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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