Resonance modes of a flute with one open tone hole

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 o±, which are respectively higher and lower than those of the upper and lower bores oU and oL excited independently. The o± differ even if oU= oL. The normal effect of cross-fingering, i.e., pitch flattening, corresponds to excitation of the o--mode, which occurs when oL⪆oU and the admittance peak of the o--mode is higher than or as high as that of the o+-mode. Excitation of the o+-mode yields intonation anomaly. This occurs when oL⪅oU and the peak of the o+-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.