Modelling hierarchical key structure with pitch scapes

Lieck, Robert; Rohrmeier, Martin

doi:10.5281/zenodo.4245558

Cited by 4 publications

(5 citation statements)

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“…For instance, studying the smoothness and curvature of these families might reveal interesting insights. Applying the concept not only to singular pieces but entire corpora of compositions might moreover reveal clusters of pieces with similar overall tonal plans and hence similar families of phantom curves (up to transposition), a related direction was explored in [7]. A naturally follow-up question would be whether such similarities are effected by factors such as genre, historical period, composer, or instrumentation.…”

Section: Discussionmentioning

confidence: 99%

“…First, the interdisciplinary work in the intersection of music theory and mathematics provided the initial impetus. Second, we were able to draw on prior work, most importantly recent work in mathematical music theory on the application of the discrete Fourier transform to pitch-class sets [1] and the combination of this line of research with hierarchical representations of tonal structure [7,12] that culminated in the development of a visualization method called wavescapes [16]. The colored triangle in Figure 1 is an example for such a wavescape.…”

Section: Documenting a Creative Processmentioning

confidence: 99%

“…The way how pitches of a piece of music are organized -the piece's harmonic structure -can be studied at multiple hierarchical levels by considering the set of pitch-class distributions for all possible segments (i.e., all possible, potentially overlapping time intervals) of the piece [7,11,16]. Let 𝑇 denote the total duration of a piece, measured in beats, quarter notes, or seconds, for example.…”

Section: Phantom Curvesmentioning

confidence: 99%

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Phantom Curves

Moss

Affatato

Harasim

2022

9th International Conference on Digital Libraries for Musicology

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Scientific discovery can arise by many means. Results can be obtained from meticulous methodological rigor and carefully plannedthrough experiments, but oftentimes insights are also gained through spontaneous associations of previously unrelated facts or exploratory work. In particular for the latter mode of discovery, a stimulating environment is quintessential in order to allow creativity to flow [4,9]. Collaborative inter-and transdisciplinary research for which knowledge and expertise from different fields is brought into interaction, is especially likely to provide such environments. Interdisciplinary work can, however, be impeded by specialized expert knowledge that is difficult to communicate to researchers with different backgrounds. In such situations, appropriately designed tools may help to build bridges between different research traditions and provide interfaces to otherwise restricted knowledge.Here, we report an instance of scientific discovery made possible through interactive exploration using an online interface for music visualization. We document how the discovery of phantom curves,

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Section: Discussionmentioning

confidence: 99%

Section: Documenting a Creative Processmentioning

confidence: 99%

Section: Phantom Curvesmentioning

confidence: 99%

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Phantom Curves

Moss

Affatato

Harasim

2022

9th International Conference on Digital Libraries for Musicology

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“…The scores were pre-processed by computing pitch-class distributions (PCDs), as used for the identification of musical keys [71,72,73], using the pitchscapes library [68]. We used a resolution of 70 equally spaced time slices per piece, resulting in sequences of 12-dimensional categorical distributions.…”

Section: C2 Hierarchical Music Analysismentioning

confidence: 99%

“…(left): Expected value of the latent variables, i.e. the mean of (18), colour-coded using a key-finding algorithm from the pitchscapes library[68]. (right): Marginal node probability, i.e.…”

mentioning

confidence: 99%

Recursive Bayesian Networks: Generalising and Unifying Probabilistic Context-Free Grammars and Dynamic Bayesian Networks

Lieck¹,

Rohrmeier²

2021

Preprint

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Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. 2) For Gaussian RBNs, we additionally derive an analytic approximation of the marginal data likelihood (evidence) and marginal posterior distribution, allowing for robust parameter optimisation and Bayesian inference. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations.

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