Mode Selection Rules for Two-Delay Systems: Dynamical Explanation for the Function of the Register Hole on the Clarinet

Abstract

Generally, time-delay systems are regarded as multi-attractor systems. We investigate mode selection rules for two-delay systems considering which oscillation mode is first excited by the Hopf bifurcation with increasing a bifurcation parameter. In particular, we focus on the case that the strength of the short time delay α1 is lower than that of the long time delay α2. In a certain range of α1/α2 in which it is sufficiently small but still not negligible, the third-harmonic mode occupies a particular range of the ratio of the two delay times such that 2 < tR2/tR1 < 4, where tR1 and tR2 denote the short and long delay times, respectively. This is the key for understanding the function of the register hole on the clarinet, which is smaller in radius than the other tone holes, but works well to raise the pitch of first register notes in a wide range more than an octave by a twelfth (19 semitones), i.e., generating third harmonics, when opened. This is confirmed using a simple model of the clarinet with two ...