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

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