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

CHAPTER 5. IMPEDANCE SPECTRA OF THE FLUTE AND CLARINET 69
5.2.4 Measurement conditions
The flute measurements were made at T = 21.5 ± 0.3 °C and relative humidity 38 ± 1%. The
clarinet measurements were made at T = 27.5 ± 0.5 °C and relative humidity 61 ± 1%.
5.3 RESULTS AND DISCUSSION
5.3.1 Flute and clarinet impedance spectra compared
A modern flute and a clarinet are, to zeroth order, similar geometrically, i.e. if one neglects
the bell of the clarinet and the variations in bore. Yet they differ significantly in the range and
timbre of sounds they produce. The instruments are of approximately equal length and the
bores of each are mostly cylindrical. The difference, of course, is that the clarinet is almost
closed by the reed at its embouchure (i.e. it is almost a closed pipe) whereas the flute is open
to the air at its embouchure (almost an open pipe). The lowest note on a flute is almost an
octave higher than the lowest note on a clarinet, the flute overblows an octave while the clarinet
overblows a twelfth and their sound spectra reflect this difference, at least for notes in the lowest
register. (The principal differences between the flute and the clarinet are due to the different
excitation mechanisms. Some—but only some—of these differences are summarised in the
low order approximation in which one is approximated by an open pipe, and the other by a
closed pipe. These approximations, of course, omit virtually all details of air jets, upon which
flutists expend hours and years of practice, and the mechanical properties of reeds, which are
of passionate interest to clarinettists. Nevertheless, it is interesting to compare and to contrast
the acoustics of idealised open and closed pipes in terms of their acoustic impedance spectra
and to compare these with the rather more complicated behaviour of the bores of flutes and
clarinets.)
Figure 5.2 shows an impedance spectrum measured using the 15 mm impedance head for
an open pipe of length 650 mm and diameter 15 mm. The broad features of this spectrum are
similar to the spectra for a modern flute and a modern clarinet, both with all finger holes closed.
The primary difference between the instruments is the condition at the embouchure. Flutes
are open at the embouchure, and hence operate at impedance minima. Clarinets operate at
impedance maxima. The first six resonances and nine antiresonances of this pipe are shown
in musical notation in Figure 5.2. While the notes shown for a flute are approximately correct,
orchestral clarinets play somewhat sharper than the notes shown here for a simple cylindrical
pipe (due primarily to the effect of the bell).
The impedance spectra of real flutes and clarinets are clearly not identical to the spectrum
of an open pipe of equivalent length. The Z spectra of the flute fingered to play C5 and the
B♭ clarinet fingered to play C4 are shown in Figure 5.3, along with the impedance of a 15 mm
diameter pipe of length 325 mm (approximately equal to the equivalent length of both instruments).
The juxtaposition of these three curves allows us to make explicit comparisons that show
interesting features of these instruments. For the clarinet, the spectrum is reasonably similar to
that of the cylindrical pipe for frequencies less than about 1.5 kHz, and the spacing between impedance
maxima (or minima) is about 2c
L
, where L is the effective length of the cylindrical pipe