Discipline: Computer Sciences and Information Management
Subcategory: Computer Science & Information Systems
Angus B. Tomlinson - Montana State University
Co-Author(s): Robin Belton, Montana State University, Bozeman, Montana; Kira Wencek, University of Rhode Island, Kingston, Rhode Island
Topological data analysis (TDA) is an emerging field that employs topology to study the structure of large, multidimensional datasets . Although TDA has been applied to many datasets, music is a dataset that has remained largely untouched, largely due to the difficulty of converting music to meaningful graphical representations. Therefore, we are seeking to develop representations that support computational methods for accurately portraying music . Music is quite structured, so we hypothesize that we can represent musical themes and variation with simplicial complexes, graphical structures useful for TDA. For our dataset, we use music scores in the MusicXML format, which enables us to easily extract the key signature, note pitches, note durations, and instrumentation from each song. The first step in our process is to extract the individual note pitches from a song. However, most people don’t possess absolute pitch, so the preservation of the note pitches is unnecessary. Instead, we convert the note pitches to their equivalent intervals, as measured from the key signature, which limits the notes to a twenty-five dimensional space. After these initial conversions, we run a sliding window algorithm, which organizes song intervals into staggered windows which span a song’s duration. We convert each window into a “bag of notes” vector (an adaption of the “bag of words” vector from NLP), which records the occurrences of each possible interval in that song segment . With this method, we can convert a song into vectors which preserve harmony and melody. Finally, we produce a Vietoris-Rips simplicial complex filtration from the song vectors and analyze its topology. Using this approach, we can capture repetitions and cycles in the data, which represent a notion of complexity within the music. Specifically, we compared the complexity of Mozart and Taylor Swift songs, and found that Mozart’s compositions are more complex. While the differences in complexity are fascinating, we want to further develop our approach to better describe musical characteristics with simplicial complexes. Possible modifications to our approach could be the separation of harmony and melody complexes using interval vector notation and a custom melody vector representation. Our initial results are promising, and through further development and research, we could develop a topological tool which could be quite useful to topologists and musicians alike. References:  Edelsbrunner, H. and Harer, J. (2009). “Computational Topology: An Introduction”. American Mathematical Society.  Bendich, P. and Tralie, C. (2015). “Cover Song Identification with Timbral Shape Sequences”. Proceedings of the International Symposium on Music Information Retrieval, pp 38-44.  Zhu, X. (2013). “Persistent homology: An introduction and a New Text Representation for Natural Language Processing”. IJCAI 2013.Not Submitted
Funder Acknowledgement(s): Funding was provided by the Ronald E. McNair Postbaccalaureate Achievement Program and Undergraduate Scholars Program at Montana State University.
Faculty Advisor: Brittany Terese Fasy; David L. Millman, email@example.com
Role: I developed the process for converting songs to simplicial complexes. Specifically, I wrote the Python scripts for converting music scores to 'bag of notes' vectors, which could in turn be transformed into Vietoris-Rips simplicial complex filtrations using existing TDA R libraries. I developed scripts for parsing MusicXML files and created algorithms for converting note pitches to intervals. I also implemented a sliding window algorithm for transforming songs into vectors.