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PROBABILISTIC TOPIC MODEL

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A PROBABILISTIC TOPIC MODEL FORUNSUPERVISED LEARNING OF MUSICAL KEY-PROFILESDiane J. Hu and Lawrence K. SaulDepartment of Computer Science and EngineeringUniversity of California, San Diego{dhu,saul}@cs.ucsd.eduABSTRACTWe describe a probabilistic model for learning musical key-profiles from symbolic files of polyphonic, classical mu-sic. Our model is based on Latent Dirichlet Allocation(LDA), a statistical approach for discovering hidden topicsin large corpora of text. In our adaptation of LDA, sym-bolic music files play the role of text documents, groupsof musical notes play the role of words, and musical key-profiles play the role of topics. The topics are discoveredas significant, recurring distributions over twelve neutralpitch-classes. Though discovered automatically, these dis-tributions closely resemble the traditional key-profiles usedto indicate the stability and importance of neutral pitch-classes in the major and minor keys of western music. Un-like earlier approaches based on human judgement, ourmodel learns key-profiles in an unsupervised manner, in-ferring them automatically from a large musical corpus thatcontains no key annotations. We show how these learnedkey-profiles can be used to determine the key of a musicalpiece and track its harmonic modulations. We also showhow the model’s inferences can be used to compare musi-cal pieces based on their harmonic structure.1. INTRODUCTIONMusical composition can be studied as both an artistic andtheoretical endeavor. Though music can express a vastrange of human emotions, ideas, and stories, composersgenerally work within a theoretical framework that is highlystructured and organized. In western tonal music, two im-portant concepts in this framework are the key and the tonic.The key of a musical piece identifies the principal set ofpitches that the composer uses to build its melodies andharmonies. The key also defines the tonic, or the most sta-ble pitch, and its relationship to all of the other pitches inthe key’s pitch set. Though each musical piece is charac-terized by one overall key, the key can be shifted within apiece by a compositional technique known as modulation.Notwithstanding the infinite number of variations possiblePermission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page.c 2009 International Society for Music Information Retrieval.Figure 1. C major (left) and C minor (right) key-profiles proposed by Krumhansl-Kessler (KK), used in theKrumhansl-Schmukler (KS) key-finding algorithm.in music, most pieces can be analyzed in these terms.Musical pieces are most commonly studied by analyz-ing their melodies and harmonies. In any such analysis,the first step is to determine the key. While the key is inprinciple determined by elements of music theory, individ-ual pieces and passages can exhibit complex variations onthese elements. In practice, considerable expertise is re-quired to resolve ambiguities.Many researchers have proposed rule-based systems forautomatic key-finding in symbolic music [2,10,12]. In par-ticular, Krumhansl and Schmuckler (KS) [8] introduced amodel based on “key-profiles”. A key-profile is a twelve-dimensional vector in which each element indicates thestability of a neutral pitch-class relative to the given key.There are 24 key-profiles in total, one for each major andminor key. Using these key profiles, KS proposed a sim-ple method to determine the key of a musical piece orshorter passages within a piece: first, accumulate a twelve-dimensional vector whose elements store the total durationof each pitch-class in a song; second, compute the key-profile that has the highest correlation with this vector. TheKS model uses key-profiles derived from probe tone stud-ies conducted by Krumhansl and Kessler (KK) [9]. Fig-ure 1 shows the KK key profiles for C major and C minor;profiles for other keys are obtained by transposition. In re-cent work [14, 15], these key-profiles have been modifiedto achieve better performance in automatic key-finding.In this paper, we show how to learn musical key-profilesautomatically from the statistics of large music collections.Unlike previous studies, we take a purely data-driven ap-proach that does not depend on extensive prior knowledgeof music or supervision by domain experts. Based on amodel of unsupervised learning, our approach bypasses theneed for manually key-annotated musical pieces, a pro-cess that is both expensive and prone to error. As an ad-ditional benefit, it can also discover correlations in the dataof which the designers of rule-based approaches are un-aware. Since we do not rely on prior knowledge, our modelcan also be applied in a straightforward way to other, non-western genres of music with different tonal systems.Our approach is based on Latent Dirichlet Allocation(LDA) [1], a popular probabilistic model for discoveringlatent semantic topics in large collections of text docu-ments. In LDA, each document is described as a mixtureof topics, and each topic is characterized by its own par-ticular distribution over words. LDA for text is based onthe premise that documents about similar topics containsimilar words. Beyond document modeling, LDA has alsobeen adapted to settings such as image segmentation [5],part-of-speech tagging [6], and collaborative filtering [11].Our variant of LDA for unsupervised learning of key-profiles is based on the premise that musical pieces in thesame key use similar sets of pitches. Roughly speaking,our model treats each song as a “document” and the notesin each beat or half-measure as a “word”. The goal oflearning is to infer harmonic “topics” from the sets of pitchesthat commonly co-occur in musical pieces. These har-monic topics, which we interpret as key-profiles, are ex-pressed as distributions over the twelve neutral pitch-classes.We show how to use these key-profiles for automatickey-finding and similarity ranking of musical pieces. Wenote, however, that our use of key-profiles differs from thatof the KS model. For key-finding, the KS model con-sists of two steps: 1) derive key-profiles and 2) predictkeys using key-profiles. In our model, these steps are nat-urally integrated by the Expectation-Maximization (EM)algorithm [3]. We do not need further heuristics to makekey-finding


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