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

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 such as bore flare, open and closed holes, taper, temperature gradients, visco-thermal effects, etc. 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 shift the position of the resonances of the instrument to higher frequencies, except of course for the note for which the hole is exactly 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. This is checked by experimental and theoretical studies of the acoustical impedance curves of a clarinet. The origin of the observed inharmonicity in playing frequencies is therefore different, maybe related to the reed or the vocal tract. 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 (without taking 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 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.