Analysis of the effect of the register hole on the tuning of a clarinet

Even if the tuning between the first and second register of a clarinet has been noticeably improved by instrument makers, the inharmonicity of the lowest twelfths (which are too large) is still an unsolved problem. In this paper, we attempt to understand the physical reasons for this problem and to explore whether or not it is possible to improve the accuracy of these twelfths. The starting point is a study of the inharmonicity associated with different bore perturbations inserted in cylindrical instruments, including bore flare, open and closed holes, etc. This study shows that the wide twelfths in question result mostly from the effects of the register hole. Using an elementary model of the clarinet as well as optimization techniques, an optimum location for the register hole is computed (we do not take into account the use of the register hole as a B flat tone hole); the result turns out to be close to the location chosen by the makers. Then small perturbations on the higher part of the cylindrical resonator are introduced and optimized in order to find out whether a simple solution exists for the improvement of the harmonic relationship between the first and quasi-third resonance frequencies. The result is negative, which probably explains why this fundamental problem of the clarinet has not yet been solved. As a consequence, one has to resort to more complicated local solutions, for instance with individual corrections for each tone hole.