As a new approach for analyzing musical scores, we present the phase-portrait method. Our purpose is to visualize objectively musical images. The phase plot approach is successfully applied to characterize the dynamical behavior of complex systems in physics. For the purposes of simplicity we illustrate this approach for the oscillation of a mass-spring system, both for a harmonic oscillation and a damped system. The displacement x and the velocity \(y=dx/dt\) of the mass are the two variables plotted in a plane defined with help of an Cartesian coordinate system (without the explicit time dependence). In the case of music the frequency of the tone is attributed to x and the change in frequency per unit time to y. Different from the physical system such a mass-spring example where the form of the trajectory is essential, in music it is necessary to visualize the details of the time dependence. For such cases introducing a third coordinate proves to be useful. Examples of such 3D portraits for pieces of music are presented.

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Complexity and Dynamics in Phase Space

  • Kinko Tsuji,
  • Stefan C. Müller

摘要

As a new approach for analyzing musical scores, we present the phase-portrait method. Our purpose is to visualize objectively musical images. The phase plot approach is successfully applied to characterize the dynamical behavior of complex systems in physics. For the purposes of simplicity we illustrate this approach for the oscillation of a mass-spring system, both for a harmonic oscillation and a damped system. The displacement x and the velocity \(y=dx/dt\) of the mass are the two variables plotted in a plane defined with help of an Cartesian coordinate system (without the explicit time dependence). In the case of music the frequency of the tone is attributed to x and the change in frequency per unit time to y. Different from the physical system such a mass-spring example where the form of the trajectory is essential, in music it is necessary to visualize the details of the time dependence. For such cases introducing a third coordinate proves to be useful. Examples of such 3D portraits for pieces of music are presented.