2017 Volume 38 Issue 1 Pages 14-22
A minimal model explaining intonation anomaly, or pitch sharpening, which can sometimes be found in baroque flutes, recorders, shakuhachis etc. played with cross-fingering, is presented. In this model, two bores above and below an open tone hole are coupled through the hole. This coupled system has two resonance frequencies ω±, which are respectively higher and lower than those of the upper and lower bores ωU and ωL excited independently. The ω± differ even if ωU= ωL. The normal effect of cross-fingering, i.e., pitch flattening, corresponds to excitation of the ω--mode, which occurs when ωL⪆ωU and the admittance peak of the ω--mode is higher than or as high as that of the ω+-mode. Excitation of the ω+-mode yields intonation anomaly. This occurs when ωL⪅ωU and the peak of the ω+-mode becomes sufficiently high. With an extended model having three degrees of freedom, pitch bending of the recorder played with cross-fingering in the second register has been reasonably explained.