A framework for exploring the relationship between nonlinear string vibration and frequency modulation
Dahlbom, David A.
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The spectra of low piano tones have long been known to display unusual, inharmonic characteristics. Surprisingly, precisely this property allowed for fairly successful generation of artificial piano tones by the frequency modulation (FM) synthesis techniques pioneered at Stanford in the 1970s. Though FM synthesis techniques are largely empirical in nature, the close correspondence between the spectra of the FM simulation of piano tones and their actual spectra suggests the hypothesis that there may be a deeper physical connection between frequency modulation and the timbre of string instruments. Recent, highly sophisticated numerical simulations of the physics of pianos have reproduced these characteristic spectra using a geometrically exact form of the equations for vibrating strings. We take these equations as a starting point for an analytical investigation into the connection between modulated waves and instrumental timbre. The content of the investigation is primarily theoretical, establishing approaches and equations which may be used for further exploration. We proceed by deriving the equations of motion from variational principles. The Lagrangian is given careful treatment so that the transverse and longitudinal directions of motion are clearly separated and their coupling is collected in a single interaction term. Approximate equations of motion are then derived. These are shown to have a structure that supports modulated wave behavior. A general form of the solution is offered which has the form a traveling wave plus a correction term. Understanding the correction term demands the solution of a complex set of nonlinear integral equations, though suggestions are made on how determining certain properties of the integral equations would be sufficient for significant insight into the role of modulation effects in the production of distinct musical timbres. Finally, another form of the solution is offered, one more amenable to iterative solution techniques but less transparent in terms of frequency-domain behavior. A discussion regarding the initial conditions and the requirements for the convergence of either of these forms of the solution is given.
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