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Acoustics of Idakk¯a: An Indian Snare Drum with Definite Pitch
Kevin Jose,1Anindya Chatterjee,1and Anurag Gupta1
Department of Mechanical Engineering, Indian Institute of Technology,
Kanpur 208016, Indiaa)
The vibration of a homogeneous circular membrane backed by two taut strings is1
shown to yield several harmonic overtones for a wide range of physical and geometric2
parameters. Such a membrane is present at each end of the barrel of an idakk¯a, an3
Indian snare drum well known for its rich musicality. The audio recordings of the4
musical drum are analysed and a case is made for the strong sense of pitch associ-5
ated with the drum. A computationally inexpensive model of the string-membrane6
interaction is proposed assuming the strings to be without inertia. The interaction es-7
sentially entails wrapping/unwrapping of the string around a curve on the deforming8
membrane unlike the colliding strings in Western snare drums. The range of param-9
eters, for which harmonicity is achieved, is examined and is found to be conforming10
with what is used in actual drum playing and construction.11
a) ag@iitk.ac.in
1
Acoustics of Idakk¯a
I. INTRODUCTION12
A few ingenious drum designs have made it possible to obtain a definite pitch, with13
several harmonic overtones, out of an otherwise inharmonic vibrating circular membrane.14
These include timpani, where the kettle is suitably constructed to produce up to four har-15
monic overtones,1and Indian drums, such as tabl¯a, pakh¯awaj, and mradangam, where the16
composite nature of the membrane yields at least ten harmonic modal frequencies.2–5Apart17
from these exceptions, drums are known to possess an indefinite pitch and hence useful only18
for generating rhythmic sounds.6In this paper, as another exception, we report the acous-19
tics of idakk¯a, an Indian bi-facial snare drum with a waisted barrel. Idakk¯a is at present20
played as a temple instrument in the south Indian state of Kerala and is found frequently21
in ancient Indian sculptures and musicology texts.7,8It is often used to play intricate r¯aga22
based melodies and can swiftly move its fundamental in a range of about two octaves. The23
instrument is worn on the left shoulder and played by striking one of the anterior drum-24
heads with a stick held in the right hand; see Figure 1. The left hand, while holding the25
barrel, pushes the waist of the barrel in order to effect the rich tonal variations in idakk¯a’s26
sound.9The purpose of this article is to establish the rich harmonic nature of idakk¯a first27
by analyzing the audio recordings and then by proposing a novel numerical model for the28
string-membrane dynamic interaction.29
The two idakk¯a drumheads are made from a thin hide, of density µ≈0.1 kg-m−2,30
obtained from the interior stomach wall of a cow. The thinness of the hide allows for large31
tension variations in the membrane. The hide is pasted onto a jackfruit wood ring, 2.5 cm32
2
Acoustics of Idakk¯a
FIG. 1. An idakk¯a being played. (color online)
thick with an internal diameter of around 20 cm. The two drumheads are connected by a33
cotton rope which threads through six equidistant holes in the rings and forms a V-shaped34
pattern between the heads. The shoulder strap has four extensions each tied to three of35
the rope segments between the drumheads; see Figure 1. This arrangement connects all36
the rope segments symmetrically to the shoulder strap, so that pushing the latter would37
uniformly change the tension in the drumheads. Four wooden pegs, each about 18 cm long,38
are inserted between the segments and sixteen spherical tassels are attached to each of these39
pegs (additional illustrations of idakk¯a are provided in the supplement10). Whereas the pegs40
keep the rope segments taut and in proper position, the purpose of hanging the tassels seem41
to be only of cultural significance.8
42
The barrel is usually made of jackfruit wood, which has a dense fibrous structure with43
low pore density and hence high elastic modulus. The barrel is around 20 cm in length,44
with two faces of diameter around 12 cm, and a waist of a slightly lower diameter as shown45
in Figure 2(bottom). The face diameter, being almost half of the drumhead size, ensures46
3
Acoustics of Idakk¯a
FIG. 2. Distinctive curved shape of the rim of the barrel (top left). A pair of strings installed
on an idakk¯a barrel (top right). Barrel of an idakk¯a (bottom). A piece of cloth may be used (as
shown) to improve the grip. The nails to which snares are tied are also visible close to the edge of
the barrel. (color online)
that a uniform state of tension indeed prevails within the vibrating membrane even when47
the drumhead tension is controlled only at six isolated points. The walls of the barrel48
are about 1 cm thick. The rim has a distinctive convex shape, as seen in Figure 2(top49
left), allowing for the membrane to wrap/unwrap over a finite obstacle during its vibration.50
This is analogous to vibration of a string over a finite bridge in Indian string instruments51
leading to a rich spectrum of overtones.11,12 Two palmyrah fibre strings, of linear density52
λ≈10−4kg-m−1, are stretched and fixed across each face of the barrel, at about 6 mm53
from the center, and tied to copper nails on the side; see Figure 2. Channels are cut into54
the rim of the barrel to ensure that the fibres sit flush with the rim and therefore with the55
4
Acoustics of Idakk¯a
drumhead. The fibres are soaked in water before they are installed. The tension in the fibres56
is estimated to be around 6.5 N; the details of this estimation are given in Section II A of57
the supplement.10 The drumheads are held tight against the barrel with the cotton rope and58
are easily disassembled when not in use. Also, whereas the drumheads can be tuned using59
the cotton rope, as described in the previous paragraph, the fibres are never tuned. The60
fibres are stretched till they are close to the breaking point and then tied onto the rim of61
the barrel. More details about the structure of idakk¯a can be found elsewhere.8,13
62
The acoustics of idakk¯a is governed by several factors, prominent among which are: (i) the63
string-membrane interaction, (ii) the curved rim of the barrel, (iii) varying membrane tension64
using the tensioning chords, (iv) the coupling between the two drumheads, and (v) the air65
loading. The string-membrane interaction in idakk¯a is tantamount to wrapping-unwrapping66
of the string around a curve of the vibrating membrane. Such a contact behavior, rather than67
an impact, is expected due to higher tension and lower mass density of the palmyrah strings68
as compared to the metallic strings (used in Western snare drums). The material used for69
the membrane and the strings, the curved rim, and the intricate method of tensioning the70
chords are all unique to idakk¯a. Out of the five factors mentioned above, our emphasis will71
be to model the string-membrane interaction using a computationally inexpensive model,72
while assuming the strings to be without mass, without damping, and forming a convex hull73
below a curve on the vibrating membrane. The influence of the curved rim on the overall74
acoustics of the drum is discussed briefly in the supplement.10 We will ignore the effects of75
drumhead coupling and air loading in the present work.76
5
Acoustics of Idakk¯a
The motivation for our study is provided in Section II by analysing the audio recordings77
of idakk¯a’s sound. A case is made for the rich harmonic sound of the drum. A mathematical78
model for the string-membrane interaction is proposed in Section III. The results of the79
model are discussed in detail in Section IV. These include finding the optimum geometric and80
material parameters for achieving harmonic overtones, recovering the obtained frequencies81
from a nonlinear normal mode analysis, and comparing our model with an existing collision82
model. The paper is concluded in Section V.83
II. MOTIVATION FOR OUR STUDY84
The audio recordings were done on a TASCAM DR-100MKII Linear PCM recorder at a85
sample rate of 48 kHz with a bit depth of 16. While splicing the audio file into individual86
samples, the sample rate was changed to 44.1 kHz. The change in the sample rate was87
a result of saving the spliced files in the default sampling rate of Audacity.14 We do not88
anticipate any difference in the results since the frequencies of our interest are of the order89
of 1 kHz, much smaller than any of these two frequencies. Although the original recordings90
were done in stereo, we have used only one channel for our analyses. The spectrograms91
and the power spectral densities (PSD) are plotted using the spectrogram and pwelch92
commands, respectively, in MATLAB.15
93
The expert musician (Mr. P. Nanda Kumar) played the seven notes of the musical scale94
in Indian classical music forward and backward with four strokes of each note. The first95
stroke, in almost every case, has a swing in frequency in the initial part of the stroke, see96
Figure 3(left), as if the musician is correcting himself to reach the correct pitch as he plays97
6
Acoustics of Idakk¯a
1234
Time (secs)
50
100
150
200
250
300
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/frequency (dB/Hz)
123456
Time (secs)
60
80
100
120
140
160
180
200
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/frequency (dB/Hz)
FIG. 3. Swings in frequency as seen in the audio recordings as correction (left) and anticipation
(right). (color online)
(and hears) the note for the first time. A swing in frequency is also observed towards the98
end of the last stroke, see Figure 3(right), possibly in anticipation of the note to be played99
next. These swings, in addition to being a corrector or anticipator, are also a reflection of100
the gamak¯as related to the r¯aga being played. It is therefore best to consider either the101
second or the third stroke of each note for further analysis. We choose the latter. We also102
note that the spectrograms contain inharmonic content for a small initial time duration; a103
typical spectrogram is shown in Figure 4(left). The inharmonicity is due to the influence104
from striking of the drum. The harmonic content however dominates the spectrogram, as105
well as the auditory experience, and sustains itself. We consider only the latter portion of106
the spectrograms for obtaining the PSD plots. The spectrograms for the case without strings107
show inharmonic content throughout as well as faster decay rates for the higher modes, see108
for instance Figure 4(right).109
7
Acoustics of Idakk¯a
100 200 300 400 500
Time (ms)
0
0.5
1
1.5
2
2.5
3
Frequency (kHz)
-140
-120
-100
-80
-60
-40
Power/frequency (dB/Hz)
100 200 300 400 500
Time (ms)
0
0.5
1
1.5
2
2.5
3
3.5
4
Frequency (kHz)
-140
-120
-100
-80
-60
-40
Power/frequency (dB/Hz)
FIG. 4. Typical spectrograms with (left) and without (right) the strings. (color online)
In Figure 5, we have three PSD plots each corresponding to the case of idakk¯a with (top110
row) and without (bottom row) strings. The fundamental frequency f0in each of these111
plots is different, representing different values of tension in the drumhead. There are three112
differences between the two cases. First, the power/frequency peaks for overtones are about113
20 dB higher in the case with strings compared to the case without strings. Second, the114
dominant overtones in the PSD plots with strings are always harmonic,16 whereas there115
is a significant inharmonic content in the other case. To show this more clearly, we have116
marked the inharmonic peaks with orange markers in the PSD plots for the case without117
strings. Some of these inharmonic peaks are present with insignificant amplitudes even in118
the top row, where the relative importance of harmonic peaks is evident. It should also119
be noted that, in the top row plots, the third overtone peak is always accompanied by a120
secondary peak, of much lower intensity, for instance those marked by a blue line at around121
3.25f0. This is indicative of a beat like phenomena. Also, the second harmonic is missing in122
these plots. This absence is not perceived when listening to an idakk¯a for well understood123
8
Acoustics of Idakk¯a
0
f0= 115
500
1000
1500
2000
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/Frequency (dB/Hz)
0
f0= 158
500
1000
1500
2000
2500
3000
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/Frequency (dB/Hz)
0
f0= 192
1000
2000
3000
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/Frequency (dB/Hz)
0
f0= 144
500
1000
1500
2000
2500
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/Frequency (dB/Hz)
0
f0= 200
1000
2000
3000
4000
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/Frequency (dB/Hz)
0
f0= 249
1000
2000
3000
4000
Frequency (Hz)
-140
-120
-100
-80
-60
-40
Power/Frequency (dB/Hz)
FIG. 5. PSD of idakk¯a drum samples with (top row) and without (bottom row) the strings in
idakk¯a. The dotted lines mark integer multiples of the corresponding fundamental frequency in
each plot. The blue line indicates 3.25f0. The orange arrows mark the inharmonic peaks. (color
online)
psychoacoustical reasons.17 Third, a harmonic distribution of overtones for the case with124
strings (top row in Figure 5) is observed over a large range of membrane tensions. This is125
noteworthy for a nonlinear problem such as the one present before us. These differences are126
sufficient to argue in favour of the distinctive role of strings in bringing about a harmonic127
character to idakk¯a’s sound. The plots without the strings indicate the combined role of the128
curved rim of the barrel, the air loading, and the drumhead coupling.129
9
Acoustics of Idakk¯a
We can also calculate the fundamental frequency of the string, seen as an isolated vibrat-130
ing structure. The tension and the linear density of the string are estimated as 6.5 N and131
10−4kg-m−1, respectively. The length of the string is 11 cm. The fundamental frequency132
can then be calculated as 1159 Hz. Since this value is larger than the frequencies for the133
fundamental in the PSD plots, in the top row of Figure 5, we can reasonably argue that the134
frequencies obtained are a result of the membrane-string interaction and not due to string135
vibration alone.136
III. MODEL137
As is evident from our analysis of the audio recordings of idakk¯a, the strings play a central138
role in bringing about harmonicity in idakk¯a’s frequency spectrum. As a first step towards139
building a mathematical model for idakk¯a, we begin by considering the transverse vibration140
of a uniformly tensed homogeneous circular membrane, with clamped edges, backed by two141
taut strings. The strings, whose ends are fixed to the circumference of the membrane, are142
under constant tension and run parallel to each other equidistant from the center of the143
membrane. At rest, the two strings sit below the plane of the membrane, with negligible144
distance between the membrane and the strings. This minimalist model is illustrated in145
Figure 6(a).146
Our model assume the strings to deform by forming a convex hull around a curve on the147
deforming membrane, thereby providing a contact force to the vibrating membrane at dy-148
namically varying contact regions; see Figures 6(b,c). We neglect the mass of the strings149
as well as any damping associated with them. The force of string-membrane interaction150
10
Acoustics of Idakk¯a
is therefore determined by statics alone. Furthermore, the strings are assumed to vibrate151
in a vertical plane orthogonal to the undeformed membrane. The plane of vibration for152
one of the strings is shown in Figure 6(b). The deformed position of the strings is derived153
directly from the shape of the membrane without solving the partial differential equation154
for the string motion. The deformed string acquires a shape of the convex hull of the curve155
obtained by intersection of the membrane with the plane. Such a deformation would entail156
contact of the strings with the membrane at dynamically varying regions. The strings, in157
this way, are understood to wrap around the curves which are the intersection of the de-158
forming membrane with the vertical planes. We compare our model with a penalty based159
contact model in Section III B. The latter, which solves a coupled system of membrane and160
string equations of motion, uses a one-sided power law to model the collision while penalising161
the physically unfeasible inter-penetration. A collision model is necessary for Western snare162
drums where the inertia of metallic snares cannot be ignored.18–21
163
A. The quasi-static string approximation164
The equation of motion of a clamped membrane, of radius R, backed by taut strings is165
given by166
µWtt =TM∆W−µσ0,M Wt+FS,(1)
where W(r, θ, t) is the transverse displacement of the membrane ((r, θ) are the polar co-167
ordinates and tis the time variable), µis its area density, TMis the uniform tension per168
unit length in the membrane, and σ0,M is the constant damping coefficient. The subscript169
tdenotes partial derivative with respect to time and ∆ is the 2-dimensional Laplacian.170
11
Acoustics of Idakk¯a
R
2b
Strings
W(R, θ, t) = 0, clamped
(a)
(b)
(c)
FIG. 6. (a) A schematic of the string-membrane configuration. The circular membrane is clamped
at the edge and the two strings sit below the membrane. Here, 2bis the distance between the
strings, Ris the radius of the membrane, and W(r, θ, t) is the transverse displacement of the
membrane. (b) An illustration of the strings forming a convex hull around the curve of intersection
between the membrane and the plane. A combination of the first two axisymmetrical modes of
a uniform circular membrane were used to generate the deformed profile. (c) A sectional view of
the string-membrane contact. The plane of section is shown in Figure (b). The grey line indicates
the intersection of the membrane with the vertical plane and the red line is the string below the
membrane. (color online)
12
Acoustics of Idakk¯a
The force density (per unit length) exerted by the strings on the membrane is denoted171
by FS. The clamped boundary requires W(R, θ, t) = 0. We represent the solutions as172
W(r, θ, t) = P∞
m=0 P∞
n=1 ηmn(t)φmn (r, θ), where ηmn (t) are the unknown time-dependent173
parts of the solution and φmn(r, θ) are the orthonormal mode shapes for a uniform circular174
membrane of radius Rclamped at the edges, i.e., φmn(R, θ) = 0 ∀m, n. Using this expan-175
sion we can convert the partial differential equation (1) into a coupled system of ordinary176
differential equations177
µ¨ηmn(t) = −γ2
mnTMηmn (t)−µσ0,M ˙ηmn(t) + 1
πR2ZA
φmnFSdA, (2)
where γmn =Bmn/R with Bmn being the nth root of the mth order Bessel function of the178
first kind. The superposed dot denotes the time derivative with respect to t. The integral179
is taken over the whole membrane such that dAis the infinitesimal area element.180
The membrane experiences a dynamic contact force density FSdue to its interaction181
with the two taut strings. We propose that182
Fs=−
2
X
i=1
fiδi
L,with fi=−TShi
ξiξi,(3)
where hi(ξi, t) is the transverse displacement of the string; δi
Lis the line delta function for183
the ith string; ξiis the intrinsic spatial coordinate on the string; TSis the uniform tension184
equal in both the strings; and the subscript ξidenotes the partial derivative of the function185
with respect to the spatial variable. We assume that the strings wrap around the membrane186
in a sense described above and illustrated in Figure 6(b). If every point of the membrane187
remains above the horizontal plane, the strings stays horizontal. Such a consideration lets us188
13
Acoustics of Idakk¯a
ignore the string dynamics making our method computationally less intensive than solving189
the full coupled problem including the equations of motion for the strings.190
We can non-dimensionalize the governing equation (2), and incorporate the force (3), by191
introducing dimensionless parameters ˜η=η/η0,˜
t=t/t0,˜
h=h/η0, and ˜
ξi=ξi/R, where192
t0=R/pTM/µ and η0is any non-zero positive real number with the dimensions of η; we193
take the magnitude of η0to be one. We obtain194
¨
˜ηmn =−B2
mn ˜ηmn −σ0,M t0˙
˜ηmn +χ
2
X
i=1 Z√1−(ψ)2
−√1−(ψ)2
˜
hi
˜
ξi˜
ξiφmn(˜
ξi) d˜
ξi!,(4)
where195
χ=TS
πRTM
and ψ=b
R.(5)
In writing the last pair of terms in (4) we have considered the geometry in accordance with196
Figure 6(a). In particular, Ris the radius of the membrane and 2bis the distance between197
strings. Also, the superposed dot is now indicative of the derivative with respect to variable198
˜
t. The non-dimensionalized governing equation has three dimensionless parameters: ψis199
purely geometric in nature, χis essentially the ratio of string to membrane tensions, and200
σ0,M t0is the dimensionless damping coefficient term. The tension TMis the only parameter201
which can be varied while playing an idakk¯a. The others are fixed once for all. The integral202
terms in (4) is the source of nonlinearity in the equation and also of coupling between the203
modes.204
14
Acoustics of Idakk¯a
-6 -4 -2 0 2 4 6
(cm)
-0.3
-0.2
-0.1
0
0.1
W (mm)
K = 1e+10, = 1.3
-6 -4 -2 0 2 4 6
(cm)
0
5
10
TS h (N/m)
(a)
-6 -4 -2 0 2 4 6
(cm)
-0.3
-0.2
-0.1
0
0.1
W (mm)
K = 1e+12, = 1.3
-6 -4 -2 0 2 4 6
(cm)
0
5
10
TS h (N/m)
(b)
FIG. 7. Comparison of displacements and forces as obtained from Equation (8) (dotted line) and
the quasi-static string approximation (black line) for two different values of Kin (a) and (b). A
typical membrane shape is shown as a grey line.
B. Comparison of the quasi-static string approximation with a penalty based205
method206
In a penalty based method we take the contact force as a one-sided power law207
Fs=−
2
X
i=1
fiδi
L,with fi=−Khi−W(t, r(ξi), θ(ξi))α,(6)
where Kand αare constants, and [x] = (1/2)(x+|x|).21 In addition the equations for string208
dynamics are also considered in the form209
λhi
tt =TShi
ξiξi−λσ0,S hi
t+λσ1,S hi
ξiξit+fi,(7)
where λis the linear mass density of the string, and σ0,S,σ1,S are constant damping coeffi-210
cients. The bending term is ignored due to low bending stiffness of the palmyrah fibres.211
15
Acoustics of Idakk¯a
For palmyrah fibres in maximal tension, the tension and the power law terms dominate212
over rest of the terms. This can be seen by writing (7) in a non-dimensionalized form213
and considering material values for the palmyrah fibres. The string equation can then be214
considered in a simplified form215
0 = TShi
ξiξi−Khi−W(t, r(ξi), θ(ξi))α(8)
for each string. It is solved numerically using the non-linear solver fsolve in MATLAB.15
216
We consider only two modes such that η01 =−0.1 mm and η02 = 0.1 mm. We fix α= 1.3.22
217
The results are shown in Figure 7. A large value of Kis needed to minimize inter-penetration218
of string and the membrane. We also compare the results with what is used in the quasi-219
static string approximation. A typical membrane shape (R= 55 mm and b= 6 mm) is used220
and the MATLAB function convhull is used to obtain the convex hull. We note that for a221
lower value of K, in Figure 7(a), there is slight inter-penetration and there is a considerable222
difference between the contact force as calculated from the penalty method and the quasi-223
static string approximation. For a higher Kvalue, the inter-penetration reduces significantly224
and the two contact forces come in better agreement. In a penalty based method the inter-225
penetration would never vanish all together unlike the present method where it is zero by226
construction. Therefore, on one hand, we can view the quasi-static string approximation as a227
simplification of the penalty based method achieved by ignoring string inertia and damping228
along with an appropriately high value of K, but on the other hand we note that the former229
is exact in enforcing avoidance of the inter-penetration of strings into the membrane.230
16
Acoustics of Idakk¯a
0 0.2 0.4 0.6 0.8 1
t (s)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
η (mm)
χ = 0.15
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
(a)
0
f0 = 384
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
χ = 0.15
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
(b)
FIG. 8. (a) Waveform and (b) PSD plot corresponding to TM= 250 N and TS= 6.5 N (such
that χ= 0.15); ψ= 0.1091. The dotted lines in (b) mark integer multiples of the fundamental
frequency. (color online)
IV. RESULTS AND DISCUSSION231
We solve Equations (4) numerically, using ode113 solver in MATLAB,15 by considering232
only the first twelve modes (including odd and even degenerate modes), such that m=233
0,1,2,3,4 and n= 1,2. These equations are coupled to each other as well as they are234
nonlinear in the unknown variables ˜ηmn , both due to the integral term present therein.235
Indeed, the string displacement ˜
hin the integrand is obtained from restricting the convex236
hull of the membrane (which is in terms of the unknown ˜ηmn ) to the one-dimensional domains237
of the two strings. In doing so, the strings are assumed to move only transversely. The238
convex hull is calculated using the convhull function in MATLAB. The evaluation of string239
displacements is done concurrently while solving the equation, in the sense that the string240
shape at time tis a function of membrane shape at time t. The integral term in (4) is241
17
Acoustics of Idakk¯a
approximated using the trapezoidal rule with 201 integration points. For all our simulations242
we use an initial condition of a small displacement (away from the strings) of the membrane243
along the shape of the fundamental mode of uniform membrane vibration. Interestingly, in244
all our simulations, only four modes, (0,1)e,(2,1)e,(0,2)eand (4,1)e(the subscript e denotes245
the even mode), remain dominant while the others are excited only weakly. Accordingly,246
we report the results for only these four modes. Most importantly, for certain parametric247
values, both significant energy transfer to higher modes and rich harmonicity in the frequency248
spectrum are observed. This is illustrated in Figure 8where the waveform and the PSD249
plot of the solution for a typical set of parameters are produced. As noted above, it is the250
interaction of the membrane with the strings which is the source of both non-linearity and251
coupling between the normal modes. Without the string-membrane interaction terms, as252
expected, only the (0,1)emode is excited, and the PSD plot shows one isolated peak for the253
fundamental.254
In the following, we begin by looking at the range of membrane tension values, keeping255
other parameters fixed, for which harmonicity is achieved. This is followed by a similar256
attempt for the distance between strings. This lets us justify the harmonicity of idakk¯a over257
a wide range of tension values, on one hand, and the optimal design of string placement on258
the other. Next, we attempt to understand the distinctiveness of idakk¯a’s drumhead design259
as compared to that of the Western snare drum, knowing well that the latter is bereft of260
harmonic rich sound.23 In Section IV C, we recover the obtained frequencies by a nonlinear261
normal mode analysis.262
18
Acoustics of Idakk¯a
A. Effect of varying χand ψ263
The solutions are obtained by varying TMover a factor of 3 such that χis varied in a range264
of 0.06 to 0.4; see Figure 9. This represents a change in the fundamental frequency by about265
1.7 times. The plots for χbetween 0.07 to 0.21 are striking for the appearance of distinct266
harmonic peaks. The peaks are sharpest for χaround 0.16. The harmonic peaks are also267
accompanied by smaller peaks of much lower intensity, suggesting a beat like phenomenon.268
This was also noted in the spectra obtained from the audio recordings. Outside this range269
of χ, several inharmonic peaks start to appear, so much so that around χ= 0.4 there is no270
definite harmonic character in the overtones. The range of desirable χvalues may be slightly271
affected if the effects due to curved rim, air loading, and bi-facial membrane coupling are also272
incorporated. While obtaining the solution for various values of χ, it must be ensured that273
the basic assumption of massless strings in the quasi-static string approximation remains274
justified. In other words, it must be ensured that inertia term in the string equation remains275
small, i.e., 1 (TSt2
0)/(R2TM) = (µTS)/(λTM) = (µπRχ)/(λ). The broad range of χwhich276
ensures a near harmonic response is testimonial of idakk¯a’s playing over a wide range of277
membrane tension values. It also ensures that the harmonic response is not too sensitive to278
the precise tension values in the two strings.279
The spectra are also obtained for different values of the geometric parameter ψ(= b/R);280
see Figure 10. Other parameters are kept fixed, including χ= 0.15. A variation in ψ281
represents a variation in both the distance bbetween the snares and the radius Rof the282
membrane. Sharp harmonic peaks are observed for a wide range of ψvarying between 0283
19
Acoustics of Idakk¯a
0
f0 = 584
1000
2000
3000
4000
5000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
χ = 0.06
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 544
1000
2000
3000
4000
5000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
χ = 0.07
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 444
1000
2000
3000
4000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
χ = 0.11
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 376
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
χ = 0.16
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 340
500
1000
1500
2000
2500
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
χ = 0.21
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 260
500
1000
1500
2000
2500
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
χ = 0.4
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
FIG. 9. The presence of harmonic overtones for a range of 0.07 <χ<0.21, ψ= 0.1091. The
dotted lines mark integer multiples of the fundamental frequency. (color online)
and 0.4. The peaks are the sharpest, with highest intensity, for ψaround 0.1, which is also284
close to the value usually used in idakk¯a construction. The secondary peaks are present285
although with a relatively lower intensity. The robustness of the system in maintaining a286
near harmonic response over a large range of geometric configurations is in confirmation287
with the existence of other drum designs, for instance the much smaller udukku which has288
only one snare passing through the center (i.e., ψ= 0).8
289
20
Acoustics of Idakk¯a
0
f0 = 384
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
ψ = 0
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 384
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
ψ = 0.1
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 384
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
ψ = 0.2
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 384
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
ψ = 0.3
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 376
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
ψ = 0.4
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0
f0 = 372
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
ψ = 0.5
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
FIG. 10. Harmonic response observed for over a range value of 0 < ψ < 0.4, χ= 0.15. The dotted
lines mark integer multiples of the fundamental frequency. (color online)
B. Comparing idakk¯a with the snare drum290
Unlike idakk¯a, the Western snare drum is not known to produce harmonic overtones.18,23
291
It is then important to understand the differentiating characteristics of the snare action in the292
former drum. The difference essentially comes from the material of the snare and the tension293
values in the two drums. Idakk¯a has natural fibres as snare strings which have lighter weight294
when compared to metallic strings used in snare drums. On the other hand, the tension in295
idakk¯a’s snares are more than three times that in the snare drum. Additionally, the tension296
21
Acoustics of Idakk¯a
0
f0 = 392
1000
2000
3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
= 0.15
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
FIG. 11. PSD plot obtained for an idakk¯a drumhead using the penalty method. The dotted lines
mark integer multiples of the fundamental frequency. (color online)
in idakk¯a’s membrane is an order of magnitude lower than in the drumhead of the snare297
drum. All of this allows us to ignore the string inertia term in the case of idakk¯a. We use the298
penalty method to compare the waveform for the two cases. The simulation for snare drum299
is performed with geometric and material parameters as given by Torin and Newton.18 The300
penalty method based solutions for idakk¯a drumhead give results close to those obtained301
using the quasi-static string approximation, with fundamental frequencies obtained within302
2% of each other, as can be observed by comparing Figure 11 with Figure 8(b). For want303
of data, the coefficients to the damping terms for palmyrah fibres are taken equal to that304
of steel. The material and geometric parameters used are TM= 250 N-m−1,TS= 6.5 N,305
σ0,M = 20 s−1,σ0,S = 2 s−1,σ1,S = 0.001 m2s−1,µ= 0.095 Kg-m−2,λ= 1.07 ×10−4Kg-306
m−1,R= 55 mm, b= 6 mm, K= 1010, and α= 1.3. Some of these parameters have been307
22
Acoustics of Idakk¯a
0 0.2 0.4 0.6 0.8 1
t (s)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
mn (mm)
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
(a)
0 0.2 0.4 0.6 0.8 1
t (s)
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
ηmn (mm)
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
(b)
FIG. 12. Waveforms obtained from a penalty method based simulation of (a) an idakk¯a drumhead
and (b) the snare drum drumhead. The physical parameters for the latter is taken from Torin and
Newton.18 (color online)
taken from Torin and Newton.18 The inclusion of string damping is a possible explanation308
for the non-appearance of higher harmonics in Figure 11.309
The waveforms for the two drum heads are shown in Figure 12. The waveform in Fig-310
ure 12(a) looks different from that in Figure 8(a) due to incorporation of damping in the311
former. The waveform of the snare drum shows abrupt changes in the amplitude of the312
mode shapes. This is caused by the collisions happening at a frequency much lower than313
the frequency of vibration of the membrane modes, which, in turn, is due to the high mass314
density and low tension of the metallic snares. On the other hand, no such changes are seen315
for the idakk¯a drumhead where the string collisions happen at a frequency higher than that316
of the membrane vibration. This difference also validates our choice of using the quasi-static317
method rather than a collision based method. A second thing to note in these waveforms is318
the amount of energy transfer that is occurring from the fundamental to the higher modes. It319
23
Acoustics of Idakk¯a
0 500 1000 1500 2000 2500
t (µs)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
f0= 384.76Hz
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0 200 400 600 800 1000 1200
t (µs)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
f0= 784.85Hz
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0 200 400 600 800 1000
t (µs)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
f0= 861.19Hz
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
0 200 400 600 800
t (µs)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
f0= 1157.35Hz
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
FIG. 13. Non-linear normal mode solutions corresponding to the zero velocity initial condition.
The variable ηon the y-axis denotes a scaled displacement amplitude. (color online)
is clear that, in the case of idakk¯a, several higher modes are predominately excited whereas,320
for the snare drum, the fundamental dominates over all other modes for all times.321
C. Non-Linear Normal Modes322
A Non-linear normal mode (NNM) of an undamped continuous system is defined as a323
synchronous periodic oscillation where all the material points of the system reach their324
extreme values or pass through zero simultaneously, thereby appearing as a closed curve in325
the configuration space.24 The purpose of this section is to determine NNMs for a system326
24
Acoustics of Idakk¯a
governed by an undamped form of Equation (4). We expect the frequencies corresponding327
to NNMs to be close to those obtained using the full dynamical solution. This will provide328
an independent validation of the nature of frequency spectra as observed in our simulations.329
In order to determine NNM we fix the initial velocity to be zero and look for a time period330
of the solution such that it crosses the zero velocity condition again in the configurational331
space. More details are provided in Section III B of the supplement.10 Fixing parameter332
values such that χ= 0.15 and ψ= 0.1091, we determine four such periodic solutions. The333
frequencies associated with these solutions are 384.76 Hz (say f), 784.85 Hz (= 2.04f),334
861.19 Hz (= 2.24f), and 1157.35 Hz (= 3.008f). The second and the fourth values clearly335
correspond to the second and the third harmonic, respectively. The third frequency value is336
close to the neighboring peak to the second harmonic as observed in most of the frequency337
spectra. The solution corresponding to these four frequencies is shown in Figure 13. The338
mode (0,1)eis seen to have a mean value greater than zero in all four NNMs. It is possible339
to find other NNMs by choosing different initial conditions. The four frequencies are also340
shown superposed on a PSD plot obtained using the quasi-static string approximation in341
Figure 14.342
The frequencies of the NNMs bear emphasis. For wide values of χ, we have already seen343
that the response of the struck drum is dominated by an almost periodic, but not sinusoidal,344
response, leading to strong harmonic content. In such cases, however, different NNMs have345
different frequencies which do not bear simple relationships with each other. For the special346
value of χ= 0.15, however, we find that three different NNMs end up with frequencies347
close to the proportions of 1 : 2 : 3, which leads to a particularly strong and rich harmonic348
25
Acoustics of Idakk¯a
0 1000 2000 3000
Frequency (Hz)
-240
-220
-200
-180
-160
-140
-120
-100
-80
Power/Frequency (dB/Hz)
Mode: (0,1)e
Mode: (2,1)e
Mode: (0,2)e
Mode: (4,1)e
FIG. 14. PSD plot of the vibration of a membrane backed by two strings using the quasi-static
string approximation superposed with dotted lines which mark the calculated NNM frequencies;
χ= 0.15 and ψ= 0.1091. (color online)
response.349
The issue can be clarified further as follows. Many conservative nonlinearities might yield350
several different periodic solutions. Each such periodic solution, if excited in isolation, would351
be non-sinusoidal and possess harmonics. However, typical nonlinearities make the frequency352
dependent on amplitude. For the idakk¯a, the nonlinearity leads to periodic solutions whose353
frequency does not depend on amplitude. Confining attention to such nonlinearities, which354
produce periodic solutions whose frequency does not depend on amplitude, in general it may355
not be possible to easily produce the initial conditions for such periodic solutions. For the356
case of idakk¯a, however, our model demonstrates that a typical strike on the drum does357
produce such initial conditions. Finally, even beyond the easy production of such initial358
conditions, there remains the issue of the existence of several possible NNMs with time359
periods that do not occur in pleasing proportions. In such cases we expect, as numerics also360
26
Acoustics of Idakk¯a
show for typical χ, that one NNM dominates. The final intriguing, or pleasing, aspect of361
idakk¯a seems to be that there is a special value of χfor which three distinct NNMs have362
frequencies in harmonic proportions. Mathematically, there is in general no reason to expect363
that multiple NNMs with rationally related frequencies can dynamically coexist in a given364
solution. However, as our simulations show, this may be occurring to some degree because,365
for the same value of χ, the time response shows a particularly strong and clean harmonic366
response; see Figure 14.367
V. CONCLUSION368
We have investigated idakk¯a as a musical drum capable of producing a rich spectrum of369
harmonic overtones. The uniqueness of idakk¯a is attributed to the nature of snare action370
due to the peculiar material of the snares. The other important aspects of idakk¯a include371
the curved nature of barrel rim and the cord tensioning mechanism. Idakk¯a’s sound is dis-372
tinctively different from that of Western snare drums, which are otherwise inharmonic,23
373
and that of African talking drums and Japanese tsuzumi, both of which have elaborate374
cord tensioning mechanisms but do not produce a definite pitch. While we have initiated375
a systematic study into the acoustics of the instrument, several important considerations376
have been left out for future investigations. These would include modeling the snare action377
combined with the curved rim, coupled drumheads, and air loading. The interesting math-378
ematical problem of membrane vibration against a unilateral boundary constraint, in the379
form of a curved rim, also needs further attention.380
27
Acoustics of Idakk¯a
ACKNOWLEDGMENTS381
We are grateful to Mr. P. Nanda Kumar, a professional idakk¯a player, for helping us in382
our audio recordings and informing us about various intricacies of the musical instrument.383
REFERENCES384
1R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing,385
“Effect of air loading on timpani membrane vibrations”, J. Acoust. Soc. Am., 76, pp.386
1336-1345, 1984.387
2C. V. Raman, “The Indian musical drums”, Proc. Indian Acad. Sci., A1, pp. 179-188,388
1934.389
3B. S. Ramakrishna and M. M. Sondhi,“Vibrations of Indian musical drums regarded as390
composite membranes”, J. Acoust. Soc. Am. 26, pp. 523-529, 1954.391
4G. Sathej and R. Adhikari,“The eigenspectra of Indian musical drums”, J. Acoust. Soc.392
Am. 125, pp. 831-838, 2009.393
5S. Tiwari, and A. Gupta, “Effects of air loading on the acoustics of an Indian musical394
drum”, J. Acoust. Soc. Am., 141, pp. 2611-2621, 2017.395
6N. H. Fletcher and T. D. Rossing The Physics of Musical Instruments (Springer, New396
York, 1998), Ch. 18.397
7B. C. Deva Musical Instruments in Sculpture in Karnataka (Indian Institute of Advanced398
Study, Shimla, 1989), pp. 33-34.399
28
Acoustics of Idakk¯a
8L. S. Rajagopalan Temple Musical Instruments of Kerala (Sangeet Natak Akademi, New400
Delhi, 2010), Ch. 1.401
9Listen to the sample audio at [URL to be inserted]. It is a percussive piece in ¯adi t¯ala402
usually played at specific times in the temple. The musician is Mr. P. Nanda Kumar.403
10See supplementary material at [URL to be inserted] for further details on idakk¯a design,404
analysis of audio recordings, and non-linear normal modes. The supplement also contains405
a preliminary discussion of the effect of a finite curved rim on membrane vibration.406
11R. Burridge, J. Kappraff, and C. Morshedi,“The sitar string, a vibrating string with a407
one-sided inelastic constraint”, SIAM Appl. Math. 42, pp. 1231-1251, 1982.408
12A. K. Mandal and P. Wahi, “Natural frequencies, mode shapes and modal interactions for409
strings vibrating against an obstacle: Relevance to Sitar and Veena”, J. Sound Vib., 338,410
pp. 42-59, 2015.411
13K. Jose Vibration of circular membranes backed by taut strings (M.Tech. thesis, IIT Kan-412
pur, 2017).413
14https://www.audacityteam.org.414
15MATLAB. version 9.0 (R2016a). The MathWorks Inc., Natick, Massachusetts, 2016.415
16The percentage deviation of the dominant peaks is, more precisely, within 1.5% of the416
nearest integers. The error values are collected in Table 1 of the Supplement.10
417
17A. H. Benade Fundamentals of musical acoustics (Dover, New York, 1990).418
18A. Torin and M. Newton, “Collisions in drum membranes: a preliminary study on a419
simplified system”, Proc. Intern. Symp. Mus. Acoust., pp. 401-406, 2014.420
29
Acoustics of Idakk¯a
19A. Torin, B. Hamilton, and S. Bilbao, “An energy conserving finite difference scheme for421
the simulation of collisions in snare drums”, Proc. 17th Int. Conf. Dig. Audio Eff., pp.422
145-152, 2014.423
20S. Bilbao, A. Torin, and V. Chatziioannou, “Numerical modeling of collisions in musical424
instruments”, Acta Acust. united Ac., 101, pp. 155-173, 2015.425
21S. Bilbao, “Time domain simulation and sound synthesis for the snare drum”, J. Acoust.426
Soc. Am., 131, pp. 914-925, 2012.427
22Based on private communication with Dr. A. Torin.428
23T. D. Rossing, I. Bork, H. Zhao, and D. O. Fystrom, “Acoustics of snare drums”, J.429
Acoust. Soc. Am., 92, pp. 84-94, 1992.430
24A. F. Vakakis, “Non-linear normal modes (NNMs) and their applications in vibration431
theory: An overview”, Mech. Syst. Signal Pr., 11, pp. 3-22, 1997.432
30
Acoustics of Idakk¯a
LIST OF FIGURES433
1 An idakk¯a being played. (color online) ................................. 3434
2 Distinctive curved shape of the rim of the barrel (top left). A pair of strings435
installed on an idakk¯a barrel (top right). Barrel of an idakk¯a (bottom). A436
piece of cloth may be used (as shown) to improve the grip. The nails to which437
snares are tied are also visible close to the edge of the barrel. (color online) . . 4438
3 Swings in frequency as seen in the audio recordings as correction (left) and439
anticipation (right). (color online) ..................................... 7440
4 Typical spectrograms with (left) and without (right) the strings. (color online) 8441
5 PSD of idakk¯a drum samples with (top row) and without (bottom row) the442
strings in idakk¯a. The dotted lines mark integer multiples of the correspond-443
ing fundamental frequency in each plot. The blue line indicates 3.25f0. The444
orange arrows mark the inharmonic peaks. (color online) .................. 9445
31
Acoustics of Idakk¯a
6 (a) A schematic of the string-membrane configuration. The circular mem-446
brane is clamped at the edge and the two strings sit below the membrane.447
Here, 2bis the distance between the strings, Ris the radius of the membrane,448
and W(r, θ, t) is the transverse displacement of the membrane. (b) An illus-449
tration of the strings forming a convex hull around the curve of intersection450
between the membrane and the plane. A combination of the first two ax-451
isymmetrical modes of a uniform circular membrane were used to generate452
the deformed profile. (c) A sectional view of the string-membrane contact.453
The plane of section is shown in Figure (b). The grey line indicates the inter-454
section of the membrane with the vertical plane and the red line is the string455
below the membrane. (color online) .................................... 12456
7 Comparison of displacements and forces as obtained from Equation (8) (dot-457
ted line) and the quasi-static string approximation (black line) for two differ-458
ent values of Kin (a) and (b). A typical membrane shape is shown as a grey459
line. ............................................................... 15460
8 (a) Waveform and (b) PSD plot corresponding to TM= 250 N and TS= 6.5461
N (such that χ= 0.15); ψ= 0.1091. The dotted lines in (b) mark integer462
multiples of the fundamental frequency. (color online) .................... 17463
9 The presence of harmonic overtones for a range of 0.07 <χ<0.21, ψ=464
0.1091. The dotted lines mark integer multiples of the fundamental frequency.465
(color online) ....................................................... 20466
32
Acoustics of Idakk¯a
10 Harmonic response observed for over a range value of 0 < ψ < 0.4, χ= 0.15.467
The dotted lines mark integer multiples of the fundamental frequency. (color468
online) ............................................................. 21469
11 PSD plot obtained for an idakk¯a drumhead using the penalty method. The470
dotted lines mark integer multiples of the fundamental frequency. (color online) 22471
12 Waveforms obtained from a penalty method based simulation of (a) an idakk¯a472
drumhead and (b) the snare drum drumhead. The physical parameters for473
the latter is taken from Torin and Newton.18 (color online) ................ 23474
13 Non-linear normal mode solutions corresponding to the zero velocity initial475
condition. The variable ηon the y-axis denotes a scaled displacement ampli-476
tude. (color online) .................................................. 24477
14 PSD plot of the vibration of a membrane backed by two strings using the478
quasi-static string approximation superposed with dotted lines which mark479
the calculated NNM frequencies; χ= 0.15 and ψ= 0.1091. (color online) . . . . 25480
33
