Fractal analysis of songs-- Performer’s preference.pdf

Nonlinear Analysis: Real World Applications 11 (2010) 1790–1794 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: /locate/nonrwa Fractal analysis of songs: Performer’s preference Atin Das a,?, Pritha Das b a NH School, 1/257 Naktala, Kolkata 7000 047, India b Department of Maths, BESU, Shibpur, Howrah 711 103, India a r t i c l e i n f o Article history: Received 31 August 2007 Accepted 6 April 2009 Keywords: Song Music Time series analysis Fractal dimension a b s t r a c t In this paper, we shall attempt fractal analysis of time series derived from two popular song types. Each type of song has its own notation, which is to be followed by an artist singing it. We found earlier that the fractal dimension (D) calculated from a song is a good measure of the complexity of its notation. The purpose of the present study is to find whether, if the same song is sung bydifferent performers,Dwill vary, andhowan individual’s performance varies when singing songs with the same notation and same wording. We shall employ nonlinear time series methods to calculate D for different samples of the said two types to address this question. ? 2009 Elsevier Ltd. All rights reserved. 1. Introduction There is a fundamental relationship between our perception of beauty in melody and musical form and the presence of fractal qualities in music. Aspects of certain theories about the origins and fundamental structures of melodies suggest that much—perhaps all—beautiful music is, in some essential sense, fractal in its melodicmaterial and internal self-similarity [1]. Musical analysis is a complex task, yet mathematical analysis of music is not new, though there is no standard method. On the basis of spectral time series analysis of recordedmusic, Voss and Clarke [2,3] concluded thatmusic is 1/f -noise. The facts that music exhibits 1/f -power spectra at low frequencies and music possesses self-similarity allow us to consider music as a time