Deciphering the Complexities in the Perception, Performance and Related Comparison of Musical Phrases Via Statistical Shape Analysis of the Fractal Nature of Music (Data)
Makarand V. Ratnaparkhi
Mathematics and Statistics, Wright State University, Dayton, OH, United States

In musicology, the understanding of the perception and performance of music as a neurological activity is known to be a challenging problem [1]. Any contribution for such understanding is useful in music education. Keeping this aspect of musicology in view the live experimental data was collected for the performance of a musical phrase played by the intermediate level students taking violin lessons. This sample of students represented a population of students who take violin lessons in Indian Classical Music.

For modeling these data, first, we introduce the concept of shape of the musical phrase. Then, for the live recordings of a musical phrase, played on violin, we discuss the use of fractal dimensions (FD) of each musical note as a descriptive statistic that provides the information about the qualitative nature of the performance of that note. Basically, these FDs, corresponding to a sequential (multivariate) times series representing the notes in this phrase, provide a performance profile of the shape of that phrase. Then, the fractal dimensions (FDs), the durations and tempi of these notes are collectively used for defining the statistical size and shape description of the musical performance. The shape of the phrase, thus defined, is of interest for comparing the performance of the same phrase by different music students participated in this study. The shape descriptors, thus obtained, are further used for the discrimination and clustering of the performances by the subjects. Other descriptive statistics such as the correlations among the FDs were calculated and related interpretations are discussed.

In music training the emphasis is on practicing any phrase by playing it repeatedly. Therefore, the behavior of the FDs corresponding to such repetitions was studied and related results are presented. Also, using the results from the above analyses, an effort has been made to associate and interpret, the (observed) between and within variation of the fractal dimensions and related shape descriptors of the performances by the students, as a result of underlying (unobservable) neurological phenomena. Further, the advantages of using the concept of fractal dimension as compared to the use of traditional FFT and wavelet decomposition for such analyses are discussed briefly.


[1] Zatorre,R.J., Chen,J.L., and Penhune,V.B. (2007). When the brain plays music: auditory–motor interactions in music perception and production. Nature Reviews Neuroscience, 8, 547-558

Keywords: Shape of a musical phrase; multivariate time series; fractal dimension; classification and clustering

Biography: Makarand Ratnaparkkhi is a Professor of Statistics at Wright State University, USA for the past 30 years with research interest in theorical and applied statistics. He has a formal training in Indian Classical Music (vocal and instrumental). Therefore, at present, he is devoting some of his research time for exploring the statistical methodologies that can be used in music education, in general, and for violin playing in particular. Besides the principle investigator, he is also one of the subjects for the collectiion of music data that is considered in this presentation.