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Synthesizing Musical Sounds by Solving the Wave Equation for Vibrating Objects: Part 2
In Part 1 of this paper, derivations from the wave equation of difference equations describing the oscillations of vibrating objects were presented. The vibrations of ideal and nonideal strings set into motion by different types of excitation were considered in particular. In Part 2, the following is discussed: 1) The resulting difference equations are solved by means of a standard iterative procedure with the aid of a computer, 2) Discrete values of the solution corresponding to the motion of a selected point of the object are written on a file of digital magnetic tape, 3) The numerical values written on this tape are converted into analog signals by means of D/A conversion and recorded on audio tape, 4) A few cycles of the solution are also plotted by means of a microfilm plotter in order to compare the visual appearance of the vibration with the sound on tape. The simulation was successful in reproducing the main features of string tones. It was found that losses play a very important role in defining the timbres of these particular tones. More generally, however, this research illustrates how musical sounds of all sorts might be generated with a digital computer by utilizing a conceptual approach that heretofore has not been attempted.
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