用拓扑分析揭示乐器音色的谐波结构，关键在时间延迟选择
Time delay embeddings to characterize the timbre of musical instruments using Topological Data Analysis: a study on synthetic and real data
- 通过不同时间延迟嵌入，优化声音数据的拓扑表征
- 特定延迟（基频分数周期）能显著提升谐波结构识别效果
- 适用于合成与真实乐器音色，适合音频分析与音乐信息检索研究者
音色使我们能在相同音高和响度下区分不同声音，在音乐、乐器识别和语音中至关重要。传统方法如频域分析或机器学习常忽略声音的细微特征。拓扑数据分析（TDA）可捕捉复杂模式，但其在音色中的应用受限，因声音的有效表征尚不明确。本研究探讨不同时间延迟嵌入对TDA结果的影响。基于合成与真实音频信号，我们发现与基频周期分数相关的特定延迟能增强谐波结构检测能力。结果表明，这些延迟使TDA能够有效揭示关键谐波特征，并区分整数与非整数谐波。该方法在合成及真实乐器声音上均有效，为未来研究拓展至更复杂声音提供了可能，可通过高维嵌入与额外持久性统计实现。
原文摘要 · Abstract (English)
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.
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