# Sciences cognitives, mathématiques et perception musicale

### Samedi 8 mars 2003

Ircam, Salle I. Stravinsky

Entrée libre dans la mesure des places disponibles

Voir aussi: Pré-séance sur l'analyse assistée par ordinateur (vendredi 7 mars 2003)

## Programme

- 10h - 10h15 :
**Moreno Andreatta**: Introduction. - 10h15 - 11h :
**Guerino Mazzola**: The Cognitive Relevance of the Mathematical Counterpoint Model in Human Depth EEG [ppt] - 11h05 - 11h50 :
**Edward Gollin**: Musical transformation and musical experience - 12h00 - 12h30 :
**Thomas Noll**: Wundtian Apperception and Musical Cognition - 12h30 - 13h00 :
**Anja Fleischer**: Inner metric analysis and its perceptual evaluation [ppt]

Buffet (Niveau -2)

- 14h30 - 15h15 :
**Stephen McAdams**: Contributions and limits of computational models of music cognition - 15h15 - 16h :
**Jean Petitot**: On segmentation models - 16h15 - 17h :
**PerAage Brandt**: Conceptual Integration and Dynamic Schematization in Tonal Phenomenology - 17h15 - 18h30 :
**Michael Leyton**: Musical Works are Maximal Memory Stores

## Abstracts

#### Guerino Mazzola (CSD University of Zürich/MultiMedia Lab)

**The Cognitive Relevance of the Mathematical Counterpoint Model in Human Depth EEG**

We first describe the counterpoint model which views intervals as tangent tones on the torus of pitch classes. The counterpoint rules are deduced from deformations of the tangent torus by local symmetries and show a nearly complete congruence with the classical rules of Fux. This model resides on the unique autocomplementarity symmetry between consonances and dissonances. We next discuss results of an empirical investigation at the Zurich Neurology Department, showing the cognitive presence of the autocomplementarity symmetry within the response of human Depth EEG, mainly in the Heschl gyri of the auditory cortex and in the hippocampal formations of the emotional brain, to musical inputs. Third, we give an outlook to generalized counterpoint theories deduced from more exotic autocomplementarity symmetries and associated interval dichotomies. This talk refers to chapters 29-31 of the author's book "The Topos of Music" (Birkhäuser, Basel 2002).

#### Edward Gollin (University of Harvard)

**Musical transformation and musical experience**

The talk explores premises concerning the relationship between certain finite mathematical groups (and their attendant group structure) and musical structure in transformational music theories, and in particular, neo-Riemannian accounts of harmonic structure and progression. The idea that group structure endows certain musical relationships with some degree of coherence or intelligibility (an idea expressed implicitly or explicitly in the writings of, among others, Brian Hyer and Richard Cohn) will be questioned. An alternative view, one in which the intelligibility of transformational relationships is borne by certain generative group elements (i.e. certain fundamental relationships within the group), will be offered. From this latter perspective, a family of group generators and an understanding of the constraining relationships among them (mathematically expressed as the presentation of a group) constitute a model of a listener's musical context. The ways of experiencing transformational relationships within that context are then expressible as words within that presentation (rather than as simply elements of the group). The model thereby captures the different ways one might experience the same transformation (understood as a group element) in different musical settings (recognizing that transformations are instantiated by elements from a class of equivalent words in a group presentation).

#### Thomas Noll (TU-Berlin)

**Wundtian Apperception and Musical Cognition**

To experience music is not merely to notice musical patterns and relations. It rather seems to be relevant to investigate processes that give our mind access to relations on several levels of musical description. The Wundtian concept of apperception is dedicated to the assumption that we do not access all ideas in our consciousness simultaneously and that we therefore voluntarily regulate the scope of our attention in order to access them. Tonal pieces seem to carry a kind of dramaturgy for such apperceptive processes. Therefore it is promising to interpret models for "apperceptive kinematics" on the background of music-theoretical phenomena. I report on investigations that are motivated by the hypothesis that the group SL(2,Z) is a reasonable ingredience to such a model. I will compare arguments in favour of two different interpretations of this group. In the first case the individual group elements carry concrete music-theoretical interpretations (fifth step, fifth alterations, minor-third-substitution, etc…).

I briefly recall basic ideas about the "active tone system"-model, where SL(2,Z) is embedded into a larger apperception group acting on itself. In the second case the group is interpreted as a group of symplecto-morphisms acting on a 2-dimensional musical phase-space. This latter interpretation is more closely related to standard situations in geometrical optics and Hamiltonian mechanics. Wundt's visual metaphor for apperception literally encourages models from geometrical optics. Mazzola's treatment of musical intervals as tangent vectors (c.f. his lecture) encourages the model of a musical phase space. My own earlier attempt to relate Counterpoint to Harmony by extrapolating interval-vectors into affine endomorphims of Z12 formally reverses the way how light rays in lense systems are encoded by elements of R^2 (such that suitable SL(2,R) operations describe acts of translation and refraction). Therefore the symplectic interpretation of SL(2) = Sp(2) seems to be more "standard" than the active-tone-system interpretation. Dispite of this choice whithin the theory we admit that the whole approach is speculative and a careful discussion of music-theoretical interpretations is essential to its further development.

#### Anja Fleischer (TU-Berlin)

**Inner metric analysis and its perceptual evaluation**

We want to discuss the model of the Analytic Interpretation regarding cognitive aspects of understanding metric structures of a piece of music. The idea of the Analytic Interpretation dates back to Riemann [1884] and Adorno [1976] and has been enriched by various studies (see [Berry 1989], [Epstein 1987], [Sundberg 1993], [Mazzola 2002]) which define the performer‚s role in the following way. The task of the performer is to elucidate the structure of a piece of music to the audience, in other words, to communicate his understanding of the piece to the listener. In Fleischer [2002] investigations concerning the Analytic Interpretation have been carried out within an experimental approach based on the RUBATO Workstation for Musical Analysis and Performance. It models the transformative process from the symbolic reality of the signs of the score into the physical reality of sounds. In the concept of the workstation the performance transformation is modeled on the basis of analytical data and the method of applying analytical weights in order to shape the performance. In [Fleischer 2002] RUBATO's tool for inner metric analysis has been tested extensively by applying it to a large amount of musical pieces, a notion of metric coherence has been introduced. Inner metric analysis studies the metric structure of the notes of a given piece without considering the time signature and bar lines and is opposed to outer metric structure given by the accent hierarchy of the bars. The analysis results in a metric weight for each note of the score. The notion of metric coherence describes the correspondences of varying degrees between the outer and inner metric structure. As a result of the explorative work with the model, a higher degree of coherence was detected within those works, which are typical representations of the accent scheme given by the time signature.

Experiments concerning the Analytic Interpretation have been carried out in a second step in order to gain a precise description in how far metric weights might help to shape a performance that elucidates the metric structure to the listeners. Performances shaped on the basis of metric weights have been evaluated within listening experiments. Metric weights of higher degree of coherence led to a more convincing interpretation regarding the question in how far the metric structure was expressed properly. These results may be taken as an indication for a relationship between analytical structures of the score and the understanding of the performed music by listeners through suitable expression of these structures within a performance.

#### Stephen McAdams (Equipe Perception et Cognition Musicale, Ircam-CNRS and Département d'Etudes Cognitives, Ecole Normale Supérieure)

**Contributions and limits of computational models of music cognition**

Computational models are important tools for formalizing experimentally derived knowledge about cognitive processes involved in music listening and music performance. They become tools for thought to the extent that they can make predictions about musical behavior that go beyond the experimental conditions upon which they are initially based. Examples will be drawn from pitch and harmony processing and timbre perception, which will be examined critically for what they can and can't help us to understand about music.

#### Per Aage Brandt (Center for Semiotic Research, University of Aarhus, Copenhagen)

**Conceptual Integration and Dynamic Schematization in Tonal Phenomenology**

In the default phenomenology of trained musicians, intelligible tonal organizations are structured by the ongoing formation of conceptually integrated semantic wholes, in which certain dynamic schemas are crucially important. Integration and schematization are real time processes that a cognitive semantics of music finsd to be at the origin of ‚wellformedness‚ or ‚meaningfulness‚ independently of specific aesthetic preferences. Tonality is a form of what I suggest to call symbolic perception, and is thus per se potentially significant (signifying).

The aim of cognitive semantics of music is to study the pre-aesthetic grounding of musical experiences in the very architecture of our percept/concept-formative mind.

#### Michael Leyton (Center for Discrete Mathematics & Theoretical Computer Science, DIMACS-Rutgers University)

**Musical Works are Maximal Memory Stores**

Voir l'article "Musical Works are Maximal Memory Stores" in *Perspectives in Mathematical and Computational Music Theory*, 2004.

The book *A Generative Theory of Shape* (Michael Leyton, Springer-Verlag, 2001) develops new foundations to geometry in which *shape is equivalent to memory storage*. With respect to this, the argument is given that *art-works are maximal memory stores*. The present paper reviews some of the basic principles concerning our claim that, in particular, musical works are maximal memory stores. The argument is that *maximizing memory storage explains the structure of musical works*. We first review the basic geometric theory of the book: A generative theory of shape is developed that has two properties regarded as fundamental to *intelligence* – maximization of transfer and maximization of recoverability. *Aesthetic structuration is taken to be equivalent to intelligence*. Thus aesthetics is brought into the very foundations of the new theory of geometry. A mathematical theory of transfer and recoverability is developed, using a structure we define, called *symmetry-breaking wreath products*. From this, it becomes possible to develop a theory of musical composition, as follows: Musical works are complex shapes. A theory of *complex-shape generation* is presented, in which any structure is described as *unfolded* from a maximally collapsed version of that structure, called an *alignment kernel*. This process is formalized by proposing a new class of groups called unfolding groups. The alignment kernel is a subgroup of that structure, consisting of symmetry ground-states which are themselves formalized by a new class of groups called *iso-regular groups*. In music, the iso-regular groups represent the anticipation hierarchies, for example the regular meters of the work. The process of musical composition is then described by an unfolding group, which "unfolds" the work, by successively breaking the iso-regular groups of the alignment kernel.

## Quelques références

- Adorno, T. W. (1976). Der getreue Korrepetitor. Interpretationsanalysen neuer Musik. In: Gesammelte Schriften 12, Frankfurt M.
- Berry, W. (1989). Musical Structure and Performance. Yale University Press, New Haven and London.
- Brandt, P. A. (1994). Dynamiques du sens, Aarhus University Press.
- Fleischer, A. (2002). Die analytische Interpretation. Schritte zur Erschließung eines Forschungsfeldes am Beispiel der Metrik. Dissertation at Humboldt.University Berlin.
- Epstein, D. (1987). Beyond Orpheus, Oxford U niversity Press.
- Gollin, E. (2000). Representations of Space and Conceptions of Distance in Transformational Music Theories, PhD, Harvard University.
- Leyton, M. (2002). A Generative Theory of Shape, Springer, 2001.
- Mazzola, G. (2002). Topos of Music, Birkhäuser, Basel.
- McAdams, S. and I Déliège (ed.). (1989). La musique et les sciences cognitives, Bruxelles, Mardaga.
- Noll, T. (2002). Tone Apperception, Weber-Fechner's Law and the GIS-Model (MaMuX Seminar, december 2002)
- Petitot, J. et al. (éd.) (2002). Naturaliser la phénoménologie. Essais sur la phénoménologie contemporaine et les sciences cognitives, CNRS Editions.
- Riemann, H. (1884). Musikalische Dynamik und Agogik, Leipzig.
- Sundberg, J. (1993). How can music be expressive? In: Speech communication. 13 (1993), North-Holland, 239-253.