Generative statistical models of chord sequences play crucial roles in music processing. To capture syntactic similarities among certain chords (e.g. in C major key, between G and G7 and between F and Dm), we study hidden Markov models and probabilistic context-free grammar models with latent variables describing syntactic categories of chord symbols and their unsupervised learning techniques for inducing the latent grammar from data. Surprisingly, we find that these models often outperform conventional Markov models in predictive power, and the self-emergent categories often correspond to traditional harmonic functions. This implies the need for chord categories in harmony models from the informatics perspective.
In Ref. @cite_0 , unsupervised learning was applied to a variant of PCFG models for chord symbols in which nonterminals describe harmonic functions. In this study, the nonterminals and the production rules were manually determined in accordance with existing data of harmony analysis and only the probabilities were learned unsupervisedly. Our PCFG models can be considered as extensions of this model as we allow larger numbers of nonterminals and they have no prior labels or restricted production rules.
@cite_0
2152561660 2394683507 2165446401 2150415460
This paper defines a generative probabilistic model of parse trees, which we call PCFG-LA. This model is an extension of PCFG in which non-terminal symbols are augmented with latent variables. Fine-grained CFG rules are automatically induced from a parsed corpus by training a PCFG-LA model using an EM-algorithm. Because exact parsing with a PCFG-LA is NP-hard, several approximations are described and empirically compared. In experiments using the Penn WSJ corpus, our automatically trained model gave a performance of 86.6 (F1, sentences ≤ 40 words), which is comparable to that of an unlexicalized PCFG parser created using extensive manual feature selection. 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. An intensional model for the programming language PCF is described in which the types of PCF are interpreted by games and the terms by certain history-free strategies. This model is shown to capture definability in PCF. More precisely, every compact strategy in the model is definable in a certain simple extension of PCF. We then introduce an intrinsic preorder on strategies and show that it satisfies some striking properties such that the intrinsic preorder on function types coincides with the pointwise preorder. We then obtain an order-extensional fully abstract model of PCF by quotienting the intensional model by the intrinsic preorder. This is the first syntax-independent description of the fully abstract model for PCF. (Hyland and Ong have obtained very similar results by a somewhat different route, independently and at the same time.) We then consider the effective version of our model and prove a universality theorem: every element of the effective extensional model is definable in PCF. Equivalently, every recursive strategy is definable up to observational equivalence. An unsupervised learning algorithm for the separation of sound sources in one-channel music signals is presented. The algorithm is based on factorizing the magnitude spectrogram of an input signal into a sum of components, each of which has a fixed magnitude spectrum and a time-varying gain. Each sound source, in turn, is modeled as a sum of one or more components. The parameters of the components are estimated by minimizing the reconstruction error between the input spectrogram and the model, while restricting the component spectrograms to be nonnegative and favoring components whose gains are slowly varying and sparse. Temporal continuity is favored by using a cost term which is the sum of squared differences between the gains in adjacent frames, and sparseness is favored by penalizing nonzero gains. The proposed iterative estimation algorithm is initialized with random values, and the gains and the spectra are then alternatively updated using multiplicative update rules until the values converge. Simulation experiments were carried out using generated mixtures of pitched musical instrument samples and drum sounds. The performance of the proposed method was compared with independent subspace analysis and basic nonnegative matrix factorization, which are based on the same linear model. According to these simulations, the proposed method enables a better separation quality than the previous algorithms. Especially, the temporal continuity criterion improved the detection of pitched musical sounds. The sparseness criterion did not produce significant improvements