Tuning a musical instrument with vibrato system: A mathematical framework to study mechanics and acoustics and to calculate optimal tuning strategies.

String instruments such as electric guitars are often equipped with a vibrato system, which allows varying the pitch of all strings as a musical effect. It is usually based on a mobile bridge that is kept in balance by the strings and a coiled spring. Tuning such an instrument is complex, since adjusting the tension on one string will alter all other strings' tensions. In practice, a heuristic method is used where all strings are repeatedly tuned to the desired pitch, which appears to reliably yield correct pitches after a while. It is unclear why this method works; an analysis is lacking. This paper presents a mathematical model to study this subject in detail; the model captures the underlying mechanics and acoustics and can be used to simulate a typical tuning process. The paper then verifies the model with experimental data and shows that the model permits calculation of optimal tuning strategies that use the least number of adjustment steps.