The logical clarinet: numerical optimization of the geometry of woodwind instruments

TL;DR

This paper presents a numerical optimization method for clarinet tone hole geometry, achieving near-chromatic tuning accuracy and revealing new design features, validated by experimental testing with a prototype instrument.

Contribution

It introduces a novel numerical approach to optimize clarinet tone hole geometry, incorporating resonance calculations and least squares minimization, resulting in innovative design insights.

Findings

Differences less than 10 cents across the clarinet range.

Regular variation of tone hole positions, diameters, and lengths.

Good agreement between numerical predictions and experimental tests.

Abstract

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.