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Probabilistic Sequential Patterns for Singing Transcription

Eita Nakamura∗, Ryo Nishikimi∗, Simon Dixon† and Kazuyoshi Yoshii∗ ∗ Kyoto University, Kyoto, Japan

E-mail: [enakamura,nishikimi,yoshii]@sap.ist.i.kyoto-u.ac.jp † Queen Mary University of London, London, United Kingdom

E-mail: [email protected]

Abstract—Statistical models of musical scores play an impor- tant role in various tasks of music information processing. It has been an open problem to construct a score model incorporating global repetitive structure of note sequences, which is expected to be useful for music transcription and other tasks. Since repetitions can be described by a sparse distribution over note patterns (segments of music), a possible solution is to consider a Bayesian score model in which such a sparse distribution is first generated for each individual piece and then musical notes are generated in units of note patterns according to the distribution. However, straightforward construction is impractical due to the enormous number of possible note patterns. We propose a probabilistic model that represents a cluster of note patterns, instead of explicitly dealing with the set of all possible note patterns, to attain computational tractability. A score model is constructed as a mixture or a Markov model of such clusters, which is compatible with the above framework for describing repetitive structure. As a practical test to evaluate the potential of the model, we consider the problem of singing transcription from vocal f0 trajectories. Evaluation results show that our model achieves better predictive ability and transcription accuracies compared to the conventional Markov model, nearly reaching state-of-the-art performance.

I. INTRODUCTION Computational models of musical scores, sometimes re-

ferred to as music language models, play a vital role in various tasks of music information processing [1]–[6]. For automatic music transcription [7], for example, a musical score model is necessary to induce an output score to be an appropriate one that respects musical grammar, style of the target music, playability, etc. Conventional musical score models include variants of Markov models [8], hidden Markov models (HMMs) and probabilistic context-free grammar mod- els [9], recurrent neural networks [10], and convolutional neural networks [11]. These models are typically learned supervisedly and the obtained generic score model is applied to transcription of all target musical pieces, in combination with an acoustic model and/or a performance model.

A musical piece commonly consists of groups of musical notes that are repeated several times, and detecting repeated note patterns has been a topic of computational music analysis [12]. Repetitive structure can complement local sequential dependence of notes, on which conventional score models have focused, and thus can be useful for transcription. This is because if the repetitive structure can be inferred correctly, one can recover part of the score or can account for timing

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Fig. 1. The proposed sequential pattern model for musical scores. (a) A PSP generates note patterns for a bar based on a metrical Markov model (rhythm probability) and a distribution of pitches defined for each beat position (pitch probability). (b) A generative process of musical scores with multiple PSPs; for each bar a PSP is chosen according to a mixture/Markov model. See section III for details. (For clear illustration, the 6/8 time in this example is different from the 4/4 time considered in the main text.)

deviations, acoustic variations, and other types of “noise” in performances. To realize this scenario, we need a score model that induces repetitions in generated scores. Since repetitive structure is a global feature, it is challenging to construct a computationally tractable model with such a property.

A framework for constructing a score model with repetitive structure has been proposed in [13]. The idea is to consider a distribution (or a Markov model) over a set of possible note patterns and implicitly represent repetitive structure with sparseness of the distribution. A note pattern is defined here as a subsequence of musical notes; if a piece consists of a small number of note patterns then it has repetitions and vice versa. The model is described as a Bayesian extension of a Markov model of note patterns where the Dirichlet parameters of the prior distributions are assumed to be small to induce repetitions. In addition, to deal with approximate repetitions (i.e. repetitions with modifications), which are com- mon in music practice [14], an additional stochastic process of modifying patterns was incorporated. In this framework, an individual score model is learned unsupervisedly in the transcription process for each input signal, in contrast to the above supervised learning scheme. That study focused on monophonic music and only rhythms were modelled, and it has been shown that the framework is indeed effective for music transcription.

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Proceedings, APSIPA Annual Summit and Conference 2018 12-15 November 2018, Hawaii

978-988-14768-5-2 ©2018 APSIPA APSIPA-ASC 2018

Although it is theoretically possible to extend this frame- work for more general forms of music, including pitches and multiple voices, the model becomes so large that computa- tional tractability will be lost. First, with Ω ∼ O(10–102) unique pitches and ` notes, the number of possible pitch patterns is Ω`, which soon becomes intractable as ` increases. Second, by adding the process of modifying patterns, the complexity increases further: with Λ unique patterns one should consider Λ2 possible ways of modification in general. Third, joint modelling of pitches and rhythms further increases the size of the state space.

In this study, we propose a new approach for modelling note patterns including pitches and rhythms that is compatible with the above framework for describing repetitive structure. To re- alize this, we treat a set of patterns related by modifications as a cluster, named probabilistic sequential pattern (PSP), without explicitly dealing with the set of all possible patterns (Fig. 1). A PSP is formulated so that it can naturally accommodate insertion, deletion, and substitution of notes as modifications of patterns. We construct score models based on a mixture (possibly with a Markov structure on the mixture weights) of such PSPs and its Bayesian extension. As a practical problem, we consider the problem of singing transcription of vocal f0 trajectories [15], [16].

The main contributions of this study are: • Construction of computationally tractable score models

based on note patterns; a Bayesian framework for describ- ing approximate repetitive structure of musical scores.

• Our model achieves better predictive ability and transcrip- tion accuracies than the conventional Markov model.

An additional contribution is proposing a framework for inte- grating a note-level score model and a beat-level f0 model. Such a framework is nontrivial due to different time units in acoustic/f0 signals and musical scores and has not been realized in most previous studies; in some studies frame-level musical score models have been considered [17], [18], which cannot properly describe musical rhythms, and in other studies an unrealistic situation that the number of musical notes is known in advance has been assumed [19].

The rest of the paper is organized as follows. In the next sec- tion, we review a simple Markov model generating melodies (pitches and rhythms) and its extension for use in singing transcription. In section III, we formulate our PSP model and explain an inference method. Numerical experiments and evaluations are presented in section IV. The last section is dedicated to conclusions and discussion.

II. MELODY MARKOV MODEL

Here we specify the problem of singing transcription con- sidered in this study. A simple model for statistical singing transcription is presented as a baseline method.

A. Singing Transcription from F0 Trajectories

Singing transcription is a task of converting a singing voice signal into a musical score representation [19], [20]. Here, we consider a situation that a singing-voice f0 trajectory is given

Musical score Y

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Generative processTranscription F0 trajectory X

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Fig. 2. Singing transcription from a vocal f0 trajectory and the corresponding generative process.

in advance as well as the information about the location of beats and bar onsets (downbeats) (Fig. 2). The former can be estimated by using an f0 estimation method for singing voices (e.g. [21]) and the latter can be estimated by a beat tracking method (e.g. [22]). In the following, time frames are indexed by t ∈ {1, . . . , T} and score times in units of 16th notes are indexed by τ ∈ {τs, . . . , τe}, where T represents the signal length and τs and τe respectively denote the first and the last beats. For each score time τ ,