Introduction to Acoustics

642 Part E Music, Speech, Electroacoustics
Part E 15.4
and Rossing ([15.5], Sect. 21.15). Inscriptions on the
set of 65 tuned chime bells in the tomb of Zeng Hou
Yi (433 BC), show that the Chinese had already established
a 12-note scale, closely related to our present, but
much later, western scale. The ceremonial use of bells
and gongs is widespread in religious cultures all over the
world and in western countries has the traditional use of
summoning worshippers to church and accompanying
the dead to their graves.
In the 18th century classical symphony orchestra of
Haydn and Mozart’s time, the timpani helped emphasise
the beat and pitch of the music being played, particularly
in loud sections, with the occasional use of cymbals
and triangle to emphasise exotic and often Turkish influences.
The percussion section of today’s symphony
orchestra may well be required to play up to 100 different
instruments for a single piece, as in Notations
I–IV by Boulez [15.203]. This typical modern score includes
timpani, gongs, bells, metals and glass chimes,
claves, wooden blocks, cowbells, tom-toms, marimbas,
glockenspiels, xylophones, vibraphones, sizzle and suspended
cymbals, tablas, timbales, metal blocks, log
drums, boobams, bell plates in C and B flat, side drums,
Chinese cymbals, triangles, maracas, a bell tree, etc.
Modern composers can include almost anything that
makes a noise – everything from typewriters to vacuum
cleaners. The percussion section of the orchestra is
required to play them all – often simultaneously.
We will necessarily have to be selective in the instruments
that we choose to consider and will concentrate
largely on the more familiar instruments of the modern
classical symphony orchestra. We will also constrict
our attention to instruments that are struck and will ignore
instruments like whistles, rattles, scrapers, whips
and other similar instruments that percussion players are
also often required to play.
01 11 21 02 31 12
1 1.59 2.14 2.30 2.65 2.92
41 22 03 51 32 61
3.16 3.50 3.60 3.65 4.06 4.15
Fig. 15.114 The first 12 modes of a circular membrane illustrating
the mode nomenclature, nodal lines and frequencies
relative to the fundamental 01 mode
15.4.1 Membranes
Circular Membrane
A uniform membrane with areal density σ, stretched
with uniform tension T over a rigid circular supporting
frame, supports acoustically important transverse displacements
z perpendicular to the surface described by
the wave equation
�
∂2z 1 ∂z
T +
∂r2 r ∂r
+ 1
r 2
∂2z ∂φ2 �
= σ ∂2z , (15.147)
∂t2 which has solutions of the form
� �
z (r,φ,t) = Jm(kmnr)
A cos mφ
+B sin mφ
e iωt .
(15.148)
Jm(kmnr) are Bessel functions of order m, where
n denotes the number of radial nodes and m the
number of nodal diameters. The eigenfrequencies
ωmn = kmn
√ T/σ are determined by the requirement that
Jm(kmna) = 0 on the boundary at r = a. The frequency
of the fundamental (01) mode is (2.405/2πa) √ T/σ,
where J0(k01a) = 0. The relative frequencies of the first
12 modes and associated nodal lines and circles are
showninFig.15.114.
For ideal circular symmetry, the independent azimuthal
cosine and sine solutions result in degenerate
modes having the same resonant frequencies. The
degeneracy of such modes can be lifted by a nonuniform
tension, variations in thickness when calfskin
is used, and a lack of ideal circularity of the supporting
rim. Any resulting splitting of the frequencies
of such modes can result in beats between the
sound of given partials, which the player can eliminate
by selectively adjusting the tension around the
perimeter of the membrane or by hitting the membrane
at a nodal position of one of the contributing
modes.
Unlike transverse waves on a stretched string, the
modes of a circular membrane are inharmonic. As a consequence,
the waveforms formed by the combination
of such partials excited when the drumhead is struck
are non-repetitive. Audio illustrates the frequencies
of the first 12 modes played in sequence.
illustrates their sound when played together
as a damped chord, which already produces the realistic
sound of a typical drum, having a reasonably
well-defined sense of pitch, despite the inharmonicity of
the modes.
Although percussion instruments may often lack
a particularly well-defined sense of pitch, one can nev-