Analysis of the effect of the register hole on the tuning of a clarinet
Résumé
Even if the tuning between the first and second register of aclarinet has been noticeably improved by instrument makers, theinharmonicity of the lowest twelfths (which are too large) isstill an unsolved problem. In this paper, we attempt to understandthe physical reasons for this problem and to explore whether ornot it is possible to improve the accuracy of these twelfths. Thestarting point is a study of the inharmonicity associated withdifferent bore perturbations inserted in cylindrical instruments,including bore flare, open and closed holes, etc. This study showsthat the wide twelfths in question result mostly from the effectsof the register hole. Using an elementary model of the clarinet aswell as optimization techniques, an optimum location for theregister hole is computed (we do not take into account the use ofthe register hole as a B flat tone hole); the result turns out tobe close to the location chosen by the makers. Then smallperturbations on the higher part of the cylindrical resonator areintroduced and optimized in order to find out whether a simplesolution exists for the improvement of the harmonic relationshipbetween the first and quasi-third resonance frequencies. Theresult is negative, which probably explains why this fundamentalproblem of the clarinet has not yet been solved. As a consequence,one has to resort to more complicated local solutions, forinstance with individual corrections for each tone hole.