Introduction to Acoustics

646 Part E Music, Speech, Electroacoustics
Part E 15.4
proximately to a perfect fifth and an octave above the
fundamental. As noted by Rayleigh in relation to church
bells ([15.3] Vol. 1, Sect. 394), the pitch of a note is
often determined by the higher quasi-harmonically related
partials rather than the lowest partial present. This
is demonstrated by the second drum beat in ,
which has all frequency components below 250 Hz removed.
The perceived pitch at long times is unchanged,
though there is a considerable loss in depth or body of
the resulting sound.
Modal frequencies for a typical kettle drum have already
been listed in Table 15.7, which includes a set of
nearly harmonic modes indicated by asterisks achieved,
in part, by empirical design of the coupled membrane
and kettle air vibrations. To a first approximation, the
modal frequencies are determined by the volume of the
kettle rather than its shape. The smaller the enclosed
volume, the larger its effect on the lowest-order drumhead
modes. Nevertheless, there are distinct differences
in the sounds of timpani used by orchestras in Vienna
and those used elsewhere in Europe (Bertsch [15.204]).
Such differences can be attributed to the Viennese preference
for calfskin rather than mylar drum heads, a small
shape dependence affecting the coupling to the internal
air resonances, and a different tuning mechanism. The
modal frequencies of the Viennese timpani measured
by Bertsch were similar to those listed in Table 15.7,
with the (11), (21), (31) and (41) modes again forming
a quasi-harmonic set of partials, in the approximate ratios
1:1.5:2.0:2.4:2.9. Rather surprisingly, the relative
frequencies of the lower two modes could be interchanged
with tuning.
Indian Tabla and Mrdanga
Another way of achieving a near harmonic set of resonances
of a vibrating drumhead is to add mass to
the drum head and hence change the frequencies of
its normal modes of vibration. For the single- and
double-headed Indian tabla and mrdanga drums, this
is achieved by selectively loading the drum skin with
several coatings of a paste of starch, gum, iron oxide
and other materials – see Fletcher and Rossing ([15.5],
Sect. 18.5). The acoustics of the tabla was first investigated
by Raman [15.205], who obtained Chladni
patterns for many of the lower-frequency modes of the
drum head. Rossing and Sykes [15.206] measured the incremental
changes in frequency of the loaded membrane
as each additional layer was added. A 100 layers lowered
the fundamental mode by about an octave. The resulting
five lowest modes were harmonically related and
including several degenerate modes derived from the
smoothly transformed modes of the original unloaded
membrane. The results were very similar to those obtained
earlier by Raman. Investigations by Ramakrishna
and Sondhi [15.207] and by De [15.208] showed that, to
achieve a quasi-harmonic set of low-frequency modes,
the areal density at the centre of such drums should be
approximately 10 times that of the unloaded sections.
Figure 15.117 illustrates the decaying waveform and
spectra of a well-tuned tabla drum (audio )
from 200 ms FFTs of the initial sound and after 0.5s.
The spectra show three prominent partials at 549, 826
and 1107 Hz, in the near-harmonic ratios 1:1.51:2.02,
which results in a well-defined sense of pitch. In contrast
to the timpani, these partials dominate the sound and
determine the pitch from the very beginning of the note.
Note too the very wide spectrum of the rapidly decaying
initial transient.
Side and Snare Drum
The side or snare drum is the classical two-headed
drum of the modern symphony orchestra. It is usually
played with either very short percussive beats or
as a roll, with rapidly alternating notes from two alternating
drumsticks. This results in a quasi-continuous
(dB)
0
–20
–40
–60
–80
–100
0 500 1000 1500 2000
(Hz)
500 ms
Fig. 15.117 Decaying waveform of a tabla drum with initial
FFT spectrum (upper trace) and after 0.5 s (lower) illustrating
the weakly damped, near-harmonic, resonances of the
loaded drumhead