🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.001056s)
  1. A Framework for Topological Music Analysis (TMA) (2022)

    Alberto Alcalá-Alvarez, Pablo Padilla-Longoria
    Abstract In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be associated to chord sequences, and calculate their main homological descriptors. These complexes lead us to the definition of dynamical systems modeling harmonic progressions. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from three Brandenburg Concertos by J.S. Bach and two Graffiti by Mexican composer Armando Luna.
  2. Musical Stylistic Analysis: A Study of Intervallic Transition Graphs via Persistent Homology (2022)

    Martín Mijangos, Alessandro Bravetti, Pablo Padilla
    Abstract Topological data analysis has been recently applied to investigate stylistic signatures and trends in musical compositions. A useful tool in this area is Persistent Homology. In this paper, we develop a novel method to represent a weighted directed graph as a finite metric space and then use persistent homology to extract useful features. We apply this method to weighted directed graphs obtained from pitch transitions information of a given musical fragment and use these techniques to the study of stylistic trends. In particular, we are interested in using these tools to make quantitative stylistic comparisons. As a first illustration, we analyze a selection of string quartets by Haydn, Mozart and Beethoven and discuss possible implications of our results in terms of different approaches by these composers to stylistic exploration and variety. We observe that Haydn is stylistically the most conservative, followed by Mozart, while Beethoven is the most innovative, expanding and modifying the string quartet as a musical form. Finally we also compare the variability of different genres, namely minuets, allegros, prestos and adagios, by a given composer and conclude that the minuet is the most stable form of the string quartet movements.