  The tone hole geometry of a clarinet is optimized numerically. The instrument
is modeled as a network of one dimensional transmission line elements. For each
(non-fork) fingering, we first calculate the resonance frequencies of the input
impedance peaks, and compare them with the frequencies of a mathematically even
chromatic scale (equal temperament). A least square algorithm is then used to
minimize the differences and to derive the geometry of the instrument. Various
situations are studied, with and without dedicated register hole and/or
enlargement of the bore. With a dedicated register hole, the differences can
remain less than 10 musical cents throughout the whole usual range of a
clarinet. The positions, diameters and lengths of the chimneys vary regularly
over the whole length of the instrument, in contrast with usual clarinets.
Nevertheless, we recover one usual feature of instruments, namely that
gradually larger tone holes occur when the distance to the reed increases. A
fully chromatic prototype instrument has been built to check these
calculations, and tested experimentally with an artificial blowing machine,
providing good agreement with the numerical predictions.
