Composition and Analysis of Music Using Mathematica
Marianella P. Machado, Christopher W. Kulp, Dirk Schlingman · 2007
In this paper we demonstrate how to create and analyze musical compositions using Mathematica. In Section 1, we begin by demonstrating how to create a musical composition for an orchestra based on the butterfly curve using traditional means. In Section 2, we then show how a computer generated piece based on any curve can be composed using Mathematica. Finally, in Section 3 we show how Mathematica can be used to analyze musical compositions using the methods from nonlinear time series analysis. 1. The composition of an orchestral piece using the butterfly curve 1.1 The relationship between music and mathematics In the composition of a musical work, it is common to find that musical transformations are closely related to geometric transformations [1]. For example, in musical terms, a geometric reflection occurs when notes are symmetric about some line in a staff. In this paper, we study the relationship between music and mathematics through the use of Mathematica. This approach is based on the premise that the compositional process of a musical work could be generated from a graph of a mathematical expression by using geometric transformations as patterns for musical transformations. In this attempt, the graphical expression of a parametric curve executed through Mathematica is taken as a starting point for determining the compositional features and creating form in an acoustical piece. The following section contains a description of the compositional process of an orchestral work based on a butterfly curve [2]. In[1]:= r@t_D := 8Sin@tD HE^Cos@tD − 2 Cos@4 tD + Sin@te12D^5L, Cos@tD HE^Cos@tD − 2 Cos@4 tD + Sin@te12D^5 L<; ParametricPlot@r@tD, 8t, 0, 48<, AspectRatio → Automatic, PlotStyle → RedD;