Analysis and optimisation of the tuning of the twelfths for a clarinet resonator

Abstract : Even if the tuning between the first and second register of a clarinet has been optimized by instrument makers, the lowest twelfths remain slightly too large (inharmonicity). In this article, we study the problem from two different points of view. First, we systematically review various physical reasons why this inharmonicity may take place, and the effect of different bore perturbations inserted in cylindrical instruments. Applications to a real clarinet resonator and comparisons with impedance measurements are then presented. A commonly accepted idea is that the register hole is the dominant cause for this inharmonicity: it is natural to expect that opening this hole will raise the resonance frequencies of the instrument, except for the note for which the hole is at the pressure node. We show that the real situation is actually more complicated because other effects, such as open holes or bore taper and bell, introduce resonance shifts that are comparable but with opposite sign, so that a relatively good overall compensation takes place. The origin of the observed inharmonicity in playing frequencies is therefore different. In a second part, we use an elementary model of the clarinet in order to isolate the effect of the register hole: a perfect cylindrical tube without closed holes. Optimization techniques are then used to calculate an optimum location for the register hole; the result turns out to be close to the location chosen by clarinet makers. Finally, attempts are made numerically to improve the situation by introducing small perturbations in the higher part of the cylindrical resonator, but no satisfactory improvement is obtained.
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Soumis le : mercredi 30 juin 2004 - 11:50:48
Dernière modification le : lundi 10 décembre 2018 - 01:24:35
Document(s) archivé(s) le : vendredi 17 septembre 2010 - 18:07:39

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Vincent Debut, Jean Kergomard, Franck Laloë. Analysis and optimisation of the tuning of the twelfths for a clarinet resonator. Applied Acoustics, Elsevier, 2005, 66, pp.365-409. 〈hal-00000589v2〉

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