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Quantum approaches to music cognition
Abstract
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.
... We can go from the blue one to the red and to the gray one or from the blue one directly to the gray one, indifferently. With this in mind, we can imagine transforming the inner pattern of one Qwalala brick into the inner pattern of another parallelepiped, creating a typical brick of Qwalala (Figs. 17,18). The same idea can be ideally applied to bricks: We can draw commutative diagrams where each Qwalala's brick is transformed into another one. ...
... The most simple as possible sketch exists because it can be sketched. 18 By proving the unicity of the morphism a, mapping D 1 to the "form of (waved) large brick wall" L in Diagram (7), the proof follows immediately by the commutativity of the diagram. ...
... The quantum paradigm is more and more used in sound analysis and processing[31,58], in music theory[10,17,18,43], in cognition[32], and in image processing[64]. Thus, the quantum formalism can be used to formalize some relationships between music and images with a cognitive flavor[45]. ...
Qwalala (2017) is an installation by Pae White, inspired by a river. It is a wave-shaped wall with a modular structure of colored glass blocks, built for the Venice Biennale. In Qwalala, visual patterns and forms meet the aesthetics of abstract geometries and the elegance of glass. We investigate Qwalala in light of mathematics, with patterns within the bricks, and the bricks seen themselves as patterns within the overall wall structure. We apply some developments of mathematical music theory, Gestural Similarity and Quantum GestArt, envisaging musical renditions of Qwalala’s structure. Then, we extend this formalism to other glass artworks, comparing them via the language of categories. The overall article faces some pedagogical questions: How can visual arts inspire us to develop mathematical formalizations? How can such a mathematical formalization inspire us to create new artworks?
... These remarks indicate that pitch class profiles enter music theory in connection with various approaches to the investigation of tonality. Upon this wider background it is desirable to understand the implications of a more recent approach to the mathematical interpretation of the KK profiles, proposed by Reinhard Blutner and Peter beim Graben in [5,6] using ideas from quantum theory. ...
... Starting point of the present study is the observation in [14, Fig. 1] that the deformed quantum cosine model ψ : R/2πZ → C over the continuous circle of fifths [6] can be equivalently described by a Gaussian wave function ψ : R → C over Regener's line of fifths R as configuration space [15], retaining the probe tone positions along the circle of fifths up to enharmonic equivalence, such that the KK profiles emerge as mixed quantum states, i.e. as weighted sums ...
... is a normal distribution density with variance σ 2 ∈ R, whose mean 0 is shifted to a tone position a k ∈ R in each corresponding summand for the given tonality, altogether constituting a tonal context {a 1 , . . . , a N } ⊂ R [6]. The coefficients α k ∈ R are the respective weights in the convex linear combination (5) with N k=1 α k = 1. ...
We adopt some basic ideas on quantum-theoretical modeling of tonal attraction and develop them further in an alternative direction. Fitting Gaussian Mixture Models (GMM) to the Krumhansl-Kessler (KK) probe tone profiles for static attraction opens the possibility to investigate the underlying wave function as the stationary ground state of an anharmonic quantum oscillator with a schematic Hamiltonian involving a perturbation potential. We numerically verify the fulfillment of the associated stationary Schrödinger equation and also inspect its excited states as a solution basis for the corresponding time-dependent Schrödinger equation. With their help we calculate the temporal evolution of any initial state. As an example, we study the dynamics of transpositions of the stationary KK wave function across Regener's line of fifths. This offers potential models for dynamic tonal attraction and also for the behavior of deflected key profiles within the Hamiltonian dynamics of a given tonality. Finally, we sketch further directions of investigation within this mathemusical experimental playground.
... These remarks indicate that pitch class profiles enter music theory in connection with various approaches to the investigation of tonality. Upon this wider background it is desirable to understand the implications of a more recent approach to the mathematical interpretation of the KK profiles, proposed by Reinhard Blutner and Peter beim Graben in [5], [6] using ideas from quantum theory. ...
... Starting point of the present study is the observation in [14, Fig. 1] that the deformed quantum cosine model ψ : R/2πZ → C over the continuous circle of fifths [6] can be equivalently described by a Gaussian wave function ψ : R → C over Regener's line of fifths R as configuration space [15], retaining the probe tone positions along the circle of fifths up to enharmonic equivalence, such that the KK profiles emerge as mixed quantum states, i.e. as weighted sums ...
... is a normal distribution density with variance σ 2 ∈ R, whose mean 0 is shifted to a tone position a k ∈ R in each corresponding summand for the given tonality, altogether constituting a tonal context {a 1 , . . . , a N } ⊂ R [6]. The coefficients α k ∈ R are the respective weights in the convex linear combination (5) with N k=1 α k = 1. ...
We adopt some basic ideas on quantum-theoretical model-ing of tonal attraction and develop them further in an alternative direction. Fitting Gaussian Mixture Models (GMM) to the Krumhansl-Kessler (KK) probe tone profiles for static attraction opens the possibility to investigate the underlying wave function as the stationary ground state of an anharmonic quantum oscillator with a schematic Hamiltonian involving a perturbation potential. We numerically verify the fulfilment of the associated stationary Schrödinger equation and also inspect its excited states as a solution basis for the corresponding time-dependent Schrödinger equation. With their help we calculate the temporal evolution of any initial state. As an example, we study the dynamics of transpositions of the stationary KK wave function across Regener's line of fifths. This offers potential models for dynamic tonal attraction and also for the behavior of deflected key profiles within the Hamiltonian dynamics of a given tonality.
... On the other hand, some common observed properties of human cognition and quantum mechanics (superposition, non-classical probability) have given universal value to the quantum-theoretic formalism to explain cognitive acts [61], including actions of human creation, such as music. The explanatory power of a quantum approach to music cognition has been demonstrated to describe tonal attraction phenomena in terms of metaphorical forces [2,4]. The theory of open quantum systems has been applied to music to describe the memory properties (non-Markovianity) of different scores [35]. ...
... which is orthogonal to the linear combination (2). In vector form, we have ...
... Given an audio scene such as that of the two crossing glides interrupted by noise (figure 7), we may follow the Hamiltonian evolution from an initial state that is known only probabilistically. For example, at time zero we may start from a mixture of 1 2 pitch-up and 2 3 pitch-down. The density matrix (18) would evolve according to equation (24), where the unitary operator U(0, t) is defined as in (25). ...
Concepts and formalism from acoustics are often used to exemplify quantum mechanics. Conversely, quantum mechanics could be used to achieve a new perspective on acoustics, as shown by Gabor studies. Here, we focus in particular on the study of human voice, considered as a probe to investigate the world of sounds. We present a theoretical framework that is based on observables of vocal production, and on some measurement apparati that can be used both for analysis and synthesis. In analogy to the description of spin states of a particle, the quantum-mechanical formalism is used to describe the relations between the fundamental states associated with phonetic labels such as phonation, turbulence, and supraglottal myoelastic vibrations. The intermingling of these states, and their temporal evolution, can still be interpreted in the Fourier/Gabor plane, and effective extractors can be implemented. The bases for a Quantum Vocal Theory of Sound, with implications in sound analysis and design, are presented.
... We call the corresponding model the phase model. And second, SO(2), the group of vector rotations in a two-dimensional (real) Hilbert space, which is related to the spatial deformation model of beim Graben and Blutner (2019). The combined model makes full use of the group SU(2). ...
... 5 Next, one can ask for suitable choices of the deformation function γ (x). beim Graben and Blutner (2019) argue that plausible interpolation conditions imply that γ (0) = 0 and γ (π) = π/2. A consequence of this choice is that the tonic (at 0) should not be deformed while the tritone (at π ) receives maximal deformation. ...
... With this deformation function, beim Graben and Blutner (2019) found excellent agreement with the static attraction data of Krumhansl and Kessler (1982). Figure 3 also presents the comparison between our two quantum models and the kernel function reconstructed from Lerdahl's (1988) classical hierarchical model (dotted line). ...
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.
... The relations of tonalities in music according to the classical for theoretical musicology circle of fifths have become the subject of studies for specialists in the field of quantum physics. The study by K. Graben and R. Blutner (2019) [13] aimed at clarifying and coordinating the static and dynamic phenomena of tonal attraction from the standpoint of musical psychology within the framework of quantum cognition. Quantum models of static and dynamic tonal attraction are presented and compared with conventional computational algorithms in musicology. ...
... Quantum models of static and dynamic tonal attraction are presented and compared with conventional computational algorithms in musicology. Such an approach discloses the character of symbolic models of music perception from the perspective of quantum mechanics (Graben and R. Blutner, 2019) [13]. The classical composition "The Well-Tempered Clavier" by Johann Sebastian Bach is ideal for representing the researchers' ideas. ...
Music is a complex and multifaceted phenomenon, serving purposes ranging from aesthetic education to therapy and emotional expression. The purpose of this study is to examine the semantic aspects of musical language as an integrated system, focusing on their ongoing development and content complexity. This research employs a comprehensive approach, drawing on the “general theory of systems” developed in the 20th century. It systematically analyses various elements and properties within the musical language, considering their historical evolution and adaptation to contemporary contexts. The study also explores the integration of music with fields such as quantum physics, neuroscience, and computer technologies. The analysis reveals that the semantic aspects of musical language encompass a rich array of elements, including intonation systems, modes, intervals, rhythms, tempo, dynamics, texture, genres, and compositional structures. While some aspects have undergone radical transformations in terms of content and emotional resonance, others remain fundamentally unchanged. The integration of music with diverse scientific and technological domains has expanded the scope of musical art and enriched its semantic dimensions. This study demonstrates that musical art, with its semantic aspects, continues to evolve and adapt to the ever-changing cultural and scientific landscape. The integration of music with fields beyond its traditional boundaries opens up new opportunities for interdisciplinary research and innovation.
... entailed a similar picture as that shown in Table 1. Since these data have been modelled by the quantum gauge approach of Blutner and beim Graben (2021) and beim Graben and Blutner (2019), we address this issue in more detail in Section 4.1. In his own model of musical forces, Larson also suggested the semitone topology of the chromatic scale in the first row of Table 1 as a distance measure, which is also one pillar in the theory of Costère's cardinality relationships (the other one is given as angular distance along the circle of fifths) (Costère, 1954;Ellard, 1973). ...
... In this context, it is tempting to speculate about statistical learning theory as the origin of culturally acquired musical schemata considered as empirical deformations (Huron, 2006;Krumhansl & Cuddy, 2010;Pearce, 2018). Another open question asks how the asymmetric deformation accounting for the major-minor dichotomy in beim Graben and Blutner (2019) fits into the framework of either SO(2), or, more general, SU(2) gauge symmetries (Blutner & beim Graben, 2021). ...
After reviewing the physicalistic or metaphorical accounts to musical and visual forces by Arnheim and Larson, respectively, which were inspired by the basic tenets of gestalt psychology, I present a novel, naturalistic, mathematical framework, based on symmetry principles and gauge theory. In musicology, this approach has already been applied to the phenomenon of tonal attraction, leading to a deformation of the circle of fifths. The underlying gauge symmetry turns out as the SO(2) Lie group of a musical quantum model. Here, I present an alternative description in terms of Riemannian geometry. Its essential constraint of invariance of the infinitesimal line element leads to a deformation of the circle of fifths into a heart of fifths. In vision, the same approach is applied to Fraser's twisted cord illusion where concentric circles are deformed to squircle objects by means of an optical gauge field induced through a checkerboard background.
... Inspiring presentations concerning qubits in music have been posted on the web (EPiQC 2019; Cervera-Lierta 2018). Beim Graben and Blutner (2019) presented quantum models for static and dynamic tonal attraction and compared them with traditional computational models in musicology. A quantum vocal theory of sound was presented by Rocchesso and Mannone (2020). ...
... Therefore, the probability of the electron going upwards or downwards is equal to cos 2 (θ /2) or sin 2 (θ /2), respectively. Analogous results can be found in Blutner 2017, Beim Graben and Blutner 2019, and Blutner and Graben 2021 concerning the theory of music. Interestingly, the experimental data shown in Figure 3 of Shepard 1964 suggest that the squared amplitude moduli |α k | 2 and |β k | 2 for both acoustical qubit states in Equation [3] could be also expressed by squared cosine and squared sine functions, respectively. ...
I propose a quantum-like approach to the description of melody perception where classic intervals that constitute a melody are replaced by acoustical qubits, i.e. two-level acoustic systems, using Shepard tones for this purpose. Each of such qubits is considered to be a superposition of two intervals, ascending and descending, that form an octave when put together. Any melody perception can thus be treated analogously to a sequence of quantum measurements. Because of an acoustical collapse, analogous to the wave function reduction in quantum mechanics, just a single interval, ascending or descending, can be heard each time. Different melodies generated by the same sequence of acoustical qubits can be then perceived.
... The starting point of the theoretical approach of Blutner (2017), Blutner and beim Graben (2020), beim Graben and Blutner (2017), and beim Graben and Blutner (2019) are experimental findings on static and dynamic tonal attraction (Krumhansl and Shepard 1979;Krumhansl and Kessler 1982;Woolhouse 2009;Woolhouse and Cross 2010). These are music-psychological priming experiments where subjects are asked to rate how well probe tone pitches fit into an earlier presented priming context. ...
... Choosing several polynomial deformations γ , beim Graben and Blutner (2019) were able to fit both static (Krumhansl and Kessler 1982) and dynamic (Woolhouse 2009) tonal attraction data. Moreover, Blutner and beim Graben (2020) have shown how different deformations can be ubiquitously unified as local gauge symmetries. ...
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.
... For instance, Steve Larson [8] analyzed musical structures as the result of shaping forces. This metaphorical approach, with an emphasis on tonal relationships, was later mathematically formalized by Peter beim Graben and Reinhard Blutner [4]. This approach is probably more focused on the "reasons" behind specific structures, while the non-Markovian and gesture analysis is inherently descriptive. ...
... Another analogy between the world of quantum and of music had been made already in 1947 by the Nobel prize Dennis Gabor, who used wavelets and quantum formalism to describe the basics of acoustics [30]. More recent works are about synthesizers based on quantum measurements [18]; rules in tonal music analyzed in terms of quantum forces [9]; melodies and notes studied in a quantized way and chords as superpositions of states [29]; music cognition studies with quantum formalism [8]; quantum-based systems [64]; criteria to measure the degree of memory of quantum states adapted to musical structures [54], and voice analyzed with quantum measures [76], design and ambiguity in the arts investigated in light of the quantum paradigm [87]. A collection of quantum perspectives on humanities is presented in the book edited by Eduardo R. Miranda [65]. ...
Movements of robots in a swarm can be mapped to sounds, highlighting the group behavior through the coordinated and simultaneous variations of musical parameters across time. The vice versa is also possible: sound parameters can be mapped to robotic motion parameters, giving instructions through sound. In this article, we first develop a theoretical framework to relate musical parameters such as pitch, timbre, loudness, and articulation (for each time) with robotic parameters such as position, identity, motor status, and sensor status. We propose a definition of musical spaces as Hilbert spaces, and musical paths between parameters as elements of bigroupoids, generalizing existing conceptions of musical spaces. The use of Hilbert spaces allows us to build up quantum representations of musical states, inheriting quantum computing resources, already used for robotic swarms. We present the theoretical framework and then some case studies as toy examples. In particular, we discuss a 2D video and matrix simulation with two robo-caterpillars; a 2D simulation of 10 robo-ants with Webots; a 3D simulation of three robo-fish in an underwater search&rescue mission.
... MIDI format songs make it possible to investigate the musical structure of pieces of music. Graben and Blutner worked on music cognition by symbolic analysis and Xiaobin proposed a music evaluation system based on symbolic analysis from MIDI files [20,58]. Ferreira and Whitehead designed a generative deep learning model to compose and classify sentiment with MIDI format [16]. ...
Listening to music can evoke different emotions in humans. Music emotion recognition (MER) can predict a person’s emotions before listening to a song. However, there are three problems with MER studies. First, the brain is the seat of music perception, but the simulation of MER based on the brain’s limbic system has not been examined so far. Secondly, although the effect of individual differences is recognized on the perception and induction of music emotion in the literature, less attention has been paid to the personalization of the model. Finally, most previous studies have emphasized the classification of music pieces into emotional groups, while often a piece of music creates several emotions with different values. The purpose of the present study is to introduce an optimized model of brain emotional learning (BEL) which is combined with Thayer’s psychological model to predict the quantitative value of all emotions that hat would reach a specific person by listening to a new piece of music. The proposed model consists of 12 emotional parts that work in parallel where each part is responsible for evaluating one Thayer’s specific emotion. Four neural areas of the emotional brain are simulated for each part. The input signal is adjusted using Thayer’s dimensions and a fuzzy system. The average of the results obtained with the proposed model were: R2 = 0.69 for arousal, R2 = 0.36 for valence, and MSE = 0.051, which was better and faster than the multilayer network models and even the original BEL model for all emotions.
... In case of analyzing the phenomenon of 'tonal attraction', a number of different empirical observations can be introduced in terms of a musical gauge field based on the internal symmetry group SU(2) (beimGraben & Blutner, 2019). ...
Abstract: The idea of complementarity is one of the key concepts of quantum mechanics. Yet, the idea was originally developed in William James' psychology of consciousness. Recently, it was reapplied to the humanities and forms one of the pillars of modern quantum cognition. I will explain two different concepts of complementarity: Niels Bohr's ontic conception, and Werner Heisenberg's epistemic conception. Furthermore, I will give an independent motivation of the epistemic conception based on the so-called operational interpretation of quantum theory, which has powerfully been applied in the domain of quantum cognition. Finally, I will give examples illustrating the potency of complementarity in the domains of bounded rationality and survey research. Concerning the broad topic of consciousness, I will focus on the psychological aspects of awareness. This closes the circle spanning complementarity, quantum cognition, the operational interpretation, and consciousness.
... On the other hand, some common observed properties of human cognition and quantum mechanics (superposition, non-classical probability) have given universal value to the quantum-theoretical formalism to explain cognitive acts [11], including actions of human creation, such as music. The explanatory power of a quantum approach to music cognition has been demonstrated to describe tonal attraction phenomena in terms of metaphorical forces [41,42]. The theory of open quantum systems has been applied to music to describe the memory properties (non-Markovianity) of different scores [43]. ...
Concepts and formalism from acoustics are often used to exemplify quantum mechanics. Conversely, quantum mechanics could be used to achieve a new perspective on acoustics, as shown by Gabor studies. Here, we focus in particular on the study of human voice, considered as a probe to investigate the world of sounds. We present a theoretical framework that is based on observables of vocal production, and on some measurement apparati that can be used both for analysis and synthesis. In analogy to the description of spin states of a particle, the quantum-mechanical formalism is used to describe the relations between the fundamental states associated with phonetic labels such as phonation, turbulence, and supraglottal myoelastic vibrations. The intermingling of these states, and their temporal evolution, can still be interpreted in the Fourier/Gabor plane, and effective extractors can be implemented. The bases for a quantum vocal theory of sound, with implications in sound analysis and design, are presented.
... The literature includes a few works about Physics and Psychology [45,46,47], which are monographic essays where the authors hold conceptual discussions about the possibilities of using Physics to explain psychological phenomena. Some of them develop models, for instance, that use quantum theory to approach brain-related phenomena [48,49,50,51,52,53]. Another group of works use Physics concepts to understand psychological data, such as those applied to understand cognition experiments [39,40]. ...
An individual's reaction time data to visual stimuli have usually been represented in Experimental Psychology by means of an ex-Gaussian function (EGF). In most previous works, researchers have mainly aimed at finding a meaning for the three parameters of the EGF in relation to psychological phenomena. We will focus on interpreting the reaction times (RT) of a group of individuals rather than a single person's RT which is relevant for the different contexts of social sciences. In doing so, the same model as for the Ideal Gases (IG) emerges from the experimental reaction time data. We show that the law governing the experimental RT of a group of individuals is the same as the law governing the dynamics of an inanimate system, namely, a system of non-interacting particles (IG). Both systems are characterized by a collective parameter which is k_BT for the system of particles and what we have called life span parameter for the system of brains. The dynamics of both systems is driven by the interaction with their respective thermostats, which are characterized by a temperature in the system of particles and by a thermostat-like entity, that we have called time driver for the group of individuals. Similarly, we come across a Maxwell-Boltzmann-type distribution for the system of brains which provides a more complete characterization of the collective time response than has ever been provided before. Another step taken is that now we are able to know about the behavior of a single individual in relation to the coetaneous group to which they belong and through the application of a physical law. This leads to a new entropy-based methodology for the classification of the individuals forming the system which emerges from the physical law governing the system of brains. To our knowledge, this is the first work reporting on the emergence of a physical theory (IG) from human RT experimental data.
... Despite its deep roots in the physics of the twentieth century, the sound field has not yet embraced the quantum signal processing framework [24] to seek practical solutions to sound scene representation, separation and analysis. A quantum approach to music cognition has not been proposed until very recently [25], when its explanatory power has been demonstrated to describe tonal attraction phenomena in terms of metaphorical forces. The theory of open quantum systems has been applied to music to describe the memory properties (non-Markovianity) of different scores [26]. ...
Concepts and formalism from acoustics are often used to exemplify quantum mechanics. Conversely, quantum mechanics could be used to achieve a new perspective on acoustics, as shown by Gabor studies. Here, we focus in particular on the study of human voice, considered as a probe to investigate the world of sounds. We present a theoretical framework that is based on observables of vocal production, and on some measurement apparati that can be used both for analysis and synthesis. In analogy to the description of spin states of a particle, the quantum-mechanical formalism is used to describe the relations between the fundamental states associated with phonetic labels such as phonation, turbulence, and slow myoelastic vibrations. The intermingling of these states, and their temporal evolution, can still be interpreted in the Fourier/Gabor plane, and effective extractors can be implemented. This would constitute the basis for a Quantum Vocal Theory of sound, with implications in sound analysis and design.
... Further, there are good arguments to assume that the merging mechanism (discrete convolution) is not learnt but innate, since it appears in many independent cognitive domains (including language and perception). Finally, let me make a remark concerning recent approaches of applying the field of quantum cognition to cognitive musicology ( Blutner, 2015Blutner, , 2016beim Graben & Blutner, 2017, 2018). Based on the lead of Guerino Mazzola ( Mazzola, 1990Mazzola, , 2002), who was the first to see the analogy between physics and music in connection with the existence of symmetries and musical forces, this approach develops an account of tonal attraction in particular and of musical forces in general. ...
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
To arbitrate theories of consciousness, scientists need to understand mathematical structures of quality of consciousness, or qualia. The dominant view regards qualia as points in a dimensional space. This view implicitly assumes that qualia can be measured without any effect on them. This contrasts with intuitions and empirical findings to show that by means of internal attention qualia can change when they are measured. What is a proper mathematical structure for entities that are affected by the act of measurement? Here we propose the mathematical structure used in quantum theory, in which we consider qualia as “observables” (i.e., entities that can, in principle, be observed), sensory inputs and internal attention as “states” that specify the context that a measurement takes place, and “measurement outcomes” with probabilities that qualia observables take particular values. Based on this mathematical structure, the Quantum-like Qualia (QQ) hypothesis proposes that qualia observables interact with the world, as if through an interface of sensory inputs and internal attention. We argue that this qualia-interface-world scheme has the same mathematical structure as observables-states-environment in quantum theory. Moreover, within this structure, the concept of a “measurement instrument” in quantum theory can precisely model how measurements affect qualia observables and states. We argue that QQ naturally explains known properties of qualia and predicts that qualia are sometimes indeterminate. Such predictions can be empirically determined by the presence of order effects or violations of Bell inequalities. Confirmation of such predictions substantiates our overarching claim that the mathematical structure of QQ will offer novel insights into the nature of consciousness.
The idea of complementarity is one of the key concepts of quantum mechanics. Yet, the idea was originally developed in William James’ psychology of consciousness. Recently, it was re-applied to the humanities and forms one of the pillars of modern quantum cognition. I will explain two different concepts of complementarity: Niels Bohr’s ontic conception and Werner Heisenberg’s epistemic conception. Furthermore, I will give an independent motivation of the epistemic conception based on the so-called operational interpretation of quantum theory, which has powerfully been applied in the domain of quantum cognition. Finally, I will give examples illustrating the potency of complementarity in the domains of bounded rationality and survey research. Concerning the broad topic of consciousness, I will focus on the psychological aspects of awareness. This closes the circle spanning complementarity, quantum cognition, the operational interpretation, and consciousness.
Prevailing theories of genre, derived primarily from literary and musical scholarship, differ in characteristics they ascribe to genre itself. Here, the temporally dynamic and culturally contingent nature of genre informs a computational framework that is reducible to extant theories of genre and connected to psychological theories of perceptual categorization. This framework, called genredynamics, interprets genres as perceptual categories in a space defined by aesthetic and sociocultural variables, and characterizes the behaviour and structure of genres using concepts from differential topology. Its existence demonstrates that disparate theoretical approaches to genre can be unified and implies that genre is best understood as both a psychological and musicological phenomenon. Classifications' temporal fluidity and incorporating sociocultural variables alongside sensory ones are necessary for this framework to be generalizable. Together, these theoretical results have broad implications for potential applications of genre theory, including the study of mental representations, social and cultural psychology, and cognition.
Music is an essential component of a promotional video since it helps to establish a brand's or entity's identity. Music composition and production, on the other hand, is quite costly. The expense of engaging a competent team capable of creating distinctive music for your firm could be prohibitively expensive. In the last decade, artificial intelligence has accomplished feats previously unimaginable to humanity. Artificial intelligence can be a lifesaver, not only in terms of the amount of money a company would have to spend on creating their own unique music but also in terms of the amount of time and work required on the firm's part. A web-based platform that can be accessed from anywhere in the world would help the product obtain customers without regard to geography. AI algorithms can be taught to recognize which sound combinations produce a pleasing melody (or music). Multiple machine learning algorithms can be used to accomplish this.
We re-examine the long-held postulate that there are two modes of thought, and develop a more fine-grained analysis of how different modes of thought affect conceptual change. We suggest that cognitive development entails the fine-tuning of three dimensions of thought: abstractness, divergence, and context-specificity. Using a quantum cognition modeling approach, we show how these three variables differ, and explain why they would have a distinctively different impacts on thought processes and mental contents. We suggest that, through simultaneous manipulation of all three variables, one spontaneously, and on an ongoing basis, tailors one's mode of thought to the demands of the current situation. The paper concludes with an analysis based on results from an earlier study of children's mental models of the shape of the Earth. The example illustrates how, through reiterated transition between mental states using these three variables, thought processes unfold, and conceptual change ensues. While this example concerns children, the approach applies more broadly to adults as well as children.
Motivated through recent applications of quantum theory to the music-theoretical conceptualisation of tonal attraction, the paper recapitulates basic facts about quantum wave functions over the finite configuration space , and proposes a particular musical application.After an introduction of position and momentum operators, the Fourier transform as well as the translation and ondulation operators, particular attention is plaid to the Quantum Harmonic Oscillator via its Hamilton operator and its eigenstates. In this setup the time development of chosen wave functions is applied to the control of moving sound sources in a Spatialisation scenario.KeywordsQuantum theoryMusic theoryPitch class profiles
Mathematics can help analyze the arts and inspire new artwork. Mathematics can also help make transformations from one artistic medium to another, considering exceptions and choices, as well as artists' individual and unique contributions. We propose a method based on diagrammatic thinking and quantum formalism. We exploit decompositions of complex forms into a set of simple shapes, discretization of complex images, and Dirac notation, imagining a world of “prototypes” that can be connected to obtain a fine or coarse-graining approximation of a given visual image. Visual prototypes are exchanged with auditory ones, and the information (position, size) characterizing visual prototypes is connected with the information (onset, duration, loudness, pitch range) characterizing auditory prototypes. The topic is contextualized within a philosophical debate (discreteness and comparison of apparently unrelated objects), it develops through mathematical formalism, and it leads to programming, to spark interdisciplinary thinking and ignite creativity within STEAM.
Mathematics can help analyze the arts and inspire new artwork. Mathematics can also help make transformations from one artistic medium to another, considering exceptions and choices, as well as artists' individual and unique contributions. We propose a method based on diagrammatic thinking and quantum formalism. We exploit decompositions of complex forms into a set of simple shapes, discretization of complex images, and Dirac notation, imagining a world of "prototypes" that can be connected to obtain a fine or coarse-graining approximation of a given visual image. Visual prototypes are exchanged with auditory ones, and the information (position, size) characterizing visual prototypes is connected with the information (onset, duration, loudness, pitch range) characterizing auditory prototypes. The topic is contextualized within a philosophical debate (discreteness and comparison of apparently unrelated objects), it develops through mathematical formalism, and it leads to programming, to spark interdisciplinary thinking and ignite creativity within STEAM.
Musical gestures connect the symbolic layer of the score to the physical layer of sound. We focus here on the mathematical theory of musical gestures, and we propose its generalization to include braids and knots. In this way, it is possible to extend the formalism to cover more case studies, especially regarding conducting gestures. Moreover, recent developments involving comparisons and similarities between gestures of orchestral musicians can be contextualized in the frame of braided monoidal categories. Because knots and braids can be applied to both music and biology (it is the case of knotted proteins for example), we end the article with a new musical rendition of DNA.
THE FULL-TEXT ARTICLE CAN BE FOUND HERE (with the updated title: Knots, Music and DNA): http://jcms.org.uk/issues/Vol2Issue2/knots-music-and-dna/article.html
We consider several puzzles of bounded rationality. These include the Allais- and Ellsberg paradox, the disjunction effect, and related puzzles. We argue that the present account of quantum cognition—taking quantum probabilities rather than classical probabilities—can give a more systematic description of these puzzles than the alternate treatments in the traditional frameworks of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always provide a deeper understanding and a true explanation of these puzzles. One reason is that quantum approaches introduce additional parameters which possibly can be fitted to empirical data but which do not necessarily explain them.Hence, the phenomenological research
has to be augmented by responding to deeper foundational issues. In this article, we make the general distinction between foundational and phenomenological research programs, explaining the foundational issue of quantum cognition from the perspective of operational realism. This framework is motivated by assuming partial Boolean algebras (describing particular perspectives). They are combined into a uniform system (i.e. orthomodular lattice) via a mechanism preventing the simultaneous realization of perspectives. Gleason’s theorem then automatically leads to a distinction between probabilities that are defined by pure states and probabilities arising from the statistical mixture of pure states. This formal distinction relates to the conceptual distinction between risk and ignorance. Another outcome identifies quantum aspects in dynamic macro-systems using the framework of symbolic dynamics. Finally, we discuss several ideas that are useful for justifying complementarity in cognitive systems.
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.
THIS PAPER UNDERTAKES A COMPARATIVE STUDY OF concepts and visual representations of hierarchical aspects of musical structure. After consideration of the rhythmic components of grouping and meter, the discussion turns to pitch-event hierarchies and the tonal hierarchy (or pitch space). Contrasting notations are evaluated in terms of the efficacy of the concepts they exemplify.
In this paper, we introduce a small family of novel bottom-up (sensory) models of the Krumhansl and Kessler (1982) probe tone data. The models are based on the spectral pitch class similarities between all twelve pitch classes and the tonic degree and tonic triad. Cross-validation tests of a wide selection of models show ours to have amongst the highest fits to the data. We then extend one of our models to predict the tonics of a variety of different scales such as the harmonic minor, melodic minor, and harmonic major. The model produces sensible predictions for these scales. Furthermore , we also predict the tonics of a small selection of microtonal scales—scales that do not form part of any musical culture. These latter predictions may be tested when suitable empirical data have been collected.
We consider several puzzles of bounded rationality. These include the Allais‐ and Ellsberg paradox, the disjunction effect, and related puzzles. We argue that the present account of quantum cognition – taking quantum probabilities rather than classical probabilities – can give a more systematic description of these puzzles than the alternate treatments in the traditional frameworks of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always provide a deeper understanding and a true explanation of these puzzles. One reason is that quantum approaches introduce additional parameters which possibly can be fitted to empirical data but which do not necessarily explain them. Hence, the phenomenological research has to be augmented by responding to deeper foundational issues. In this article, we make the general distinction between foundational and phenomenological research programs, explaining the foundational issue of quantum cognition from the perspective of operational realism. This framework is motivated by assuming partial Boolean algebras (describing particular perspectives). They are combined into a uniform system (i.e. orthomodular lattice) via a mechanism preventing the simultaneous realization of perspectives. Gleason's theorem then automatically leads to a distinction between probabilities that are defined by pure states and probabilities arising from the statistical mixture of pure states. This formal distinction relates to the conceptual distinction between risk and ignorance. Another outcome identifies quantum aspects in dynamic macro‐systems using the framework of symbolic dynamics. Finally, we discuss several ideas that are useful for justifying complementarity in cognitive systems.
The doubly circular relations of the major and minor keys based on all twelve pitch-classes can be depicted in toroidal models. We demonstrate a convergence of deriva-tions from the different bases of conventional harmonic theory and recent experi-ments in music psychology. We present a formalization of the music-theoretical derivation from Gottfried Weber's 1817 chart of tone-centers by using a topographic ordering map. We find the results to be consistent with Krumhansl and Kessler's 1982 visualization of perceptual ratings. Tonal Theory for the Digital Age (Computing in Musicology 15, 2007), 73–98.
The prolongational component in A Generative Theory of Tonal Music assigns tensing and relaxing patterns to tonal sequences but does not adequately describe degrees of harmonic and melodic tension. This paper offers solutions to the problem, first by adapting the distance algorithm from the theory of tonal pitch space for the purpose of quantifying sequential and hierarchical harmonic tension. The method is illustrated for the beginning of the Mozart Sonata, K. 282, with emphasis on the hierarchical approach. The paper then turns to melodic tension in the context of the anchoring of dissonance. Interrelated attraction algorithms are proposed that incorporate the factors of stability, proximity, and directed motion. A distinction is developed between the tension of distance and the tension of attraction. The attraction and distance algorithms are combined in a view of harmony as voice leading, leading to a second analysis of the opening phrase of the Mozart in terms of voiceleading motion. Connections with recent theoretical and psychological work are discussed.
psychoacoustic theories of dissonance often follow Helmholtz and attribute it to partials (fundamental frequencies or overtones) near enough in frequency to affect the same region of the basilar membrane and therefore to cause roughness, i.e., rapid beating. In contrast, tonal theories attribute dissonance to violations of harmonic principles embodied in Western music. We propose a dual-process theory that embeds roughness within tonal principles. The theory predicts the robust increasing trend in the dissonance of triads: major < minor < diminished < augmented. Previous experiments used too few chords for a comprehensive test of the theory, and so Experiment 1 examined the rated dissonance of all 55 possible three-note chords, and Experiment 2 examined a representative sample of 48 of the possible four-note chords. The participants' ratings concurred reliably and corroborated the dual-process theory. Experiment 3 showed that, as the theory predicts, consonant chords are rated as less dissonant when they occur in a tonal sequence (the cycle of fifths) than in a random sequence, whereas this manipulation has no reliable effect on dissonant chords outside common musical practice.
Major and minor triads emerged in western music in the 13th to 15th centuries. From the 15th to the 17th centuries, they increasingly appeared as final sonorities. In the 17th century, music-theoretical concepts of sonority, root, and inversion emerged. I propose that since then, the primary perceptual reference in tonal music has been the tonic triad sonority (not the tonic tone or chroma) in an experiential (not physical or notational) representation. This thesis is consistent with the correlation between the key profiles of Krumhansl and Kessler (1982; here called chroma stability profiles) and the chroma salience profiles of tonic triads (after Parncutt, 1988). Chroma stability profiles also correlate with chroma prevalence profiles (of notes in the score), suggesting an implication-realization relationship between the chroma prevalence profile of a passage and the chroma salience profile of its tonic triad. Convergent evidence from psychoacoustics, music psychology, the history of composition, and the history of music theory suggests that the chroma salience profile of the tonic triad guided the historical emergence of major-minor tonality and continues to influence its perception today.
No other study has had as great an impact on the development of the similarity literature as that of Tversky (1977), which provided compelling demonstrations against all the fundamental assumptions of the popular, and extensively employed, geometric similarity models. Notably, similarity judgments were shown to violate symmetry and the triangle inequality and also be subject to context effects, so that the same pair of items would be rated differently, depending on the presence of other items. Quantum theory provides a generalized geometric approach to similarity and can address several of Tversky's main findings. Similarity is modeled as quantum probability, so that asymmetries emerge as order effects, and the triangle equality violations and the diagnosticity effect can be related to the context-dependent properties of quantum probability. We so demonstrate the promise of the quantum approach for similarity and discuss the implications for representation theory in general. (PsycINFO Database Record (c) 2013 APA, all rights reserved).
The analyses presented in this paper use interval-cycle profiles to show that there is a statistical link between interval cycles and Krumhansl and Kessler's (1982) major- and minor-key
tonal hierarchies. An interval cycle is the minimum number of additive iterations of an interval that are required for the
original pitch classes to be restated—a formal property that we hypothesize leads to perceptual grouping referred to as interval-cycle proximity. The interval cycles that best explain the extent to which any chord's interval-cycle profile correlates with the tonal hierarchies are found to be those belonging to dominant harmonies, particularly the dominant seventh.
The results of the analyses and the interval-cycle proximity hypothesis are discussed with respect to Browne's (1981) rare-interval hypothesis and Brower's (2000) musical pattern mapping schema.
We explored how musical culture shapes one's listening experience. Western participants heard a series of tones drawn from either the Western major mode (culturally familiar) or the Indian thaat Bhairav (culturally unfamiliar) and then heard a test tone. They made a speeded judgment about whether the test tone was present in the prior series of tones. Interactions between mode (Western or Indian) and test tone type (congruous or incongruous) reflect the utilization of Western modal knowledge to make judgments about the test tones. False alarm rates were higher for test tones congruent with the major mode than for test tones congruent with Bhairav. In contrast, false alarm rates were lower for test tones incongruent with the major mode than for test tones incongruent with Bhairav. These findings suggest that one's internalized cultural knowledge may drive musical expectancies when listening to music of an unfamiliar modal system.
The perception of consonance/dissonance of musical harmonies is strongly
correlated to their periodicity. This is shown in this article by consistently
applying recent results from psychophysics and neuroacoustics, namely that the
just noticeable difference of human pitch perception is about 1% for the
musically important low frequency range and that periodicities of complex
chords can be detected in the human brain. The presented results correlate
significantly to empirical investigations on the perception of chords. Even for
scales, plausible results are obtained. For example, all classical church modes
appear in the front ranks of all theoretically possible seven-tone scales.
Musical compositions could be characterized by a certain degree of memory,
that takes into account repetitions and similarity of sequences of pitches,
durations and intensities (the patterns). The higher the quantity of
variations, the lower the degree of memory. This degree has never
quantitatively been defined and measured. In physics, mathematical tools to
quantify memory (defined as non-Markovianity) in quantum systems have been
developed. The aim of this paper is to extend these mathematical tools to
music, defining a general method to measure the degree of memory in musical
compositions. Applications to some musical scores give results that agree with
the expectations.
Classical (Bayesian) probability (CP) theory has led to an influential research tradition for modeling cognitive processes. Cognitive scientists have been trained to work with CP principles for so long that it is hard even to imagine alternative ways to formalize probabilities. However, in physics, quantum probability (QP) theory has been the dominant probabilistic approach for nearly 100 years. Could QP theory provide us with any advantages in cognitive modeling as well? Note first that both CP and QP theory share the fundamental assumption that it is possible to model cognition on the basis of formal, probabilistic principles. But why consider a QP approach? The answers are that (1) there are many well-established empirical findings (e.g., from the influential Tversky, Kahneman research tradition) that are hard to reconcile with CP principles; and (2) these same findings have natural and straightforward explanations with quantum principles. In QP theory, probabilistic assessment is often strongly context- and order-dependent, individual states can be superposition states (that are impossible to associate with specific values), and composite systems can be entangled (they cannot be decomposed into their subsystems). All these characteristics appear perplexing from a classical perspective. However, our thesis is that they provide a more accurate and powerful account of certain cognitive processes. We first introduce QP theory and illustrate its application with psychological examples. We then review empirical findings that motivate the use of quantum theory in cognitive theory, but also discuss ways in which QP and CP theories converge. Finally, we consider the implications of a QP theory approach to cognition for human rationality.
The paper presents fMRI results from experiments of subjects listening to musical stimuli. In this study we examine the neural correlates of tonality by presenting a set of stimuli with key changes of different distances along the circle-of-fifths, along with atonal control stimuli. Results are presented using both conventional statistical analysis across subjects, together with experiments using support vector machines. We find that a number of areas are significantly more active for tonal than atonal processing, and further that the response is significantly stronger in some of these areas for more distant key changes.
Models of the perceived distance between pairs of pitch collections are a core component of broader models of music cognition. Numerous distance measures have been proposed, including voice-leading, psychoacoustic, and pitch and interval class distances; but, so far, there has been no attempt to bind these different measures into a single mathematical or conceptual framework, nor to incorporate the uncertain or probabilistic nature of pitch perception.
This paper embeds pitch collections in expectation tensors and shows how metrics between such tensors can model their perceived dissimilarity. Expectation tensors indicate the expected number of tones, ordered pairs of tones, ordered triples of tones, etc., that are heard as having any given pitch, dyad of pitches, triad of pitches, etc.. The pitches can be either absolute or relative (in which case the tensors are invariant with respect to transposition). Examples are given to show how the metrics accord with musical intuition.
One of the most pervasive structural principles found in music historically and cross-culturally is a hierarchy of tones.
Certain tones serve as reference pitches; they are stable, repeated frequently, are emphasized rhythmically, and appear at
structurally important positions in musical phrases. The details of the hierarchies differ across styles and cultures. Variation
occurs in the particular intervals formed by pitches in the musical scale and the hierarchical levels assigned to pitches
within the scale. This variability suggests that an explanation for how these hierarchies are formed cannot be derived from
invariant acoustic facts, such as the harmonic structure (overtones) of complex tones. Rather, the evidence increasingly suggests
that these hierarchies are products of cognition and, moreover, that they rely on fundamental psychological principles shared
by other domains of perception and cognition.
This book builds on and in many ways completes the project of Fred Lerdahl and Ray Jackendoff's influential A Generative Theory of Tonal Music. Like the earlier volume, this book is both a music-theoretic treatise and a contribution to the cognitive science of music. After presenting some modifications to Lerdahl and Jackendoff's original framework, the book develops a quantitative model of listeners' intuitions of the relative distances of pitches, chords, and regions from a given tonic. The model is used to derive prolongational structure, trace paths through pitch space at multiple prolongational levels, and compute patterns of tonal tension and attraction as musical events unfold. The consideration of pitch-space paths illuminates issues of musical narrative, and the treatment of tonal tension and attraction provides a technical basis for studies of musical expectation and expression. These investigations lead to a fresh theory of tonal function and reveal an underlying parallel between tonal and metrical structures. Later portions of the book apply these ideas to highly chromatic tonal as well as atonal music. In response to stylistic differences, the shape of pitch space changes and psychoacoustic features become increasingly important, while underlying features of the theory remain constant, reflecting unvarying features of the musical mind. The theory is illustrated throughout by analyses of music from Bach to Schoenberg, and frequent connections are made to the music-theoretic and psychological literature.
A context-independent model of tonal attraction is presented based on the formal musical property of interval cycles. An interval cycle is the minimum number of additive iterations of an interval that are required for the original pitch classes to be re-stated. Interval cycles are conjectured to give rise to an abstract grouping property, interval cycle proximity, which in turn is responsible for tonal attraction. The model was tested using a probe tone experiment requiring subjects to rate the probe for strength of attraction/resolution with respect to a preced- ing context chord. The results, displayed as `attraction profiles', agreed with the predictions of the model, and showed that even diatonic chords, such as dominant sevenths, can be heard chromatically. The results are discussed in relation to examples of real music and previous research within the field.
Listeners rated test tones falling in the octave range from middle to high C according to how well each completed a diatonic C major scale played in an adjacent octave just before the final test tone. Ratings were well explained in terms of three factors. The factors were distance in pitch height from the context tones, octave equivalence, and the following hierarchy of tonal functions: tonic tone, other tones of the major triad chord, other tones of a diatonic scale, and the nondiatonic tones. In these ratings, pitch height was more prominent for less musical listeners or with less musical (sinusoidal) tones, whereas octave equivalence and the tonal hierarchy prevailed for musical listeners, especially with harmonically richer tones. Ratings for quarter tones interpolated halfway between the halftone steps of the standard chromatic scale were approximately the averages of ratings for adjacent chromatic tones, suggesting failure to discriminate tones at this fine level of division.
Investigated the cognitive representation of harmonic and tonal structure in Western music using a tone-profile technique in 2 experiments with 24 undergraduates and community adults. Listeners rated how well single tones (any one of the 12 tones of the chromatic scale) followed a musical element such as a scale, chord, or cadence. Stable rating profiles reflecting the tonal hierarchies in major and minor keys were obtained, which, when intercorrelated and analyzed using multidimensional scaling, produced a 4-dimensional spatial map of the distances between keys. Listeners integrated harmonic functions over multiple chords, developing a sense of key that needed to be re-evaluated as additional chords were sounded. It is suggested that the perceived relations between chords and keys and between different keys are mediated through an internal representation of the hierarchy of tonal functions of single tones in music. (56 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The aim of the present study was the investigation of neural correlates of music processing with fMRI. Chord sequences were presented to the participants, infrequently containing unexpected musical events. These events activated the areas of Broca and Wernicke, the superior temporal sulcus, Heschl's gyrus, both planum polare and planum temporale, as well as the anterior superior insular cortices. Some of these brain structures have previously been shown to be involved in music processing, but the cortical network comprising all these structures has up to now been thought to be domain-specific for language processing. To what extent this network might also be activated by the processing of non-linguistic information has remained unknown. The present fMRI-data reveal that the human brain employs this neuronal network also for the processing of musical information, suggesting that the cortical network known to support language processing is less domain-specific than previously believed.
Western tonal music relies on a formal geometric structure that determines distance relationships within a harmonic or tonal
space. In functional magnetic resonance imaging experiments, we identified an area in the rostromedial prefrontal cortex that
tracks activation in tonal space. Different voxels in this area exhibited selectivity for different keys. Within the same
set of consistently activated voxels, the topography of tonality selectivity rearranged itself across scanning sessions. The
tonality structure was thus maintained as a dynamic topography in cortical areas known to be at a nexus of cognitive, affective,
and mnemonic processing.
Exploring the application of Bayesian probabilistic modeling techniques to musical issues, including the perception of key and meter.
In Music and Probability, David Temperley explores issues in music perception and cognition from a probabilistic perspective. The application of probabilistic ideas to music has been pursued only sporadically over the past four decades, but the time is ripe, Temperley argues, for a reconsideration of how probabilities shape music perception and even music itself. Recent advances in the application of probability theory to other domains of cognitive modeling, coupled with new evidence and theoretical insights about the working of the musical mind, have laid the groundwork for more fruitful investigations. Temperley proposes computational models for two basic cognitive processes, the perception of key and the perception of meter, using techniques of Bayesian probabilistic modeling. Drawing on his own research and surveying recent work by others, Temperley explores a range of further issues in music and probability, including transcription, phrase perception, pattern perception, harmony, improvisation, and musical styles.
Music and Probability—the first full-length book to explore the application of probabilistic techniques to musical issues—includes a concise survey of probability theory, with simple examples and a discussion of its application in other domains. Temperley relies most heavily on a Bayesian approach, which not only allows him to model the perception of meter and tonality but also sheds light on such perceptual processes as error detection, expectation, and pitch identification. Bayesian techniques also provide insights into such subtle and advanced issues as musical ambiguity, tension, and "grammaticality," and lead to interesting and novel predictions about compositional practice and differences between musical styles.
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.
Much of our understanding of human thinking is based on probabilistic models. This innovative book by Jerome R. Busemeyer and Peter D. Bruza argues that, actually, the underlying mathematical structures from quantum theory provide a much better account of human thinking than traditional models. They introduce the foundations for modelling probabilistic-dynamic systems using two aspects of quantum theory. The first, 'contextuality', is a way to understand interference effects found with inferences and decisions under conditions of uncertainty. The second, 'quantum entanglement', allows cognitive phenomena to be modeled in non-reductionist ways. Employing these principles drawn from quantum theory allows us to view human cognition and decision in a totally new light. Introducing the basic principles in an easy-to-follow way, this book does not assume a physics background or a quantum brain and comes complete with a tutorial and fully worked-out applications in important areas of cognition and decision.
Musical performance and composition imply hypergestural transformation from symbolic to physical reality and vice versa. But most scores require movements at infinite physical speed that can only be performed approximately by trained musicians. To formally solve this divide between symbolic notation and physical realization, we introduce complex time (-time) in music. In this way, infinite physical speed is “absorbed” by a finite imaginary speed. Gestures thus comprise thought (in imaginary time) and physical realization (in real time) as a world-sheet motion in space-time, corresponding to ideas from physical string theory. Transformation from imaginary to real time gives us a measure of artistic effort to pass from potentiality of thought to physical realization of artwork. Introducing -time we define a musical kinematics, calculate Euler-Lagrange equations, and, for the case of the elementary gesture of a pianist’s finger, solve corresponding Poisson equations that describe world-sheets which connect symbolic and physical reality.
In this work, Eugene Narmour continues to develop the unique theories of musical perception and cognition first set forth in The Analysis and Cognition of Basic Melodic Structures. The two books together constitute the first comprehensive theory of melody founded on psychological research. Narmour explains the cognitive operations by which listeners assimilate and ultimately encode complex melodic structures, and goes on to show how sixteen melodic archetypes can combine to form some 200 complex structures that, in turn, can chain together in a theoretically infinite number of ways. Of particular importance to music theorists and music historians is Narmour's argument that melodic analysis and formal analysis, though often treated separately, are in fact indissolubly linked. Illustrated with over 250 musical examples, The Analysis and Cognition of Melodic Complexity will also appeal to ethnomusicologists, psychologists, and cognitive scientists.
The first sentence in the preface to Kohler's The Place of Value in a World of Facts proclaims boldly that the purpose of the book is philosophical. It is dedicated to Ralph Barton Perry, and ranges widely over areas which at the time most American psychologists would have feared to tread, perhaps because during their student days they had become infected by Titchener's pontifical proclamation that science has nothing to do with values or by the behaviorists' thumping insistence on facts, facts, facts, nothing but facts. Some twenty years earlier Kohler had published Die physischen Gestalten in Ruhe und im stationiiren Zustand (1920), dedicated to Carl Stumpf, a brilliant work full of new facts which he and the other members of the triumvirate, Wertheimer and Koffka, and their students were bringing to light, all set in relation to a rigid framework of physical field theory. Gestaltpsychologie was already on the way to becoming Gestalttheorie, for Kohler insisted throughout his life that the phenomenal world is for science the only world open to inspection and that the initial data of this world are Gestalten no matter from what angle or branch of science they may be reported.
Steve Larson draws on his 20 years of research in music theory, cognitive linguistics, experimental psychology, and artificial intelligence-as well as his skill as a jazz pianist-to show how the experience of physical motion can shape one's musical experience. Clarifying the roles of analogy, metaphor, grouping, pattern, hierarchy, and emergence in the explanation of musical meaning, Larson explains how listeners hear tonal music through the analogues of physical gravity, magnetism, and inertia. His theory of melodic expectation goes beyond prior theories in predicting complete melodic patterns. Larson elegantly demonstrates how rhythm and meter arise from, and are given meaning by, these same musical forces.
Conducted 4 experiments with 10 undergraduates, each of whom had at least 5 yrs of musical training. Ss judged the similarities between pairs of tones presented in an explicitly tonal context. Results suggest that musical listeners extract a pattern of relationships among tones that is determined by (a) pitch height and chroma, and (b) membership in the major triad chord and the diatonic scale associated with the tonal system of the context. Multidimensional scaling of ratings gave a 3-dimensional conical structure around which the tones were ordered according to pitch height. Results also suggest that, in psychological representation, tones less closely related to tonality are less stable than tones closely related to tonality, and that the representation incorporates the tendency for unstable tones to move toward the more stable tones in time, reflecting the dynamic character of musical tones. In the similarity ratings of the scaling study, tones less related to tonality were judged more similar to tones more related to tonality than the reverse temporal order. Memory performance for nondiatonic tones was less accurate than for diatonic tones, and nondiatonic tones were more often confused with diatonic tones than diatonic tones were confused with nondiatonic tones. Results indicate the tonality-specific nature of the psychological representation and argue that the perception of music depends both on psychoacoustic properties of the tones, and on processes that relate the tones through contact with a well-defined and complex psychological representation of musical pitch. (9 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The question of how tonal structures in music are perceived and represented by the human mind has been approached by multiple disciplines, primarily music theory and cognitive psychology, and more recently, neuroscience. A parsimonious model of tonal space as the surface of a torus has emerged from various types of theoretical considerations and empirical data. Here I provide a brief overview of different vari-ants of a very data-driven approach to modeling tonal space based on self-organizing maps (SOMs), focusing primarily on an ecologically inspired model (Leman and Car-reras 1997) that allows one to project any desired auditory stimulus to the toroidal surface. I illustrate this with examples of the tonal trajectories charted by short chord progressions. Tonal Theory for the Digital Age (Computing in Musicology 15, 2007), 39–50.
This study presents a probabilistic model of melody perception, which infers the key of a melody and also judges the probability of the melody itself. The model uses Bayesian reasoning: For any "surface" pattern and underlying "structure," we can infer the structure maximizing P(structure|surface) based on knowledge of P(surface, structure). The probability of the surface can then be calculated as ∑ P(surface, structure), summed over all structures. In this case, the surface is a pattern of notes; the structure is a key. A generative model is proposed, based on three principles: (a) melodies tend to remain within a narrow pitch range; (b) note-to-note intervals within a melody tend to be small; and (c) notes tend to conform to a distribution (or key profile) that depends on the key. The model is tested in three ways. First, it is tested on its ability to identify the keys of a set of folksong melodies. Second, it is tested on a melodic expectation task in which it must judge the probability of different notes occurring given a prior context; these judgments are compared with perception data from a melodic expectation experiment. Finally, the model is tested on its ability to detect incorrect notes in melodies by assigning them lower probabilities than the original versions.
In this study, we examine through electrophysiological measures three alternative mechanisms underlying musical chord priming: psychoacoustic distance, common parent-key, and distance along the circle of fifths. In contrast with previous behavioral studies, we present complex tones which do not blur the melodic component, we present various chord arrangements, and we focus on nonmusicians. Target major chords, in three different harmonic conditions (1, 2, and 4 steps along the circle of fifths between prime and target chords), elicited two centro-anterior negativities labeled N5E (early) and N5L (late) suggesting a dissociation between an earlier psychoacoustic process based on pitch commonality and proximity and a later cognitive process based on a common parent-key.
This article is a study of melodic expectancy in North Sami yoiks, a style of music quite distinct from Western tonal music. Three different approaches were taken. The first approach was a statistical style analysis of tones in a representative corpus of 18 yoiks. The analysis determined the relative frequencies of tone onsets and two- and three-tone transitions. It also identified style characteristics, such as pentatonic orientation, the presence of two reference pitches, the frequency of large consonant intervals, and a relatively large set of possible melodic continuations. The second approach was a behavioral experiment in which listeners made judgments about melodic continuations. Three groups of listeners participated. One group was from the Sami culture, the second group consisted of Finnish music students who had learned some yoiks, and the third group consisted of Western musicians unfamiliar with yoiks. Expertise was associated with stronger veridical expectations (for the correct next tone) than schematic expectations (based on general style characteristics). Familiarity with the particular yoiks was found to compensate for lack of experience with the musical culture. The third approach simulated melodic expectancy with neural network models of the self-organizing map (SOM) type (Kohonen, T. (1997). Self-organizing maps (2nd ed.). Berlin: Springer). One model was trained on the excerpts of yoiks used in the behavioral experiment including the correct continuation tone, while another was trained with a set of Finnish folk songs and Lutheran hymns. The convergence of the three approaches showed that both listeners and the SOM model are influenced by the statistical distributions of tones and tone sequences. The listeners and SOM models also provided evidence supporting a core set of psychological principles underlying melody formation whose relative weights appear to differ across musical styles.
This review article highlights state-of-the-art functional neuroimaging studies and demonstrates the novel use of music as a tool for the study of human auditory brain structure and function. Music is a unique auditory stimulus with properties that make it a compelling tool with which to study both human behavior and, more specifically, the neural elements involved in the processing of sound. Functional neuroimaging techniques represent a modern and powerful method of investigation into neural structure and functional correlates in the living organism. These methods have demonstrated a close relationship between the neural processing of music and language, both syntactically and semantically. Greater neural activity and increased volume of gray matter in Heschl's gyrus has been associated with musical aptitude. Activation of Broca's area, a region traditionally considered to subserve language, is important in interpreting whether a note is on or off key. The planum temporale shows asymmetries that are associated with the phenomenon of perfect pitch. Functional imaging studies have also demonstrated activation of primitive emotional centers such as ventral striatum, midbrain, amygdala, orbitofrontal cortex, and ventral medial prefrontal cortex in listeners of moving musical passages. In addition, studies of melody and rhythm perception have elucidated mechanisms of hemispheric specialization. These studies show the power of music and functional neuroimaging to provide singularly useful tools for the study of brain structure and function.
Wagner in the round: using interval cycles to model chromatic harmony
- M Woolhouse
Woolhouse, M. (2012). Wagner in the round: using interval cycles to model
chromatic harmony. In E. Cambouropoulos, C. Tsougras, P. Mavromatis, &
K. Pastiadis (Eds.), Proceedings of the 12th international conference on music
perception and cognition and the 8th triennial conference of the european
society for the cognitive sciences of music (pp. 1142-1145).
Computational music science, Cool math for hot music
- G Mazzola
- M Mannone
- Y Pang
Mazzola, G., Mannone, M., & Pang, Y. (2016). Computational music science, Cool
math for hot music. Cham: Springer.
On the sensations of tones
- H Helmholtz
Helmholtz, H. (1877). On the sensations of tones. Dover, New York (NY)
Translated by A. J. Ellis.