Intonation and Compensation of Fretted String Instruments

In this paper we present mathematical and physical models to be used in the analysis of the problem of intonation of musical instruments such as guitars, mandolins and the like, i.e., we study how to improve the tuning on these instruments. This analysis begins by designing the placement of frets on the fingerboard according to mathematical rules and the assumption of an ideal string, but becomes more complicated when one includes the effects of deformation of the string and inharmonicity due to other string characteristics. As a consequence of these factors, perfect intonation of all the notes on the instrument can never be achieved, but complex compensation procedures are introduced and studied to minimize the problem. To test the validity of these compensation procedures, we have performed extensive measurements using standard monochord sonometers and other basic acoustical devices, confirming the correctness of our theoretical models. In particular, these experimental activities can be easily integrated into standard acoustics courses and labs, and can become a more advanced version of basic experiments with monochords and sonometers.