Resonance modes of a flute with one open tone hole

抄録

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 ω_U and ω_L excited independently. The ω_± differ even if ω_U= ω_L. The normal effect of cross-fingering, i.e., pitch flattening, corresponds to excitation of the ω_--mode, which occurs when ω_L⪆ω_U 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 ω_L⪅ω_U 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.