Time delay embeddings to characterize the timbre of musical instruments using Topological Data Analysis: a study on synthetic and real data

Abstract

Timbre allows us to distinguish between sounds even when they share the same pitch and loudness, playing an important role in music, instrument recognition, and speech. Traditional approaches, such as frequency analysis or machine learning, often overlook subtle characteristics of sound. Topological Data Analysis (TDA) can capture complex patterns, but its application to timbre has been limited, partly because it is unclear how to represent sound effectively for TDA. In this study, we investigate how different time delay embeddings affect TDA results. Using both synthetic and real audio signals, we identify time delays that enhance the detection of harmonic structures. Our findings show that specific delays, related to fractions of the fundamental period, allow TDA to reveal key harmonic features and distinguish between integer and non-integer harmonics. The method is effective for synthetic and real musical instrument sounds and opens the way for future works, which could extend it to more complex sounds using higher-dimensional embeddings and additional persistence statistics.

References in corpus (19)

• Networks beyond pairwise interactions: structure and dynamics
• The physics of higher-order interactions in complex systems
• What are higher-order networks?
• Higher-order interactions in complex networks of phase oscillators promote abrupt synchronization switching
• Random walks on hypergraphs
• Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
• Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
• Synchronization induced by directed higher-order interactions
• Multistable attractors in a network of phase oscillators with three-body interaction
• Turing patterns in systems with high-order interactions
• Persistent Dirac for molecular representation
• Topological Data Analysis for Multivariate Time Series Data
• Dirac signal processing of higher-order topological signals
• Higher-order interactions induce anomalous transitions to synchrony
• Turing patterns on discrete topologies: from networks to higher-order structures
• Computational Topology Techniques for Characterizing Time-Series Data
• Some mathematical and computational relations between timbre and color
• Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
• A Lightweight Speaker Recognition System Using Timbre Properties