  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.
