features
Can
CanOne
OneShape
Shapethe
theSound
SoundofofaaDrum?
Drum?
SirSir
Frederick
Frederick
Crawford
Crawford
FIMA
FIMA
JJ
ustust
50 50
years
years
ago,
ago,
thethe
Polish
Polish
mathematician
mathematician
KacKac
[1][1]
posed
posedtrombone
trombone
andand
human
human
voice,
voice,
forfor
example,
example,
cancan
vary
vary
frequency
frequency
conconthethe
intriguing
intriguing
question,
question,
‘Can
‘Can
oneone
hear
hear
thethe
shape
shape
of aofdrum?’
a drum?’tinuously
tinuously
over
over
limited
limited
ranges,
ranges,
whereas
whereas
keyed
keyed
instruments,
instruments,
such
such
It stimulated
It stimulated
a variety
a variety
of of
demonstrations
demonstrations
thatthat
drums
drums
with
withas the
as the
piano
piano
or clarinet,
or clarinet,
favour
favour
discrete
discrete
setssets
of frequencies.
of frequencies.
Per-Perdiffering
differing
irregular
irregular
polygonal
polygonal
boundaries
boundaries
could
could
have
have
identical
identical
fre-fre-cussion
cussion
instruments,
instruments,
such
such
as drums,
as drums,
cymbals
cymbals
andand
bells,
bells,
have
have
fre-frequency
quency
spectra
spectra
[2].[2].
So,So,
too,too,
could
could
circular
circular
drums,
drums,
if only
if only
they
theyquency
quency
spectra
spectra
determined
determined
by by
their
their
shapes
shapes
andand
sizes,
sizes,
andand
describdescribcould
could
be be
made
made
with
with
inhomogeneous
inhomogeneous
membranes
membranes
having
having
differing
differingable
able
adequately
adequately
in terms
in terms
of ratios
of ratios
between
between
a few
a few
overtones
overtones
andand
thethe
spatially-varying
spatially-varying
density
density
or thickness
or thickness
profiles
profiles
[3].[3].
TheThe
answer
answer
to tofundamental,
fundamental,
so freeing
so freeing
a sober
a sober
scientist
scientist
from
from
thethe
intoxicating
intoxicating
vo-voKacKac
waswas
therefore
therefore
an an
emphatic
emphatic
‘No!’
‘No!’
cabulary
cabulary
of tone
of tone
colour.
colour.
What
What
should
should
those
those
ratios
ratios
be be
forfor
a drum
a drum
to sound
harmonious?
harmonious?
This
This
article
article
derives
derives
from
from
twotwo
related
related
thoughts:
thoughts:first,
first,
thatthatto sound
thethe
sounds
sounds
of musical
of musical
instruments
instruments
to be
to described,
be described,
written
written
plastic
plastic
membranes,
membranes,
now
now
widely
widely
used
used
as as
an an
alternative
alternative
to tradito tradi- ForFor
down,
down,
composed
composed
with
with
and
and
reproduced,
reproduced,
an
an
agreed
agreed
categorisation
categorisation
tional
tional
materials
materials
such
such
as calfskin,
as calfskin,
could
could
be be
made
made
with
with
prescribed
prescribed
of of
thethe
frequencies
frequencies
involved
involved
(as(as
notes)
notes)
thickness
thickness
profiles
profiles
by by
3D3D
printing
printing
and,
and,
is
required.
is
required.
Within
Within
any
any
such
such
categoricategorisecondly,
secondly,
thatthat
these
these
profiles
profiles
could
could
shape
shape … drums have frequency spectra
sation,
sation,
it is
it is
to to
be be
expected
expected
thatthat
cer-certhethe
frequency
frequency
spectrum.
spectrum.
sequences
sequences
(scales)
(scales)
andand
combinacombinaA A
circular
circular
drum
drum
with
with
a homogea homoge- determined by their shapes andtaintain
tions
of of
notes
notes
(chords),
(chords),
andand
of of
instruinstruneous
neous
membrane
membrane
hashas
a frequency
a frequency
specspec- sizes, … so freeing a sobertions
ments
ments
(groups
(groups
andand
orchestras)
orchestras)
willwill
be be
trum
trum
determined
determined
by by
thethe
zeros
zeros
of Bessel
of Bessel
scientist from the intoxicatingmore
more
appealing
appealing
to the
to the
human
human
earear
(har(harfunctions.
functions.
To To
an an
occidental
occidental
ear,ear,
thethe
ra- ramonious)
monious)
than
than
others.
others.
In different
In different
cul-cultiostios
between
between
frequencies
frequencies
so determined
so determined vocabulary of tone colour …
tures,
tures,
a variety
a variety
of of
categorisations
categorisations
of of
sound
sound
discordant.
discordant.
Could
Could
those
those
ratios
ratios
be be
notes,
scales
scales
andand
chords
chords
have
have
evolved,
evolved,
determined
determined
by by
thethe
capaccapacmade
made
harmonious
harmonious
by by
shaping
shaping
thethe
profile
profile
of of
an an
inhomogeneous
inhomogeneousnotes,
ities
of their
of their
instruments,
instruments,
composers
composers
andand
performers.
performers.
What
What
hashas
membrane?
membrane?
WeWe
shall
shall
show
show
thatthat
thethe
answer
answer
is aisqualified
a qualified
‘Yes’,
‘Yes’,ities
come
come
to be
to regarded
be regarded
as harmonious
as harmonious
to occidental
to occidental
earsears
over
over
thethe
lastlast
with
with
implications
implications
forfor
orchestral
orchestral
drums.
drums.
fewfew
centuries
centuries
derives
derives
from
from
a categorisation
a categorisation
founded
founded
on on
a ‘wella ‘welltempered’
tempered’
12-note
12-note
scale,
scale,
with
with
a frequency
a frequency
ratio
ratio
between
between
consecconsecA musical
A musical
interlude
interlude
utive
utive
notes
notes
of 2of1/12
21/12
, described
, described
as one
as one
semitone
semitone
[6].[6].
Within
thisthis
scale,
scale,
three
three
particular
particular
frequency
frequency
ratios
ratios
thatthat
areare
WeWe
have
have
introduced
introduced
some
some
rather
rather
vague
vague
terms:
terms:
‘discordant’,
‘discordant’,
‘har‘har- Within
considered
to be
to be
harmonious
harmonious
areare
thethe
octave
octave
(=(=
2),2),
thethe
perfect
perfect
monious’,
monious’,
andand
‘occidental
‘occidental
ear’.
ear’.
Behind
Behind
them
them
lieslies
an extensive
an extensive
mu-mu-considered
fifth
(=(=3/2)
3/2)
andand
thethe
perfect
perfect
third
third
(=(=5/4).
5/4).A A
prominent
prominent
sical
sical
background
background
to which
to which
thethe
encyclopaedic
encyclopaedic
works
works
by by
Scholes
Scholesfifth
chord
in in
musical
musical
composition
composition
is the
is the
major
major
triad,
triad,
consisting
consisting
of of
[4][4]
andand
Latham
Latham
[5][5]
offer
offer
access.
access.
More
More
specifically,
specifically,
Savage
Savage
[6][6]chord
a fundamental
frequency
frequency
andand
its its
perfect
perfect
fifth
fifth
andand
third.
third.HapHaphashas
discussed
discussed
thethe
ratios
ratios
between
between
frequencies
frequencies
thatthat
have
have
come
come
to toa fundamental
pily,
twotwo
notes
notes
in ain12-note
a 12-note
scale
scale
approximate
approximate
closely
closely
to the
to the
per-perdefine
define
those
those
terms.
terms.
ForFor
ourour
purposes,
purposes,
it will
it will
suffice
suffice
to say
to say
a little
a littlepily,
fifth,
fifth,
3/23/2
≈≈
27/12
27/12
==
1.498
1.498
(the(the
discrepancy
discrepancy
being
being
called
called
about
about
tone
tone
colour
colour
(timbre),
(timbre),
andand
thereby
thereby
to introduce
to introduce
some
some
particpartic-fectfect
thethe
Pythagorean
Pythagorean
comma
comma
[5, [5,
p. 1019]),
p. 1019]),
andand
to the
to the
perfect
perfect
third,
third,
ularular
ratios
ratios
relevant
relevant
to this
to this
article.
article.
==
1.260.
1.260.
As As
Savage
Savage
points
points
outout
[6],[6],
butbut
forfor
thatthat
≈≈
24/12
24/12
TheThe
sounds
sounds
of musical
of musical
instruments
instruments
generally
generally
consist
consist
of aoffuna fun-5/45/4
‘amazing
mathematical
mathematical
fluke’,
fluke’,
thethe
beautiful
beautiful
music
music
thatthat
wewe
have
have
damental
damental
frequency
frequency
accompanied
accompanied
by by
a spectrum
a spectrum
of of
overtones,
overtones,‘amazing
learnt
to enjoy
to enjoy
would
would
notnot
exist,
exist,
nornor
could
could
instruments
instruments
such
such
as the
as the
constituting
constituting
its its
tone
tone
colour,
colour,
andand
identifying
identifying
thethe
instrument
instrument
to an
to anlearnt
piano
be be
constructed.
constructed.
experienced
experienced
musical
musical
ear.ear.A note
A note
played
played
on on
a piano,
a piano,
forfor
examexam-piano
In what
follows,
follows,
wewe
shall
shall
respond
respond
to our
to our
question,
question,
‘Can
‘Can
oneone
ple,ple,
cancan
be be
distinguished
distinguished
from
from
a note
a note
with
with
thethe
same
same
fundamenfundamen- In what
shape
thethe
sound
sound
of of
a drum?’,
a drum?’,
by by
attempting
attempting
to design
to design
an an
inhoinhotal tal
frequency
frequency
played
played
by by
plucking
plucking
a violin
a violin
string
string
or by
or by
striking
striking
a ashape
mogeneous
membrane
membrane
whose
whose
overtones
overtones
relate
relate
to the
to the
fundamental
fundamental
drum.
drum.
Subtler
Subtler
areare
thethe
distinctions
distinctions
between
between
thethe
tone
tone
colours
colours
forfor
a amogeneous
through
these
these
three
three
harmonious
harmonious
ratios.
ratios.
given
given
fundamental
fundamental
played
played
by by
bowing
bowing
a cello,
a cello,
a viola
a viola
or aorviolin.
a violin.through
Instrument-makers
Instrument-makers
influence
influence
tone
tone
colour
colour
through
through
application
application
of of
Testing
Testing
thethe
test
test
their
their
artsarts
to design
to design
andand
construction,
construction,
andand
a few
a few
achieve
achieve
thethe
leg-legendary
endary
status
status
of aofStradivari.
a Stradivari.
Players
Players
do do
so through
so through
their
their
personal
personalTo To
assess
assess
thethe
feasibility
feasibility
of of
what
what
wewe
areare
attempting,
attempting,
wewe
require
require
techniques.
techniques.
In particular,
In particular,
percussionists
percussionists
vary
vary
tone
tone
colour
colour
by by
their
theirthethe
simplest
simplest
plausible
plausible
model
model
forfor
thethe
membrane,
membrane,
andand
an an
adequate
adequate
choices
choices
of of
beaters
beaters
(for(for
example,
example,
wooden
wooden
sticks
sticks
with
with
or or
without
withouttesttest
forfor
thethe
effect
effect
of aofspecified
a specified
inhomogeneity
inhomogeneity
profile
profile
on on
a given
a given
felt,felt,
cloth,
cloth,
leather
leather
or cork
or cork
covers,
covers,
or brushes)
or brushes)
andand
of where
of where
on on
thetheovertone.
overtone.
To To
thisthis
end,
end,
wewe
start
start
from
from
thethe
Helmholtz
Helmholtz
equation
equation
forfor
membrane
membrane
a drum
a drum
is struck.
is struck.
propagation
propagation
across
across
a stretched
a stretched
flexible
flexible
membrane
membrane
[7, [7,
p. 739]:
p. 739]:
Tone
Tone
colour
colour
inspires
inspires
adjectives
adjectives
worthy
worthy
of aofseed
a seed
catalogue
catalogue
or or
2
a wine
a wine
connoisseur:
connoisseur:
brilliant,
brilliant,
coarse,
coarse,
mellow,
mellow,
muddy,
muddy,
lush,
lush,
shrill,
shrill,
η 2+η (ω
+ 2(ω
σ/T
σ/T
)η )η
= 0,
= 0,
∇2∇
velvety,
velvety,
etc.,
etc.,
together
together
with
with
an an
extensive
extensive
palette
palette
of colours
of colours
as nuas nuwhere
η isηthe
is the
displacement
displacement
of the
of the
membrane
membrane
normal
normal
to its
to its
plane,
plane,
anced
anced
as soft-red,
as soft-red,
red-purple,
red-purple,
green-purple
green-purple
andand
emerald,
emerald,
perhaps
perhapswhere
is the
frequency
frequency
of vibration,
of vibration,
σ isσ the
is the
areal
areal
mass
mass
density
density
at aat a
influenced
influenced
by by
synaesthesia
synaesthesia
[4, [4,
pp.pp.
202–210],
202–210],
[5, [5,
pp.pp.
272–274].
272–274].ω isω the
given
point
point
within
within
thethe
membrane
membrane
andand
T is
T the
is the
areal
areal
surface
surface
ten-tenInstruments
Instruments
differ
differ
in their
in their
capacity
capacity
to vary
to vary
thethe
fundamental
fundamental
fre-fre-given
sion,
assumed
assumed
uniform.
uniform.
quency
quency
andand
tone
tone
colour
colour
of the
of the
notes
notes
thatthat
they
they
produce.
produce.
TheThe
violin,
violin,sion,
Mathematics TODAY
DECEMBER 2016 280
© Furtseff | Dreamstime.com
Multiplying this equation by η, integrating over the area, G,
of the membrane and imposing η = 0 at the boundary gives
2
D[η] − (ω /T )H[σ, η] = 0,
2
D[η] =
|∇η| dG,
G
H[σ, η] =
ση 2 dG.
G
This expression can be shown to be variational [7, pp. 790–793],
so substitution for η of a trial function satisfying the boundary
conditions will give an approximate value for ω 2 exceeding its exact value. As a trial function for an inhomogeneous membrane,
we shall use the exact solution to the Helmholtz equation for a
homogeneous membrane with areal mass density σ0 , for which
η = η0 and ω = ω0 , and we have
D[η0 ]
ω02
=
,
T
H[σ0 , η0 ]
ω2
D[η]
D[η0 ]
=
≤
,
T
H[σ, η]
H[σ, η0 ]
which reduce to
H[σ, η0 ]
ω02
.
≥
2
ω
H[σ0 , η0 ]
This expression relates ω and σ, but will only be an adequate
test of the effect on ω of varying σ if there is approximate equality
between the two sides. Without solving the Helmholtz equation
exactly, some reassurance on this point can be inferred by employing a remarkable result that has lain idle in the literature on
the Kac question for several decades [3]. First, we note that if the
conformal transformation w = f (z) is applied to the Helmholtz
equation for a homogeneous membrane, expressed in rectangular
coordinates (u, v) in the w-plane,
∂2η
∂2η
+
+ (ω 2 σ0 /T )η = 0,
∂u2
∂v 2
we obtain
∂2η
∂2η
+ (ω 2 σ0 /T )|f (z)|2 η = 0,
2 +
∂x
∂y 2
expressed in rectangular coordinates (x, y) in the z-plane [8,
p. 242]. This equation can be interpreted as describing a membrane with an inhomogeneous areal mass density profile, but having a frequency spectrum and mass identical to that of the homogeneous membrane.
Gottlieb’s key step in [3] was to choose
w = f (z) =
z−a
,
1 − az
where the parameter a lies in the range 0 < a < 1. This transformation maps the unit disc onto itself, and maps concentric circles
of radius R in the w-plane into nested circles of radius ra , with
centres displaced by xa from the origin, in the z-plane:
u2 + v 2 = R 2 ,
2
(x − xa ) + y 2 = ra2 ,
1 − R2
,
1 − a2 R 2
1 − a2
.
ra = R
1 − a2 R 2
xa = a
The areal density inhomogeneity profile in the z-plane is
2
σ(x, y) = σ0 |f (z)| = σ0
σ(r, θ) = σ0
(1 − a2 )
2
2
((1 − ax) + a2 y 2 )
(1 − a2 )
2,
2
2
(1 − 2ar cos θ + a2 r2 ) .
Mathematics TODAY
DECEMBER 2016 281
Candombe drummers
© Kobby Dagan | Dreamstime.com
Taking a = 0.3, for example, at the centre and cardinal points
on the boundary of the unit disc, we have, respectively,
2
σ(0, θ)/σ0 = (91/100) ,
it is adequate for our purposes. The closeness of these approximations must also be interpreted, however, as a warning that substantial inhomogeneities may be required to tune the frequencies
of overtones by only small amounts.
2
σ(1, π)/σ0 = (7/13) ,
Tuning the overtones
2
σ(1, π/2)/σ0 = (91/109) = σ(1, 3π/2)/σ0 ,
2
4
σ(1, 0)/σ0 = (13/7) = (13/7) σ(1, π)/σ0
≈ 12σ(1, π)/σ0 .
Exact solutions, η0 , for the homogeneous membrane in the
w-plane may now be used in the z-plane as trial functions,
Jn (jn,k r) cos nθ, to obtain ω/ω0 approximately, where jn,k is
the kth root of the Jn Bessel function, and ω0 = ωn,k . The
fundamental and third overtones are axisymmetric (n = 0) with
j0,1 = 2.4048 and j0,2 = 5.5201. Substituting in the test expression above, and assuming exact equality, we calculate:
a=0
ω/ω0,1 = 1
ω/ω0,2 = 1
0.1
0.2
0.3
0.4
0.5
1.0057
1.0038
1.0234
1.0154
1.0551
1.0358
1.1050
1.0666
1.1811
1.1115
Even for inhomogeneities as strong as a = 0.3, with its 12-fold
variation in areal mass density around the membrane, these calculations approximate closely the exact values of unity that would
have been obtained if the exact solutions for η in the z-plane had
been used in the test expression. We shall therefore assume that
Mathematics TODAY
DECEMBER 2016 282
With that pessimistic remark in mind, we start by considering
only the fundamental and first three overtones. The first and
second vary azimuthally as J1 (j1,1 r) cos θ and J2 (j2,1 r) cos 2θ,
where j1,1 = 3.8317, j2,1 = 5.1356, and the third, J0 (j0,2 r),
is axisymmetric. Since the frequencies ωn,k are proportional
to the jn,k , the frequencies of these overtones normalise to the
fundamental, as: ω1,1 /ω0,1 = 1.5933 (= 1.0622 × (3/2)),
ω2,1 /ω0,1 = 2.1355 (= 1.0678 × 2), ω0,2 /ω0,1 = 2.2954
2
(= 1.0202 × (3/2) ). The first ratio is close to 3/2 (perfect fifth),
2
the second to 2 (octave), and the third to (3/2) , which would also
qualify as harmonious [6].
How might we tune these overtones? Reducing the mass of a
homogeneous membrane will cause all of its frequencies to rise.
We shall consider the feasibility of loading a homogeneous membrane of reduced mass with additional masses in such a way that
its fundamental frequency is lowered to that of the higher mass,
while its overtones are lowered differentially to frequencies related harmoniously to that frequency.
We shall assign unit mass to an unloaded homogeneous unit
disc membrane, so that σ0 = 1/π, and load a homogeneous membrane, of mass m, with masses, mi , each of which is a uniform
flexible ring of mean radius ri , centred on the origin, and of rectangular section with radial width δri and thickness 1/δri .
To sharpen the criteria for feasibility, we shall consider first
the limit of delta-function rings (δri → 0), for each of which the
areal density contribution is
σi =
mi δ(r − ri )
.
2πri
For a particular trial function, Jn (jn,k r) cos nθ, and a single
delta-function ring, mi = mn,k , substitution in the test expression yields
1 mn,k δ(r−ri )
2
Jn (jn,k r)2 r dr
ωn,k
≥ m + 0 12πri
2
1
ωT (n,k)
Jn (jn,k r)2 r dr
0 π
= m + mn,k
Jn (jn,k ri )
2
Jn+1 (jn,k )
2,
where ωT (n,k) is the frequency to which provision of the deltafunction ring tunes ωn,k . The axisymmetric delta-function ring
affects only the radial variation of η.
The mass mn,k will exert its strongest tuning effect if it is located at radius ri = rn,k , where rn,k > 0 is the smallest radius
2
for which Jn (jn,k ri ) maximises. The optimised test expression
can then be used, assuming approximate equality:
2
ωn,k
Jn (jn,k rn,k )2
.
≈ m + mn,k
2
ωT (n,k)
Jn+1 (jn,k )2
For the fundamental to be unchanged in frequency, we require
2
2
ω0,1
J0 (j0,1 rn,k )
= 1 ≈ m + mn,k
.
2
2
ωT (0,1)
J1 (j0,1 )
Prescription of ωT (n,k) enables the masses m and mn,k to be estimated.
The first three overtones
It is straightforward to extend the analysis for a single deltafunction ring by adding two more. We locate them at the maximising radii: mass m1,1 at r1,1 = 0.4805, mass m2,1 at r2,1 =
0.5947, and mass m0,2 at r0,2 = 0.6941. After evaluating the
corresponding Bessel function coefficients, we obtain
2
ω0,1
=1
2
ωT (0,1)
≈ m + 1.7820m1,1 + 1.1241m2,1 + 0.6412m0,2 ,
2
ω1,1
1.59332
=
= 1.1283
2
ωT (1,1)
(3/2)2
2
ω2,1
2
ωT (2,1)
≈ m + 2.0871m1,1 + 1.8216m2,1 + 1.2690m0,2 ,
=
The first two overtones
With m0,2 = 0, the first three equations above can be solved satisfactorily to give m = 0.7144 ≈ 0.71, m1,1 = 0.0611 ≈ 0.06
and m2,1 = 0.1572 ≈ 0.16, all of which are positive. Because of
the optimised locations of the two delta-function rings, the total
mass is less than unity.
It is important to distinguish here between precision and accuracy: once a trial function has been specified for use in the test
expression, calculation of the coefficients in the equations above
can be arbitrarily precise, to four decimal places, for example.
The accuracy of the masses calculated depends, however, on how
closely the trial function simulates the exact solution. We recognise this distinction by approximating the masses to two decimal
places, which seems a plausible level of accuracy to infer from
our tests on isospectral membranes.
2.13552
= 1.1401
22
≈ m + 1.6889m1,1 + 2.0514m2,1 + 1.7549m0,2 ,
2
ω0,2
2.29542
=
= 1.0408
2
ωT (0,2)
(3/2)4
≈ m + 0.1267m1,1 + 1.0012m2,1 + 1.4011m0,2 .
Regrettably, the solution obtained by assuming exact equality in
each equation is a nonphysical mixture of positive and negative
masses. Can the first two overtones, at least, be tuned?
The third overtone
We now check whether or not the solution for the first two overtones tunes the third overtone fortuitously to a harmonious frequency. Substituting these masses in the fourth equation above
yields
2
ω0,2
2.29542
=
≈ m+0.1267m1,1 +1.0012m2,1 = 0.8795,
2
ωT (0,2)
X2
from which it follows that X ≈ 2.4475, a ratio close to (5/4)4 =
2.4414. Savage [6] classifies this within the perfect third criterion, so we need search no further for harmonious frequencies
with m0,2 > 0.
Summing up, our original aim for the third overtone (to have
X = (3/2)2 with m0,2 > 0) has fallen outside the tuning window with positive masses. That fatal defenestration brings to
mind President Truman’s injunction to the heat-averse, ‘get out
of the kitchen’. Having demonstrated that the two overtones most
likely to be excited could be tuned by providing two delta-function
rings, m1,1 and m2,1 , we do, however, emerge from the ‘kitchen’
(as the percussion section is familiarly known) with some satisfaction.
Practicality
Flexible rectangular section rings, in the delta-function limit of
infinite thickness and infinitesimal width, have served to demonstrate mathematically the feasibility of tuning the first two overtones, but are of course impractical. A flexible rectangular section ring of finite width δri , thickness, 1/δri and mean radius ri ,
is physically realisable, however, and contributes an areal mass
density profile of
σi = mi /2πri δri ,
ri − δri /2 ≤ r ≤ ri + δri /2.
We must now recalculate the coefficients in the equations obtained above for two delta-function masses. For a given trial function, Jn (jn,k r) cos nθ, the relevant integrals are of the form
rn,k +δrn,k /2
rn,k −δrn,k /2
mn,k
2
Jn (jn,k r) r dr.
2πrn,k δrn,k
Since the radial separation between the two rings, r2,1 − r1,1 =
0.5947 − 0.4805 ≈ 0.11, we may take δr2,1 = 0.1 = δr1,1 as an
Mathematics TODAY
DECEMBER 2016 283
example without overlap. The integrations yield
2
ω0,1
=1
ωT2 (0,1)
2
ω1,1
ωT2 (1,1)
2
ω2,1
ωT2 (2,1)
≈ m + 1.7725m1,1 + 1.1202m2,1 ,
=
1.59332
= 1.1283
(3/2)2
≈ m + 2.0692m1,1 + 1.8029m2,1 ,
=
2.13552
= 1.1401
22
≈ m + 1.6851m1,1 + 2.0259m2,1 .
Assuming exact equality, we obtain
m = 0.7090 (0.7144),
m1,1 = 0.0626 (0.0611),
m2,1 = 0.1607 (0.1572),
which agree with the delta-function solutions (shown in parentheses) to two decimal places, confirming that the first two overtones
can be tuned.
The corresponding areal mass densities are
σ/σ0 ≈ 0.71,
σ1,1 /σ0 ≈ 0.65,
σ2,1 /σ0 ≈ 1.16,
implying a variation in thickness over the membrane of a modest
2.6 (= 1.87/0.71). This should be realisable with a 3D printer
without unduly stiffening the membrane.
The rings should preferably be located on the undersides of
membranes, so that the variations in thickness do not give them
an appearance distractingly different from homogeneous membranes. It might, however, aid the percussionist in varying tone
colour if circles with radii corresponding to the displacement
maxima (r1,1 and r2,1 ) were marked on the membrane.
Conclusions
We began with the plausible hypothesis that at least the first few
overtones of a circular drum could be tuned harmoniously by suitably prescribed inhomogeneities in its membrane. ‘The tragedy
of science,’ wrote Thomas Huxley, is ‘a beautiful hypothesis slain
by an ugly fact.’ We end with the practical conclusion that only
the first two overtones can be tuned (to a perfect fifth and an octave, respectively).
Mathematics TODAY
DECEMBER 2016 284
Our recompense for the Kac-induced headaches that we have
suffered (his name aptly translates as ‘hangover’!) is, however,
substantial: these are the overtones which influence most strongly
the tone colour of orchestral drums. In modern orchestras, a percussionist controls several timpani [5, p. 1275]. Although they
are used mainly for repeated notes and drum rolls, composers
have demonstrated that use of these timpani, combined with operation of their tuning pedals, enables melodies to be contributed by
skilled percussionists [4, pp. 775–782]. Tuned membranes would
offer materially enhanced melodic potential, and new opportunities to composers and percussionists.
As soon as plastic membranes with prescribed inhomogeneities are made available by 3D printing, composers and percussionists will be able to decide whether or not to exploit these
new opportunities. On that crucial judgement, will depend the
fairy-tale outcome of an obscure opsimath becoming the Stradivari of drums, or the grim prospect of his being hunted down by
demented percussionists, baffled by the augmented complexity of
their timpani.
Acknowledgement
The earliest version of this article benefited from encouragement
and comments from Professor Uwe Grimm.
References
1 Kac, M. (1966) Can one hear the shape of a drum? Amer. Math.
Monthly, vol. 73, pp. 1–23.
2 Driscoll, T.A. (1997) Eigenmodes of isospectral drums, SIAM Rev.,
vol. 39, pp. 1–17.
3 Gottlieb, H.P.W. (1988) Density distribution for isospectral circular
membranes, SIAM J. Appl. Math., vol. 48, pp. 948–951.
4 Scholes, P.A. (1970) The Oxford Companion to Music, 10th edition,
Oxford University Press, United Kingdom.
5 Latham, A. (2002) The Oxford Companion to Music, Oxford University Press, United Kingdom.
6 Savage, M. (2014) Beautiful music: an amazing mathematical fluke,
Math. Today, vol. 50, pp. 88–90.
7 Riley, K.F., Hobson, M.P. and Bence, S.J. (2006) Mathematical Methods for Physics and Engineering, 3rd edition.
8 Spiegel M.R. (1964) Complex Variables, McGraw-Hill Book Company, New York.