Tonal structure is in part conveyed by statistical regularities between musical events, and research has shown that computational models reflect tonal structure in music by capturing these regularities in schematic constructs like pitch histograms. Of the few studies that model the acquisition of perceptual learning from musical data, most have employed self-organizing models that learn a topology of static descriptions of musical contexts. Also, the stimuli used to train these models are often symbolic rather than acoustically faithful representations of musical material. In this work we investigate whether sequential predictive models of musical memory (specifically, recurrent neural networks), trained on audio from commercial CD recordings, induce tonal knowledge in a similar manner to listeners (as shown in behavioral studies in music perception). Our experiments indicate that various types of recurrent neural networks produce musical expectations that clearly convey tonal structure. Furthermore, the results imply that although implicit knowledge of tonal structure is a necessary condition for accurate musical expectation, the most accurate predictive models also use other cues beyond the tonal structure of the musical context.
@cite_22: We examined a variety of real-time responses evoked by a single piece of music, the organ Duetto BWV 805 by J S Bach. The primary data came from a concurrent probe-tone method in which the probe tone is sounded continuously with the music. Listeners judged how well the probe tone fit with the music at each point in time. The process was repeated for all probe tones of the chromatic scale. A self-organizing map (SOM) (Kohonen 1997 Self-organizing Maps (Berlin: Springer)) was used to represent the developing and changing sense of key reflected in these judgments. The SOM was trained on the probe-tone profiles for 24 major and minor keys (Krumhansl and Kessler 1982 Psychological Review 89 334^368). Projecting the concurrent probe-tone data onto the map showed changes both in the perceived keys and in their strengths. Two dynamic models of tonality induction were tested. Model 1 is based on pitch class distribu- tions. Model 2 is based on the tone-transition distributions; it tested the idea that the order of tones might provide additional information about tonality. Both models contained dynamic components for characterizing pitch strength and creating pitch memory representations. Both models produced results closely matching those of the concurrent probe-tone data. Finally real- time judgments of tension were measured. Tension correlated with distance away from the predominant key in the direction of keys built on the dominant and supertonic tones, and also correlated with dissonance.
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@cite_21: Listeners' expectations for melodies and harmonies in tonal music are perhaps the most studied aspect of music cognition. Long debated has been whether faster response times (RTs) to more strongly primed events (in a music theoretic sense) are driven by sensory or cognitive mechanisms, such as repetition of sensory information or activation of cognitive schemata that reflect learned tonal knowledge, respectively. We analyzed over 300 stimuli from 7 priming experiments comprising a broad range of musical material, using a model that transforms raw audio signals through a series of plausible physiological and psychological representations spanning a sensory- cognitive continuum. We show that RTs are modeled, in part, by information in periodicity pitch distributions, chroma vectors, and activations of tonal space-a representation on a toroidal surface of the major minor key relationships in Western tonal music. We show that in tonal space, melodies are grouped by their tonal rather than timbral properties, whereas the reverse is true for the periodicity pitch representation. While tonal space variables explained more of the variation in RTs than did periodicity pitch variables, suggesting a greater contribution of cognitive influences to tonal expectation, a stepwise selection model contained variables from both representations and successfully explained the pattern of RTs across stimulus categories in 4 of the 7 experiments. The addition of closure-a cognitive representation of a specific syntactic relationship- succeeded in explaining results from all 7 experiments. We conclude that multiple representational stages along a sensory- cognitive continuum combine to shape tonal expectations in music. © 2014 American Psychological Association.
@cite_1: Tonal music is a highly structured system that is ubiquitous in our cultural environment. We demonstrate the acquisition of implicit knowledge of tonal structure through neural self-organization resulting from mere exposure to simultaneous and sequential combinations of tones. In the process of learning, a network with fundamental neural constraints comes to internalize the essential correlational structure of tonal music. After learning, the network was run through a range of experiments from the literature. The model provides a parsimonious account of a variety of empirical findings dealing with the processing of tone, chord, and key relationships, including relatedness judgments, memory judgments, and expectancies. It also illustrates the plausibility of activation being a unifying mechanism underlying a range of cognitive tasks. Natural environments contain highly structured systems to which we are exposed in everyday life. The human brain internalizes these regularities by passive exposure, and the acquired implicit knowledge influences perception and performance. Aspects of language and music provide two examples of highly structured systems that may be learned in an incidental manner. In each case, there is a paradox. On the one hand, a thorough formal description of the structure has proven to be extremely challenging. On the other hand, native speakers or nonmusician listeners internalize the regularities underlying linguistic or musical structures with apparent ease. A substantial corpus of research has been devoted to the learning process of language, but little has been devoted to the learning of music. The central purpose of the present article is to investigate how implicit knowledge of some basic features of Western musical grammar may be acquired and mentally
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@cite_2: In this paper, a model for tone-center perception is developed. It is based on an auditory model and principles of schema dynamics such as self-organization and association. The auditory module simulates virtual-pitch perception by transforming musical signals into auditory images. The schema- based module involves data-driven long-term learning for the self-organization of a schema for tone-center perception. The focus of this paper is on a retroactive process (called perceptual interpretation) by which the sense of tone center is adjusted according to a reconsideration of preceding perceptions in view of new contextual evidence within the schema. To this purpose, a metaphor is introduced, in which perceptual interpretation is described as the movement of a snaillike object in an attractor space. Additionally, the mathematical details of the model are presented, and the results of computer simulations are discussed.
@cite_20: The perception of tonal structure in music seems to be rooted both in low-level perceptual mechanisms and in enculturation, the latter accounting for cross-cultural differences in perceived tonal structure. Unsupervised machine learning methods are a powerful tool for studying how musical concepts may emerge from exposure to music. In this paper, we investigate to what degree tonal structure can be learned from musical data by unsupervised training of a Restricted Boltzmann Machine, a generative stochastic neural network. We show that even based on a limited set of musical data, the model learns several aspects of tonal structure. Firstly, the model learns an organization of musical material from different keys that conveys the topology of the circle of fifths (CoF). Although such a topology can be learned using principal component analysis (PCA) when using pitch-only representations, we found that using a pitch-duration representation impedes the extraction of the CoF topology much more for PCA than for the RBM. Furthermore, we replicate probe-tone experiments by Krumhansl and Shepard, measuring the organization of tones within a key in human perception. We find that the responses of the RBM share qualitative characteristics with those of both trained and untrained listeners.
Another limitation of most long-term memory models for tonal learning is that they work with stimuli that are reduced in one or more ways. For example, the input may consist of discrete representations of tones such as MIDI note numbers @cite_20 , pitch classes @cite_1 , artificial harmonic representations @cite_0 , or of artificial harmonic sounds such as Shepard tones @cite_2 . Furthermore, the musical material that a model is exposed to may be limited to monophonic melodic lines @cite_20 , sets of chords or harmonic cadences @cite_2 , or even a set of probe-tone profiles @cite_22 . A notable exception to this is @cite_36 , which uses an audio recording of Bach's Well Tempered Clavier (WTC), performed on a harpsichord, to train a SOM by converting the acoustic signal to . The work of @cite_21 also takes an ecological approach by using real audio and plausible psychological representations, with multiple representations along the sensory-cognitive spectrum, to better account for human tonal expectation.