Measured and calculated sounding frequencies of pipes coupled with free reeds

The Asian free-reed mouth organs incorporate approximately symmetric free reeds coupled to pipe resonators. Previous research has shown that the reeds in these instruments behave as “blown-open” reeds in which the playing frequency is above both the natural frequency of the reed and the first peak of the measured impedance curve. Measurements of the input impedance have been made for a variety of these instruments. Detailed calculations of input impedance have also been made using transmission matrices, taking into account the position of the reed along the pipe, tuning slots, finger holes, and nonuniform cross sections. These calculations are in good agreement with the measured impedances of the same instruments. If the reed is treated as a damped, driven harmonic oscillator, the sounding frequencies of these reed-pipes can be predicted using a phase relation between the reed vibration and the phase of the complex impedance. [Research supported by National Science Foundation REU Grant PHY-0649007.]