Investigation of strings with periodic diameter variations

The vibrational properties of musical strings play a decisive role in the sound production of stringed instruments. While conventional strings are designed to have a uniform geometry, introducing variations in their geometrical and mechanical properties offers a way to explore new mechanisms for controlling their dynamic behavior. This paper investigates the case of strings with periodic diameter variations. In order to assess the influence that such variations may have on the vibration behavior of strings, a semi-analytic model of a vibrating string is developed. The latter is based on the Transfer Matrix Method (TMM) combined with Floquet theory. A parametric study of diameter variations is carried out. This study allows to separate the influence of the spatial periodicity, the amplitude and the phase on the modal parameters. Specifically, the spatial periodicity determines which string partials will lie in a stop band, while the phase determines the location of these partials inside stop bands (and thus allows for various inharmonicity levels for the same relative standard deviation of the diameter) and the amplitude acts as a scaling factor. In addition, these results are supported by sound syntheses that provide a perceptual perspective.