoretical constructions: the Tonnetz and the the support complex of K.