Basic
physics of the ideal circular membrane
We
begin by analyzing the resonant properties of an ideal circular
mem-brane. although not restricted to a circular shape, many
drums feature this property, providing a convenient starting
point for discussion.
We solve
for the harmonic frequencies of an ideal, thin, homogeneous,
stretched, circular membrane using Bessel's functions. we
assume the outer circular edge of the membrane constitutes
a fixed boundary condi-tion, as with any standard drum. for
such a membrane we find the funda-mental frequency inversely
proportional to the radius, directly proportional to the square
root of the tension, and inversely proportional to the square
root of the mass per unit area.
Because of the nature of the material and its boundary conditions,
the vi-brational energy exhibits different observable "modes".
each of these mo-des represents a manner in which the material
moves in response to the vibrational energy. we distinguish
these modes by noting which areas of the membrane moves, and
which areas do not. we find it convenient to de-signate the
stationary points on the surface of the membrane where the
material remains in a fixed position, as "nodes".
the nodes, in effect, draw boundaries around the material
which vibrates. these boundaries, or no-des, consist of three
basic types: nodal points, nodal diameters and nodal circles.
For circular membranes, we designate the normal modes of vibration
by the notation "(x,y)", where x indicates the number
of nodal diameters, and y indicates the number of nodal circles.
we leave nodal points out of this discussion for simplicity,
due to the rarity of the phenomenon. to see the first 14 modes
of an ideal circular membrane, their mode designations, and
their relative modal frequency, click here. note that, none
of the modal frequencies consist of multiples of the fundamental,
and thus do not consti-tute a harmonic series. note here that
two headed drums complicate ana-lysis by introducing coupling
between the to resonating membranes. (click here to see some
details on that subject). also note the addition of each diametric
division of the membrane results in the next harmonic mode
(e.g. 3 diametric divisions, third harmonic). whereas the
addition of each circular node results in the next odd harmonic
(e.g. three circular nodes, fifth harmonic). when considering
the circular nodes, always consider the fixed boundary of
the membrane itself.
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Acoustic
properties of the timpani
By
modifying or introducing certain design features, we may emphasize
particular overtones, and even alter them completely. a carefully
tuned classical western timpani is known to have a strong
principle note, as well as two or more harmonic overtones,
including a prefect fifth, major se-venth, and an octave.
these overtones come from the (2,1),(3,1), and (1,2) modes
respectively. furthermore, recent measurements indicate the
mo-des (1,1), (4,1) and (5,1) have ratios 1, 2.44, and 2.9
respectively. both of these represent frequencies within a
semitones, from the ratios 2.5 and 3, respectively. thus the
first five nodal diameters (0,1),(1,1),(2,1),(3,1), and (4,1)
give the timpani the frequency profile of 1:2:3:4:5:6 -- yielding
a strong sense of pitch. the timpani employs several features
to alter the overtones of an ideal circ
The largest factor for the "correction" of the overtones,
into a close appro-ximation of a harmonic series, stems from
the mass of the air which the membrane vibrates against. the
timpani features a large surface area and thus interacts with
a large volume of air. this air mass serves to lower the frequencies
of the principle modes of vibration. the shape of the timpani's
large conical shell exhibits resonance properties of its own.
modes with similar shapes interact and reinforce each other,
though the medium of the air trapped inside it the timpani.
the stiffness of the preferred timpani membrane, raises the
frequencies of higher overtones. all of these proper-ties
shift the harmonic overtones and result in a close approxima-tion
of a harmonic series (from which to designate pitch). ular
membrane.
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Acoustical
properties of the bass drum
With
similar design features as the timpani, the bass drum also
exhibits these features. a large symphonic bass drum exhibits
a near harmonic series in the low frequency range from 32hz
to 200hz. however, the ear hears frequencies above 200hz much
better than between 32hz - 200hz. inharmonic frequencies above
200hz saturate the bass drum's frequency spectrum, and thus
gives the drum an undecernable relative pitch.
The
tabla employs many interesting features to "re-normalize"
its overto-nes into a true harmonic series. having a perfect
harmonic series, the tab-la exhibits a perfect tunable pitch.
the main feature, which "corrects" the overtones,
results from loading the membrane with a graduated weight
(heaviest at the center of the membrane, decreasing towards
the outer edge, and stops abruptly approximately half-way
towards the outer edge). this modification results from applying
concentric layers of wet flour (or rice) paste, mixed with
iron powder. a skilled tabla maker uses a small soft stone
to dry the paste, pack the material, and create a smooth sur-face.
once hard, the tabla maker applies the next layer (ontop),
with slightly smaller radius (from the center). the tabla
maker assures the harmonics become properly adjusted during
this process by monitoring the tone of the drum at each stage,
and adjusting the weight of each layer according-ly. research
demonstrates with the application of each new layer, certain
overtone frequencies which ordinarily result from different
modes, shift closer and closer towards each other (towards
an appropriate harmonic overtone). upon completion, several
ordinarily distinct overtones have the same frequency. for
example, with each application of the another layer to the
shiyahi, the (0,2) and (2,1) modes gradually approach the
third harmo-nic. to see nodal pattern of nine normal and seven
combination modes of the tabla, their mode designations, and
their relative modal frequency, click here.
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The
tabla's acoustical properties, its harmonic series, application
of the shiyahi.
When complete, this black patch (called the shiyahi, or gob),
not only re-sults in the drum's harmonic series (thus tunable
pitch), but also gives the drum a unique surface on which
to create sounds, unavailable to drums with an unmodified
membrane.
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Producing
differ
Different
strokes and their placement on the membrane emphasize diffe-rent
harmonic modes. a quick tapping motion placed at the center
of the tabla brings out the fundamental mode (the strokes
Tun, Tin). placing the third finger on the drum, and snapping
with the first finger brings out the first harmonic (the
strokes Na, Thin). this occurs because the finger resting
on the tabla create a diametrical node across the drum,
restricting the fun-damental mode. as with all drums, striking
closer and closer to the edge brings out higher harmonics,
due to the restricted amplitude of the boun-dary region
and wave reflection at the boarder. the surface tension
and membrane sheer forces the energy in to higher modes.
however, with the tabla we note the harmonic character of
these modes because of its acoustical properties (described
above).
ent
modes with different strokes.
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The
bayan's aco
With
the bayan, the shiyahi is placed off-center. the performer
rests their hand on membrane, generally on the side furthest
from the shiyahi. this results completely dampens a portion
of the membrane, and effectively re-centers the shiyahi. with
less research on this drum, we can not thoroughly discuss
the harmonics of this drum. we note, that due to the asymmetrical
nature of the bayan (without a hand resting on the drum) the
overtones will not constitute of a harmonic series and will
not result a discernible pitch, as with the dayan. this is
well known empirically. as a unique feature of the bayan,
performers change the its pitch both by siding their hand
across the drum, and sometimes by applying pressure on the
membrane. siding of the hand across the bayan decreases the
radius of area of the resonant area of the drum (which is
inversely proportional to pitch). this renders a strong sense
of pitch over a wide range of frequencies. an advanced performer
uses the lyrical qualities of this drum to embellish the rhythm.
ustical
properties, and modulation of pitch.
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The
multi-layered composite membrane design and construction.
Both
the tabla and the bayan have a multi-layered membrane. The
main layer, complete covers the "mouth" of the drum.
Two layers, one above this main layer and one below, cover
only a small outer portion of this area. An annular strip
covers approximately two centimeters of the drum. These an-nular
strips serve to dampen higher harmonics, which rely more on
the ou-ter portions of the membrane than the lower harmonics.
Performers may also place a string between the top annular
strip (accessible from the top of the drum) and the main layer,
to adjust this effect by controlling the amount of contact
between these layers.
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Fastening
the membrane to the tabla, and its tuning.
This
membrane is assembled together with a interwoven leather thread,
and given 16 holes around the edge. another leather thread
weaves these 16 points on the membrane to a leather hoop located
at the bottom of the drum. thus fastening the membrane to
the drum itself. by decreasing the length of this weave, one
may increase the tension on the membrane, and thus increase
the pitch. In addition, wooden pegs placed between the drum
and these straps increase the tension on the membrane by pulling
on the straps. this is essential for the tabla, which uses
eight pegs equally placed around the drum. a tablist tunes
the tabla by adjusting the peg's position. due to the geometry
of the tabla, lowering the pegs position in-creases the tension
on the membrane by pulling on the straps. a tablist fine tunes
the drum by tapping on the edge of the membrane, adjusting
the position of the membrane, which increases/decreases the
membrane's tension, and thus increases/decreases the pitch.
equal tension around the drum is critical to proper tuning
of the dayan.
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Properties
and materials of the shell.
The
tabla's shell consist of a very hard wood. The tabla's shell
is much thicker than most other drums, of any size. furthermore
its inner cavity is rather shallow. these features yields
a much smaller volume (of trapped air) inside than its outer
shape or appearance implies (when compared with most other
drums). the thick shell increases sustained resonance of the
membrane by minimizing energy dissipation through the shell.
recent research of making a dayan with an aluminum shell resulted
in a extremely heavy dayan with remarkable tonal quality.
The
bayan's shell on the other hand, consists of a "thin"
polished bronze layer, which accurately portrays the volume
of air trapped inside the drum. the bayan's shell seldomly
consist of clay, or even less frequently (and ge-nerally much
smaller in size) with wood -- usually found in rural areas.
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