A model for the observed decorrelation of partials in the overtone spectra of bowed stringed instruments

It has been shown that the overtone frequencies in the spectra of bowed stringed instruments played with vibrato exhibit less than perfect pairwise correlations. However, these results are inconsistent with the mechanism of performing vibrato by changing the length of the string. Since modulating the string length affects the frequencies of all string modes proportionately, it is curious that the overtones exhibit less than perfect correlations. The observed decorrelations, therefore, may be attributed to the filtering of the string’s vibrations by the mechanical-acoustic resonant modes of the instrument body. The exact frequency deviations depend upon the frequencies of instrument’s resonant modes in relation to the string’s overtone frequencies and the width and rate of the vibrato. By modelling the instrument body as a sum of resonant modes driven by a frequency modulated saw tooth wave, we develop an analytical model relating the observed frequency deviations to the modal properties of the instrument. The effect of a single resonant mode on the instantaneous frequency trajectories is found analytically and informs numerical simulations of instruments with multiple modes. The simulated results compare well with data from recorded violin tones.