Flute acoustics: measurement, modelling and design - School of ...

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74 CHAPTER 5. IMPEDANCE SPECTRA OF THE FLUTE AND CLARINET 10 8 (a) 10 7 10 6 10 5 10 8 (b) |Z| (Pa s m -3 ) 10 7 10 6 10 5 10 8 (c) 10 7 10 6 10 5 0 1 2 3 4 frequency (kHz) Figure 5.3: Measured impedance spectra of (a) an open pipe (15 mm diameter, 325 mm long), (b) a B♭ clarinet fingered to play C4 (written D4) and (c) a flute fingered to play C5. Arrows in (b) and (c) show the extrema corresponding to the played notes.
5.3. RESULTS AND DISCUSSION 75 and, for the clarinet, approximately the length from mouthpiece to the first of the array of open tone holes. The cut off frequency for this array of tone holes (Benade 1976) is approximately 1.5 kHz. At frequencies well above this, the inertia of the air in the open holes is sufficient to isolate acoustically the wave in the bore from the radiation field outside. Consequently, at frequencies well above this, the frequency spacing between maxima is much smaller: approximately 2c L clar ,whereL clar is approximately the length of the whole clarinet (with some correction for the bell). A consequence is that a note played with this fingering (C4) has impedance maxima approximately corresponding to the first, third, fifth and seventh harmonic, but there is no systematic pattern relating higher resonances to harmonics. Further, a note using a similar fingering in the next register (G5), whose fundamental involves the second maximum, has no systematic relation between higher harmonics and impedance maxima. This explains why the common oversimplification that clarinet notes have strong odd harmonics is very limited: it applies primarily to notes in the lowest register, and then only for harmonics falling below or near the cut off frequency. A similar effect occurs in the flute, whose cut off frequency is about 2 kHz, but it is clearly visible only at much higher frequencies (Wolfe & Smith 2003). The reason is that the embouchure of the flute has a Helmholtz resonator, formed by the enclosed volume of air between the embouchure and the cork (the compliance) and the air in the embouchure hole (the mass). Near its rather broad resonance (centred at about 3 kHz), this Helmholtz resonator acts as an acoustical short circuit in parallel with the bore, which explains the lack of structure in the impedance spectrum in this frequency range. Using such impedance spectra, it is possible to explain not only such general features, but also some of the particularities of individual fingerings. Brief examples of such discussions are given for many of the results on the web site of the Music Acoustics Laboratory at the University of New South Wales . 5.3.2 Clarinet impedance spectra Figure 5.4 shows the magnitude and phase of the impedance spectra (with added reed compliance) for the lowest note (written E3) on the B♭ clarinet. Both measurements were made using an output waveform with 2 15 points. 32 cycles were averaged to reduce noise. Spectra similar to that shown in Figure 5.4 were measured for all standard and many non-standard clarinet fingerings. The resulting database is available at and shows for each measured fingering the magnitude of the impedance spectrum along with a sound spectrum (the clarinet was played by Catherine Young) and fingering schematic. A brief commentary on each page explains some interesting features of the note, and any relationship with the impedance spectrum. Figure 5.5 is an image from the above-mentioned web site showing the magnitude of the impedance spectrum and a fingering diagram for the clarinet fingered to play F5. 5.3.3 Modelling the flute headjoint A flute headjoint has a cylindrical or tapered bore, closed at one end by the cork and open where it joins the flute body. A single hole (the ‘embouchure hole’) near the corked end is applied to
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Flute acoustics: measurement, model
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To Renée iii
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vi I am grateful to my parents, Ros
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340 REFERENCES Botros, A., Smith, J
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342 REFERENCES Kob, M. & Neuschaefe
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