Resonance modes of a flute with one open tone hole

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Abstract
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 ! AE , which are respectively higher and lower than those of the upper and lower bores ! U and ! L excited independently. The ! AE differ even if ! U ¼ ! L. The normal effect of cross-fingering, i.e., pitch flattening, corresponds to excitation of the ! À-mode, which occurs when ! L ' ! U and the admittance peak of the ! À-mode is higher than or as high as that of the ! þ-mode. Excitation of the ! þ-mode yields intonation anomaly. This occurs when ! L / ! U and the peak of the ! þ-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.

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