The Logical Clarinet: Numerical Optimization of the Geometry of Woodwind Instruments

Abstract

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