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   Figure 6. Semilog plots of measured amplitudes of acoustical imped- ance spectra (after Wolfe et al., 2010). The flute (second from bot- tom) and saxophone (second from top) have fingerings that play C5 (523 Hz) and the clarinet (middle) plays C4 (262 Hz). In all cases, this means tone holes open in the bottom half of the instrument, as indicated in the schematics. The length of the cylinder (bottom) was chosen to put its first maximum at C4 and its first minimum near C5. The cylinder + cone has a first maximum at C5 (top). Thus, all five ducts have the same acoustical length (~L). In the flute, the disap- pearance of resonances around 4 kHz is an interesting effect that lim- its the range of the instrument. The small volume of air in the “dead end” beyond the embouchure hole (see Figure 1b) and the air in that hole constitute, respectively, the “spring” and mass of a Helmholtz resonator, which helps the instrument play in tune. At resonance, however, this short circuits the bore.
sine and cosine functions as in Figure 4. For a complete, open cone of length L, the solutions to the wave equation can form a complete harmonic series, with nc/2L   fn = nf1. These are the same frequencies as in an open-open cylin- der of the same effective length. Of course, no instrument is a complete cone; that would leave nowhere for air to enter. Simply truncating a cone gives resonances that are inhar- monic. However, harmonicity is approximately restored if
the truncated apex is replaced by a mouthpiece having an equal effective volume. (That effective volume includes an extra volume representing the compliance of the reed.)
Acoustic Impedance
The acoustic response of the instrument bore is quanti- fied by its acoustic impedance spectrum, Z(f), the acoustic pressure at the mouthpiece divided by the acoustic current into the mouthpiece and measured in acoustic megohms or MPa∙s∙m−3. Figure 6 shows the magnitude of the measured impedance spectra Z(f) for 5 ducts having the same effec- tive length of about 33 cm. In the spectrum for the cylinder (Figure 6, bottom), the minima in Z (largest flow for given pressure) correspond to the modes of the open-open pipe and the maxima to those of a closed-open pipe. The mini- ma form a complete harmonic series with f1 near 520 Hz or about C5 (cf. Eq. 1), and the maxima form a series with f1 near 260 Hz (C4) and its odd multiples (Eq. 2).
Above that of the cylinder is the measurement for a flute fingered to play C5. In the downstream half of the instru- ment, nearly all the tone holes are open, giving it an effective length corresponding approximately to its closed upstream half, as suggested by the schematic. The impedance mini- mum corresponding to C5 is circled.
At low frequencies, Z(f) for both the flute and clarinet (see Figure 6) resemble that of a simple cylinder; for these fre- quencies, the bore is effectively terminated near the first open holes. The clarinet, however, is almost closed at the mouthpiece by its reed so that it operates at maxima in im- pedance, and with a similar closed length of bore, it plays C4, an octave lower than the flute.
In Figure 6, top, is the measurement of a truncated cone, with a cylinder of equal volume replacing the truncation. Below is the measurement of a soprano saxophone, fingered to play C5 (trill fingering), with the fundamental impedance maximum circled. As for the clarinet, the reed requires large pressure variation for small flow and so plays at impedance maxima. The varied high-frequency behavior is discussed in Tone Holes, Register Holes, End Effects, and the Cutoff Frequency.
Returning to Z(f) for the cylinder, it is worth considering the time domain. Suppose we inject a short pulse of high- pressure air at the input. It travels to the open end where, with the constraint of negligible acoustic pressure, it is re- flected with a pressure phase change of π so that it becomes a pulse of negative pressure. After one round-trip of the 33-cm
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