Project

Quantum approaches to music cognition

Goal: We are using Schrödinger wave functions and gauge theory as a mathematical model of music cognition. We already have a nice model of (static) tonal attraction (beim Graben & Blutner 2016), explaining the data from Krumhansl & Kessler (1982). Future research will include dynamic attraction and the Schönberg/Mazzola theory of modulation.

Methods: Fundamental Symmetry, quantum cognition, wave functions, circular similarity, harmony theory

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Peter Beim Graben
added a research item
How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.
Peter Beim Graben
added 2 research items
Metaphors involving motion and forces are a source of inspiration for understanding tonal music and tonal harmonies since ancient times. Starting with the rise of quantum cognition, the modern interactional conception of forces as developed in gauge theory has recently entered the field of theoretical musicology. We develop a gauge model of tonal attraction based on SU(2) symmetry. This model comprises two earlier attempts, the phase model grounded on U(1) gauge symmetry, and the spatial deformation model derived from SO(2) gauge symmetry. In the neutral, force-free case both submodels agree and generate the same predictions as a simple qubit approach. However, there are several differences in the force-driven case. It is claimed that the deformation model gives a proper description of static tonal attraction. The full model combines the deformation model with the phase model through SU(2) gauge symmetry and unifies static and dynamic tonal attraction.
Reinhard Blutner
added a research item
How well does a given pitch fit into a tonal scale or tonal key, let it be a major or minor key? A similar question can be asked regarding chords and tonal regions. Structural and probabilistic approaches in computational music theory have tried to give systematic answers to the problem of tonal attraction. We will discuss two previous models of tonal attraction, one based on tonal hierarchies and the other based on interval cycles. To overcome the shortcomings of these models, both methodologically and empirically, I propose a new kind of models relying on insights of the new research field of quantum cognition. I will argue that the quantum approach integrates the insights from both group theory and quantum probability theory. In this way, it achieves a deeper understanding of the cognitive nature of tonal music, especially concerning the nature of musical expectations (Leonhard Meyer) and a better understanding of the affective meaning of music.
Peter Beim Graben
added a research item
Quantum cognition emerged as an important discipline of mathematical psychology during the last two decades. Using abstract analogies between mental phenomena and the formal framework of physical quantum theory, quantum cognition demonstrated its ability to resolve several puzzles from cognitive psychology. Until now, quantum cognition essentially exploited ideas from projective (Hilbert space) geometry, such as quantum probability or quantum similarity. However, many powerful tools provided by physical quantum theory, e.g., symmetry groups have not been utilized in the field of quantum cognition research so far. Inspired by seminal work by Guerino Mazzola on the symmetries of tonal music, our study aims at elucidating and reconciling static and dynamic tonal attraction phenomena in music psychology within the quantum cognition framework. Based on the fundamental principles of octave equivalence, fifth similarity and transposition symmetry of tonal music that are reflected by the structure of the circle of fifths, we develop different wave function descriptions over this underlying tonal space. We present quantum models for static and dynamic tonal attraction and compare them with traditional computational models in musicology. Our approach replicates and also improves predictions based on symbolic models of music perception.
Reinhard Blutner
added a research item
Music can have an extrinsic and/or an intrinsic meaning. The former is relevant in the case of program music, i.e. music that attempts to render an extra-musical narrative. The latter conforms to pure (absolute) music, i.e. music that can be understood without reference to extrinsic sources. Taking the intrinsic content of music as basic, we have to ask about its nature. Using a term of Immanuel Kant, I propose to identify it with aesthetic emotion. As tonal music is organized by series of chords relative to the context of a tonal scale, the question is how music forms can be mapped onto aesthetic emotions. In order to get a concise account of the affective response, this paper makes several simplifications. The most important simplification is to assume that affective responses can be represented by a two-dimensional space of emotions, where one dimension refers to surprise and the other dimension refers to pleasantness. Relating pleasantness with consonance and surprise with entropic uncertainty leads to an account which directly relates structural and probabilistic aspects of tonal music with its affective content. The present bare-bone semantics of pure music proposes an explicit modelling of the affective response based on an algebraic meaning conception
Peter Beim Graben
added a research item
Quantum cognition emerged as an important discipline of mathematical psychology during the last two decades. Using abstract analogies between mental phenomena and the formal framework of physical quantum theory, quantum cognition demonstrated its ability to resolve several puzzles from cognitive psychology. Until now, quantum cognition essentially exploited ideas from projective (Hilbert space) geometry, such as quantum probability or quantum similarity. However, many powerful tools provided by physical quantum theory, e.g., symmetry groups have not been utilized in the field of quantum cognition research sofar. Inspired by seminal work by Guerino Mazzola on the symmetries of tonal music, our study aims at elucidating and reconciling static and dynamic tonal attraction phenomena in music psychology within the quantum cognition framework. Based on the fundamental principles of octave equivalence, fifth similarity and transposition symmetry of tonal music that are reflected by the structure of the circle of fifths, we develop different wave function descriptions over this underlying tonal space. We present quantum models for static and dynamic tonal attraction and compare them with traditional computational models in musicology. Our approach replicates and also improves predictions based on symbolic models of music perception.
Peter Beim Graben
added a research item
How well does a given pitch fit into a tonal scale or key, being either a major or minor key? This question addresses the well-known phenomenon of tonal attraction in music psychology. Metaphorically, tonal attraction is often described in terms of attracting and repelling forces that are exerted upon a probe tone of a scale. In modern physics, forces are related to gauge fields expressing fundamental symmetries of a theory. In this study we address the intriguing relationship between musical symmetries and gauge forces in the framework of quantum cognition.
Peter Beim Graben
added 2 project references
Peter Beim Graben
added a research item
How well does a given pitch fit into a tonal scale or key, being either a major or minor key? This question addresses the well-known phenomenon of tonal attraction in music psychology. Metaphorically, tonal attraction is often described in terms of attracting and repelling forces that are exerted upon a probe tone of a scale. In modern physics, forces are related to gauge fields expressing fundamental symmetries of a theory. In this study we address the intriguing relationship between musical symmetries and gauge forces in the framework of quantum cognition.
Peter Beim Graben
added a project goal
We are using Schrödinger wave functions and gauge theory as a mathematical model of music cognition. We already have a nice model of (static) tonal attraction (beim Graben & Blutner 2016), explaining the data from Krumhansl & Kessler (1982). Future research will include dynamic attraction and the Schönberg/Mazzola theory of modulation.