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Mathematical Representation of the Musical Passage from Beethoven's Bagatelle "Für Elise"

Mathematical Representation of the Musical Passage from Beethoven's Bagatelle "Für Elise" Using the System Identification Technique


Authors: Mário Rodrigues Fleury Rosa*, Marie Odile Rodrigues Fleury Rosa, Arthur Silveira Franco

High School Centro de Ensino Médio do Setor Oeste (CEMSO), Centro Educação Fundamental Polivalente (CEF), Brasília, Federal District, Brazil


Abstract

The purpose of this study is to investigate the mathematical representation of a 24-bar passage of Beethoven’s Bagatelle No. 25, “Für Elise”, through the technique of dynamical system identification that builds multiple models and the developer selects a model that has the highest FIT (the percentage which indicates the agreement between the model response and the measured output) and meets the analysis criteria. We selected the part of the score that contains the notes present in the A minor scale. The sound recording was made in less noise condition with a Samsung cell phone for eighteen (18) seconds. The signal was recorded in .m4a format and converted to. WAV with a duration of one minute and eight seconds. We used the software MATLAB®free trial version and it is toolbox System Identification in which we declared the data collected in the experiment defining the input and output, these being the parameters of the model. Then we chose the representation method - the transfer function - which is a representation of the division of two polynomials. The result shows the equation that represents this piece of music is a polynomial division, where the numerator is a third-order polynomial and the denominator is a sixth-order polynomial. We conclude that even one of the most profound and unique forms of human expression can be represented mathematically in function form.



Mathematical Representation of the Musical Passage from Beethoven's Bagatelle _Für Elise_
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