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The chimney tube in musical acoustics: A textbook-level formulation for students and musicians.

The chimney tube, comprising connected body and chimney segments with respective diameters d1 and d2, is ubiquitous in musical acoustics and can be a useful analog for flute embouchure holes, woodwind toneholes, and organ pipes. Most treatments are complex and not readily accessible to lay audiences. Simple expressions for the input impedance of ideal chimney tubes are derived by two methods. Chimneys short compared to the overall length of the resonator are shown to have effective lengths larger than their physical length by the factor (d1/d2)2 when the chimney is open and smaller by the same factor when closed. Examples presented range from a simple cylindrical tube open at both ends to the classic Helmholtz resonator. Insights are provided about tonehole placement and the tradeoffs between a tonehole's position and diameter; how the resonance frequencies of an open-open chimney tube with a short chimney do not depend on which end is used as the input; and the interesting way in which a long chimney can change the spacing of the resonances from the 1:2:3 pattern of a cylinder open at both ends to the characteristic 1:3:5 pattern of a cylinder open at one end and closed at the other.