# Pickers’ Guide to Acoustics of the Banjo, parts I and II

David Politzer∗, Jim Woodhouse†, and Hossein Mansour+

(April 2, 2021)

###### Abstract

The results of an investigation of fundamental banjo acoustics and physics are
now readily available.acoustics ; euphonics . The papers describe measurements
and calculations which demonstrate the extent to which the sound of a banjo
can be represented by linear modeling of the basic parts common to the banjo
family: strings, drum head, light floating bridge, and break angle. This note
summarizes the conclusions of that work and outlines some of the physics
involved. While the scrutinized banjo behaviors are well-known to discerning
builders and pickers, the focus here is how these arise from the physics. A
practical application (discussed in an addendum) concerns head tap-tuning:
what’s going on, how it works, and why some people and some tuners can’t get
it right.

∗452-48 Caltech, Pasadena CA 91125; †Cambridge University Engineering
Department, Trumpington Street, Cambridge CB2 1PZ; +Dassault Systèmes –
SIMULIA, 5005 Wateridge Vista Dr., San Diego, CA 92121

††preprint: HDP: 21 – 01

Pickers’ Guide to Acoustics of the Banjo, parts I and II

##  I Introduction

What happens when you use a drum head for the soundboard of a plucked
instrument? You get a banjo — or one of its many cousins from cultures around
the world. Their sounds share features that distinguish them from instruments
that are similarly strung and played but use thin wood (or some such material)
in place of the drum head.

“What happens…?” can also mean: What’s going on from a physics perspective?
How does it work?

In all acoustic string instruments, string vibrations activate the soundboard.
The soundboard vibrations disturb the surrounding air, and that turns into the
sound we hear. The simplest distinction of the banjo from, say, an identically
strung guitar of comparable size, is that the head and bridge are much lighter
than the analogous parts on the guitar. Energy is transferred faster by the
strings to the banjo head, and it moves more. So, the banjo sound is louder at
first and decays faster. But there are many more consequences of the drum-as-
soundboard. (To hear an example, you can go to
http://www.its.caltech.edu/~politzer/pickers-guide/four-pluck-comparison.mp3
or simply click here for gentle plucks on a typical banjo and a guitar, at the
same place along equal scale-length open steel strings.) Using acoustic guitar
again for comparison, well-studied guitar physics and acoustics imply that, if
it were physically possible to build a wood soundboard and bridge as light as
a banjo’s, it would still not sound like a banjo.

Physics and structural engineering deal with numbers and equations. Ref.
acoustics  translates banjo into a form appropriate for that kind of analysis.
The approach and tools are mostly ones long used to study stringed
instruments. There is also a body of previous research that has specifically
addressed drums. However, it hadn’t all been put together to study drum-topped
string instruments. In addition, yet other physics from other contexts enters
as well. Guitar and violin, which are both more widely familiar and have been
intensively studied, provide useful comparisons. Ref. acoustics  has links to
sound files at an on-line science of musical instruments resource,
Euphonicseuphonics , put together by author J.W. These are sounds synthesized
using the physics models developed in ref. acoustics . You can judge for
yourself how well straightforward linear modeling of strings, bridge, and head
account for banjo sound. Furthermore, the Euphonics sound links include
variations of the relevant parameters, such as bridge mass and break angle.
The agreement of the variation in modeled sounds with actual players’
experience is yet further confirmation that the basics have, indeed, been
captured by the modeling. Sketched below are some of the physics details of
that work.

Strings, floating bridge, and drum-like head are the elements common to all
instruments with banjo sound. To be sure, there are many other features whose
design variations contribute discernible sound differences.coupled  To
understand those, knowledge of the basics can’t but help. There are also
certainly non-linear effects, even in the basic elements, that can be heard if
one listens closely.

##  II energy & dissipation

On an acoustic guitar (or any wood topped instrument), sound carries away only
a small fraction of the energy that is put in by a pluck. Most of the energy
is dissipated into heat. The dominance of dissipation into heat is frustrating
if you want a simple perspective on what’s going on. Those losses are
functions of frequency that cannot be deduced from simple physics models. A
given resonance of the instrument can contribute to sound production or sound
suppression depending on the degree of its loss to friction. It is possible to
study frictional dissipation empirically. By and large, higher frequency
motions lose a larger fraction of their energy to heat than lower ones.
Furthermore, the sound radiation efficiency of soundboard vibrations grows
with increasing frequency. Together, these effects make the higher frequency
sounds weak and short-lived. The total sound is dominated by the response of a
handful of low frequency resonances of the body. And their properties are
sensitive to the detailed design of the instrument.

The vibrations of a wood soundboard depend on its internal properties and the
applied forces but not particularly on the surrounding air. In contrast, drums
have long been of intellectual interest in physics precisely because the
surrounding air and produced sound have a strong impact on the actual head
motion.

In ref. acoustics , a variety of lines of inquiry show that heat loss is
relatively small in any part of banjo sound production. This means that
dissipation-free, linear equations govern most of what’s going on. The
equations imply that the drum head has increasingly more resonances than a
wooden soundboard as frequency increases and that the radiation efficiency
decreases with increasing frequency. Compared to guitar, together these give a
sound spectrum that is much richer at high frequencies, and those high
frequency components have sustain that is comparable to the low frequency
ones.

Several observations come together to support this perspective on dissipation
in the banjo and its contrast to the guitar. The simplest evidence presented
in ref. acoustics  is the success of the analytic and numerical modeling in
reproducing measured behavior with little or no energy loss besides energy
carried away by sound. Somewhat more subtle are analyses of pluck sounds into
mode frequencies and lifetimes. On guitar, recognizable string modes have much
longer lifetimes than body modes. This gives strings the time to dissipate
energy themselves without most of it proceeding to sound production. On banjo,
the string resonant frequencies die off on times comparable to body modes,
suggesting that’s how strings lose most of their energy, i.e., by transferring
it to the body.

To go further, we turn to a particular measure of how plucked strings get the
soundboard moving.

##  III bridge admittance

Imagine plucks on a banjo and a comparable size guitar, strung and tuned
identically. Following a long and successful tradition in the study of guitars
and violins, we focus on precisely what happens when the vibrating string
applies a force to the bridge.

The characterization of bridge motion in response to string force is
admittance. Admittance is the velocity at a point divided by the applied force
that produces it. For vibrations in the context of linear physics, we can
consider forces, motions, and admittance as functions of frequency. In
general, admittance is a complex quantity, with its phase reflecting the phase
difference between the driving force and the produced velocity. Viewed as a
function of frequency, the admittance at the bridge involves all of the
subsequent physics — not just the bridge itself but also the whole head and
the surrounding air. (Admittance is precisely the inverse of impedance.)

Energy considerations offer another perspective on the interpretation and
importance of admittance. The up-and-down motions of strings apply oscillating
forces to the bridge. However, energy is transferred from a string to the
bridge only to the extent that the contact point moves. Otherwise, there is no
energy transfer. So admittance is a direct measure of how much of the pluck
energy gets to the soundboard.

Figure 1: Measured bridge admittance at the location of the 1st string slot:
blue is the head without bridge & strings, red is with bridge and damped
strings, and black is guitar with damped strings.

Fig. 1 shows measured admittance of a Deering Eagle II banjo and a Martin
Woodhouse steel string acoustic guitar. The guitar is in black. The blue curve
is for the banjo without bridge and strings, forced on the bare head at the
location where the bridge ordinarily goes. The banjo with bridge and damped
strings is in red. The vertical applied force and measured velocity are at the
location of the first string on the bridge.

The contrast of guitar in black with the bare drum head in blue illustrates
what’s been said so far. The drumhead soundboard is louder overall — because
it is much lighter. And it is increasingly more responsive at higher
frequencies because there are more and more resonant modes. Specifically,
aside from their lowest several modes, wood soundboards have resonances that,
on average, are equally spaced in frequency. On a drum head, the density of
resonances increases proportional to the frequency.

Add strings and bridge (and tailpiece) to the drumhead, and the behavior
changes radically.

The up-down-up-down-up-down aspect of the curves initially obscures some
underlying trends. Imagine a running average, smoothing out the curves. The
guitar admittance is essentially flat. The bare drum head curve rises. Adding
strings, bridge, and tailpiece lowers the drum head admittance for low
frequencies and lowers it again at high frequencies. This produces a broad
intermediate region of strong response, which, in acoustics, is called a
“formant.”

Ref. acoustics  reports an explicit calculation for a rectangular banjo with
an ideal membrane for a head, damped strings, bridge, and break angle to
tailpiece. (Rectangular geometry has long been a favorite of physicists
because analytic and computer calculations are far simpler than the
corresponding ones for a circular head; they do still sound like
banjos.squared-eel ) The well-isolated lowest modes are sensitive to the
geometry, but the smoothed trends are just like in Fig. 1.

A very general perspective, pioneered by Eugen Skudrzykskudrzyk , predicts the
smooth curves and identifies the crucial elements. The smooth curves reflect
the following observations. The individual peaks of the actual admittance
occur at frequencies for which the waves reflected off the soundboard boundary
return to the point of admittance in phase with the initial disturbance and
essentially double it. The intervening troughs occur at frequencies for which
the reflections return $180^{\text{o}}$ out of phase and cancel it. The
average is the same as if there were no reflected wave coming back at all. All
of the head motion is outgoing waves initiated by the bridge motion. That is
what would happen if the soundboard were infinite in extent. So, the smooth
curve is that of an infinite head — or, equivalently, any shaped finite head
with a non-reflective edge. Therefore, the smoothed curve for any size and
shape head with the same tension and density would be the same. (Calculations
for an infinite plane are even easier than for rectangles.)

The solid red line in Fig. 2 is derived using the membrane wave equation. That
equation is simply $F=ma$ for every tiny patch of the head. The mass of each
patch is proportional to its area. The force on each patch is the vertical
(i.e., perpendicular to the head) component of the tension of the adjoining
patches. It is the same differential equation for every patch. How a patch
moves depends on what its neighbors are doing. The admittance is the motion of
the particular point where an external force is applied.

Adding the bridge and (damped) strings changes the head equation of motion
only at the location of the bridge, i.e., where it contacts the head. The
simplest approximation to a bridge is to model it as a point mass at the
location where the admittance is evaluated. So there is extra inertia at that
point and extra forces — in addition to the applied force and the uniform
membrane wave forces.

Even though banjo bridges are light, e.g., 2.2 gm, their mass is significant
compared to the nearby head, which might be 11 gm total for the whole
$11^{\prime\prime}$ D circle. An additional mass at the point of admittance
suppresses the high frequencies. Larger mass gives greater suppression. One
can say that bridge mass produces a low-pass filter, with the transition
frequency decreasing with increasing mass.

The origin of the extra forces at the admittance point due to strings and head
is a bit more complicated. At equilibrium, those forces sum to zero. However,
when the bridge moves away from its equilibrium position, there is a net
return force due to strings and head that is in addition to what is supplied
by the string and head waves. For small motions, it is a Hooke’s Law force,
i.e., linear in the displacement. It has four sources, and each is under the
builder’s and player’s control — at least to some extent. The string break
angle is non-zero to keep the strings in contact with the bridge as they
vibrate. In practice, it is typically between $5^{\text{o}}$ and
$15^{\text{o}}$ at equilibrium. 1) The bridge motion changes the break angle.
That alters the vertical component of the string tension on the bridge. And 2)
there is also a change in the string tension — a higher tension when the
bridge displacement is positive — because the string must stretch when the
bridge moves up. The increased-tension contribution to the Hooke’s Law return
force is proportional to the square of the sine of the break angle and the
stretchiness of the strings (the longitudinal consequence of the string’s
Young’s modulus). Bridge motion also produces 3) an analogous change in the
break angle of the head at the feet of the bridge (albeit less apparent to
casual observation) and 4) requires some stretching of the head, which
increases its tension.

These four effects add a springlike return force right at the admittance
point, which is in addition to the return force provided everywhere on the
head by its tension. In ref. acoustics , this is called the “stiffness.”
Stiffness is simple to implement in the Skudrzyk calculation. It produces a
suppression of admittance for low frequencies, which is increasingly effective
as the break angle increases. Equivalently, this is a high-pass filter.

The effects of the extra mass and return force at the bridge combine to yield
a central frequency region of largest admittance — a formant — and the center
of that region reflects the balance of the two — as illustrated in Fig. 2 by
the dashed blue curve. Using realistic parameter values, this compares well to
the corresponding measurement in Fig. 1, shown in red. The significant
discrepancies above 3 kHz are due to the additional formants discussed in the
next section, §IV.

Figure 2: Skrudrzyk admittance calculations — red solid: head alone; red
dashed: high frequency limit of head alone; blue solid: head with a 1.5 g
bridge mass; blue dashed: head with a 1.5 gm bridge, realistic stiffness, and
a small amount of damping; black solid: wood soundboard with mass per unit
area matching the head; black dashed: wood soundboard with mass increased by a
factor 10 (realistic for a thin piece of wood).

The same equations can be described with different words. We can consider the
bridge mass and string/head stiffness as defining a single oscillator. Its
motion is very strongly damped by the energy loss to outgoing waves in the
head. Very strong damping implies a very short decay time and a very broad
width in frequency. If the damping were independent of frequency, it would
reduce the peak frequency in the admittance relative to the undamped
oscillator value. However, the rising trend of the head admittance pushes in
the other direction, leaving the total, net peak frequency shift subject to
the detailed numerical values.

This calculation identifies the physics behind simple modifications that are
commonly used to alter the sound of a particular banjo. For example, consider
more mellow. Here, “more mellow” means lowering the central frequency of this
main formant. You can increase the mass of the bridge. And there are several
ways to decrease the stiffness. Bridge height and tailpiece design have
obvious impacts on break angle; smaller angle is more mellow. Lower string and
head tension and easier stretchiness each promote mellow.

The bridge is held in place by “pre-tensioned” strings and head. At
equilibrium, their forces cancel. But it is sometimes said that the pre-
tension enhances the communication of string motion to the head. Indeed, the
discussion above shows that higher pre-tension raises the main formant
frequency. However, this actually happens by suppressing low frequency motion
rather than enhancing the highs, as illustrated in Fig. 2.

##  IV formants from bridge flexing

Careful observation of the Deering Eagle II revealed two additional formants
at higher frequencies, i.e., centered around 3.5 kHz and 5 kHz. The basic
admittance measurements reported in ref. acoustics  were performed by tapping
downward at the $1^{\text{st}}$ string bridge slot. Two further strategies
revealed more about the higher formants’ origins. Tapping downward at the
$3^{\text{rd}}$ string slot enhanced the 5 kHz formant, and tapping sideways
on the edge of the bridge enhanced the 3.5 kHz. In those frequency regions,
finite element numerical simulations showed dramatic bridge flexing. Note that
these are not resonant motions of the bridge on its own. Rather, the resonant
motion requires a bridge flexing that fits with head motion at the same
frequency. The head motion is particularly large at the bridge feet while they
maintain contact. Fig.s 3 and 4 show the result of the numerical calculation
of resonances near 3.5 kHz and 5 kHz. (FE/BE refers to a discretization of the
solid objects, i.e., “finite element,” and a continuous fluid calculation of
the surrounding air whose specific behavior in any situation is required to
match the motion of its solid boundaries, i.e., “boundary elements.”)

Figure 3: FE/BE calculation of resonant bridge/head motion near 3500 Hz.
Figure 4: FE/BE calculation of resonant bridge/head motion near 5000 Hz.

Note that these extreme bridge flexing motions are peculiar to the bridge used
in the measurements and numerical calculations. The bridge in question is
three-foot ebony-topped maple, standard equipment on the Deering Eagle II.
Analogous effects occur at different frequencies with different bridge
designs, the relevant factors being flexibility and footprint.bridge-hills

The speed of sound in hardwood is 10 to 12 times greater than sound in air. So
the wavelengths of longitudinal waves of compression are 10 to 12 times
longer. For relevant frequencies, these are at least 20 times longer than the
vertical dimension of a typical bridge. That means, in terms of sound
traveling as compression waves through the wood, the bridge moves as one,
i.e., as a single, rigid piece. There is no real meaning to it taking a
particular path through the bridge. On the other hand, bridges can flex on
length scales comparable to their own geometry, and those frequencies can be
much lower than those of bridge-scale compression waves.

##  V some important physics

###  V.1 wave speeds

Discussing sound production by a soundboard requires establishing a bit about
the physics notions of an ideal drum head and an ideal wooden soundboard.

Just as there is a physics model and equation that gives a reasonable,
approximate account of thin, taut musical strings, there are well-established
physics models for the ideal drum head, known as a “membrane,” and the typical
soundboard of a wooden string instrument, known as a “thin plate.” For the
present purposes, members of the international banjo family are instruments
whose heads behave, to a first approximation, like ideal membranes.

The ideal string is a physics abstraction that does a reasonable job
accounting for the behavior of the thin strings of real musical instruments.
The ideal string has a uniform density and a uniform tension that provides the
only force that pushes the string back toward its equilibrium position when it
is disturbed. Simple observations confirm that this idealization is a very
good approximation for small amplitude vibrations. One need only note how well
it works. Obvious forces that are neglected are bending stiffness and internal
friction. The long lifetime (in terms of number of cycles) of a plucked string
implies that internal friction is not a primary issue. Bending stiffness makes
successive string resonances sharper and sharper relative to the ideal. This
can be observed as a small effect on solid, thick strings. And that’s why
they’re usually wound – to add mass without significant extra stiffness.

The membrane is an ideal 2-dimensional abstraction. It has a uniform,
isotropic tension that provides the only force that pushes the membrane back
toward its equilibrium position when it is disturbed. This specification of
mass and force gives rise to wave behavior that is well-understood — although
not previously explored carefully in the context of banjos. For the present
purposes, a banjo is any string instrument whose soundboard can be
approximated as a membrane. In practice, that means that the tension is high
enough, head thin enough, and amplitudes small enough that tension dominates
over other solid material forces in determining transverse motion. Inherent
stiffness or rigidity are ignored. The closest approximation to an ideal
membrane in the real world is a soap bubble film. Indeed, an ideal membrane is
actually something of a two-dimensional liquid.

The thin plate is an idealization that is known to work well for wooden
soundboards. The restoring force is a stiffness arising from the stretch and
compression inherent in bending a solid object that is extended in two
dimensions and thin in the third dimension. The resulting wave dynamics is
quite different from the membrane. Beating the general description of
deformation of an elastic solid into a convenient form for thin plates is a
more complicated story than needed for the ideal membrane. But a very useful
result emerges, and it is a standard topic in mechanical engineering
education. In physics education, it typically only appears as an advanced
example, applied only to the static equilibrium deformation of a beam under
load.

The resulting wave equations imply that on a membrane, transverse plane waves
all travel at the same speed. On a plate, the speed of waves $v$ of definite
frequency $f$ is higher for higher frequencies, i.e., $v\propto\sqrt{f}$.

###  V.2 static deformation & low frequency limit

A constant small force applied at a point along a string produces a small
displacement. That is also true for a plate. However, for a membrane, the
displacement is infinite for any non-zero force applied at a point. The reason
is simple. Tension $T$ on a 2D surface is a force per unit length. If the
applied force $F$ is spread uniformly over a disk of circumference $2\pi r$,
the resulting static deformation requires $F=2\pi rT\times$ (the slope of the
membrane at $r$). Hence the slope at the edge of the radius $r$ disk is
${F\over{2\pi rT}}$. That implies that the deformation goes like log $r$ as
$r\to 0$. Of course, bridges have finite sized feet. But one lesson is that
small bridge feet can enhance the relevance of forces that were otherwise
negligible, e.g., bending stiffness near the bridge. Measurements as a
function of $r$ and calculations for a rectangular membrane and square
footprint are shown in Fig. 5.

Figure 5: Compliance (static displacement divided by total applied force)
versus diameter of the forcing disk. Red stars: measured static compliance at
the centre of the head of the Deering banjo, for different diameters of rigid
indenter. (See photo on page 1.) Solid line is the prediction from ideal
membrane theory. Blue stars: values deduced for a square loading region at the
centre of a square membrane, with a cutoff frequency 10 kHz. Large red symbol:
DC compliance deduced from the $\omega\to 0$ measured admittance using a
hammer with a very small area of contact.

The relatively extreme bending of the head at the feet of the bridge has a
dynamical effect, beyond the static displacement. Very high frequency drum-
head modes can impact the low frequency motion of the bridge, even in an
analysis restricted to modeling linear forces. Here is an outline of how that
comes about:

Each membrane resonant mode behaves like a forced harmonic oscillator. Each
has a resonance peak. Below its peak, the amplitude of each one approaches a
finite value as the forcing frequency goes to zero. That value is its static
equilibrium displacement. The full membrane is just the sum of its modes. The
dynamical admittance measurement, evaluated well below the frequency of the
lowest mode is essentially the static compliance. The large red wagon wheel in
Fig. 5 is an estimate of that value from admittance measurements of a real
head excited by a tiny hammer. So the static compliance at the bridge is made
finite by some combination of effects that are not part of the simple physics
that otherwise give a first approximation to everything that goes into banjo
sound production. Contributors certainly include other forces. In addition,
some connection to the infinity of modes as $f\to\infty$ may be relevant
because it is the sum over all those modes that gives the naively infinite
point compliance. And all these effects have significant impact on the
structure of the lowest frequency modes of the finite banjo head. It is
unusual in acoustics and much of physics that the lowest frequency, longest
wavelength phenomena are sensitive to very short distance physics. In this
case, stiffness at short distances enters DC compliance because of the sharp
bending localized right at the feet of the bridge. That sharp bending also
means that short wavelength drum modes can have a significant impact on low
frequency motions — and it happens because of the underlying singular behavior
of the point compliance.

These various effects produce an extreme sensitivity of frequency and lifetime
of low frequency modes to small variations of parameters, as illustrated in
$\S 3.4$ of the second paper of ref. acoustics . There, the “toy” calculation
studies the lowest two modes, modeled as damped harmonic oscillators with
their own mass, spring constant, and damping, as they are impacted by the
contribution to the static compliance due to all other, higher frequency
modes. The resulting frequencies and lifetimes are shown to be very sensitive
functions of that compliance. The take-away is that the details of the lowest
modes are very sensitive to bridge physics and the static compliance.

###  V.3 sound radiation efficiency

When a surface vibrates at a given frequency, a relevant issue is the
wavelength on the surface versus the wavelength in air. If the surface
wavelength is smaller than that in air, neighboring pieces on the surface tend
to cancel each other’s contribution to the sound. For an infinite surface, the
cancellation is complete. With a plate, there is a critical frequency at which
the two wavelengths match (i.e., where the wave velocities are equal). Above
that frequency, the plate waves are longer than air waves, and sound radiation
is far more effective than below it.

For any physically reasonable banjo head, the speed common to all waves is
less than the speed of sound in air. Hence, an infinite head would not radiate
sound at all. In more practical terms, the radiation of sound is intimately
connected to the finite size of the head. This has an important, simple
consequence. Radiation efficiency decreases with increasing frequency for a
given size of head. That’s because for high frequency (and shorter wave
length) the head is effectively closer to infinite as measured in units of the
wavelength. This means that the high harmonics in a single pluck sound do not
die off quickly relative to the rest. In fact, all harmonics have comparable
sustain.

##  VI body sound in each pluck

The simplest picture of stringed instruments is that strings drive the
soundboard at their near-harmonic frequencies. For various reasons, the string
amplitudes decay, and the sound decays as well. However, a pluck involves a
sudden initiation. That actually excites all of the modes of the
soundboard/body, not just the resonant frequencies of the string. On wood-
topped instruments, most of this energy goes into heat, and the
soundboard/body modes die away very quickly relative to the string modes.

Fig. 6 shows spectrograms of the same note plucked on a banjo (above) and a
guitar (below). The tall, thin lines are related directly to the string modes.
The other stuff at the bottom comes from the body modes.

Figure 6: Spectrograms of a single pluck on the $1^{\text{st}}$ string (G4 –
392 Hz) on a banjo (top) and guitar (bottom).

Three things are different on the banjo. 1) Almost all of the energy goes into
sound. 2) String and head/body mode decay times are not as dissimilar as with
guitar, primarily because the strings dump their energy into the head more
readily. And 3) banjo music generally consists of a steady stream of closely
spaced notes. So, the decaying head/body sounds survive for a significant
fraction of each note. Note that four notes per second, i.e., spaced by 0.25
s, is not particularly fast for banjo music. Hence, head/body modes survive
for much of the time between notes.

##  VII synthesized sounds from physical modeling

The sound files of ref. euphonics , §5.5, offer a way to listen to a generic
synthesis generated by the linear, physical modeling. You can explore how that
sound changes when the various parameter values are changed one at a time.

One important caveat is that all generated sound files give audio realizations
of the motion of the bridge and not the actual produced sound somewhere in
space. This has several advantages. It is precise and concise. Note that the
bridge motion in response to some pluck or tap reflects not only the physics
of the bridge and soundboard but also that of the strings, the surrounding
air, and all their mutual interactions. Bridge motion is crucial in
translating string vibration into produced sound. And calculating the impact
of changes in various model parameters is a feasible task. In contrast, the
sound at some point in space depends on the relation of that point to the
instrument and also to the space they are in, i.e., the room and everything
else in it. Furthermore, that is a calculationally intensive task that would
be a practical obstacle to comparing the results for ranges of the various
parameters.

To get the bridge motion “sound” to be a bit closer to real instruments’
sound, all the synthesized files include a smooth, slowly varying, monotonic
filter that boosts the highs slightly: $\times f^{0.4}$ for frequency $f$.

By and large, physically motivated parameter values give a plausibly banjo-
like sound. And sound variations produced by varying the parameters track
changes that are commonly experienced with real banjos.

##  VIII conclusion

The essential feature of banjo is the drum head soundboard. Ref. acoustics
presents efforts to understand how that works, i.e., to get a first
approximation of the linear behavior of the basic parts. Membrane physics
requires that the bridge and tailpiece (through the break angle) play
essential roles and are not merely passive conveyors of string vibration to
the membrane, as is the case with the bridge and saddle on a guitar. This note
sketched some of the results and underlying physics. Many features are in
stark contrast to the response of a similarly strung guitar. The compelling
details are in ref. acoustics . Even a perusal of the figures and their
captions and a listen to the sound files would give a more complete picture
than was presented here. For the most part, measurements were made on only one
banjo with one bridge. But its features were successfully modeled in
considerable detail. The principle input parameters were simply the design
choices of strings, head, and bridge. And the rectangular banjo model produced
a very similar basic behavior. We conclude that the physics presented here
gives a complete first approximation to banjo acoustics.

Even though ref. acoustics  studied only one bridge and one banjo, the
comparison of measurements and calculations provides a compelling picture.
That offers many suggestions of how modifications of the instrument will
change the sound. For example, bridge flexing is associated with the
production of high frequency formants, i.e., regions of enhanced sounds.
Hence, bridges with different flex characteristics are expected to sound
different.bridge-hills

ADDENDUM: HEAD TAP TUNING

Ref. acoustics  contains many observations and perspectives that did not make
it into this summary. But something of possible practical interest to players
concerns tap tuning the head for tension adjustment. The ultimate tensioning
advice is to adjust the tension to what sounds best to you. But there are many
reasons to use a method that involves less trial & error. One might want a new
head to sound like the previous one. Or one might aspire to the sound of
someone else’s banjo. So a common recommendation is to tap and adjust the
tension to a specified pitch. On an $11^{\prime\prime}$ mylar head, bluegrass
tight is G# or A. The problem is that not everyone can clearly identify the
tap pitch. Some electronic tuners will catch it, and some won’t.

A clue was identified by author and banjo novice J.W., when he had to mount a
new head to replace the torn original. He recognized a connection with
features of the many graphs of spectra and images of modes that we generated.
(E.g., consider ref. acoustics , second paper, Fig. 7.) For tap tuning, one is
instructed to tap or scratch, with a very small area of contact, near the
rim.huber  J.W. found that damping the central region of the head, e.g., by
damping the bridge, enhanced the desired effect.

Highly localized tapping near the edge along with damping the interior excites
a motion characterized by short wavelengths along the rim and extending only
one wavelength or so inward in the radial direction. These are the whispering
gallery modes studied by Rayleighwhisper  to account for the whispering
gallery of St. Paul’s Cathedral in London.

Because it’s a sharp tap or scratch near the rim, the frequency spectrum
consists of many narrow peaks, going up to substantially high frequencies.
However, the perceived tap pitch is not one of them. The key to understanding
what’s going on is to look at the differences between successive frequency
peaks. Those numbers rapidly approach a single value (asymptotic from above).
Although there is no peak whose frequency is that difference value, the
combination (“superposition”) of the motions represented by those peaks is
approximately periodic with a frequency given by the difference. (For this to
make sense, it is essential to remember that the spectrum analysis identifies
sinusoidal Fourier components and not simply all motions that repeat with a
fixed period.)

Depending on hearing acuity, brain function, and how close the actual
frequency differences are to multiples of a common factor, people tend to
identify the pitch of such a complex sound as that associated with the common
difference. In psychoacoustics, this is called the phenomenon of the “missing
fundamental.”

In some situations, hearing the missing fundamental is absolutely unambiguous.
For almost any banjo and tuning, a pluck in the lowest half-octave on the
$4^{\text{th}}$ string produces essentially no sinusoidal frequency component
equal to the pitch. The note is heard nevertheless. In fact, the timbre is not
noticeably different from the slightly higher notes on that string whose
frequency spectra do, indeed, include the fundamental.

In the whispering gallery mode description, the difference frequency described
above is the fundamental frequency of the whisper.

Here is a specific example of a very tight head on a Deering Sierra, tapped at
the edge with a pencil eraser, damped at the bridge, and recorded very nearby:
click here or go to http://www.its.caltech.edu/~politzer/pickers-
guide/whisper-1-just-one.mp3 .

Fig. 7 is a spectrogram of a series of such taps. The nearly equal frequency
spacing (i.e., in the vertical direction) of intensity peaks is obvious.

Figure 7: spectrogram of a sequence of 21 taps

Using the spectrogram, the common spacing works out to about 237 Hz, which is
an A#. Hearing-challenged author D.P. had a hard time at first to associate
the tap sound with a string pluck pitch. However, in retrospect, it’s a pretty
good match. As usual, the actual octave is ambiguous to some listeners, e.g.,
it could be 474 Hz.

Fig. 8 gives the computed spectrum of the twenty one taps in Fig. 7. Indeed,
237 Hz or 474 Hz are nowhere to be seen.

Figure 8: spectrum of the sound of the 21 tap sequence of Fig. 7 Figure 9:
waveform of a single tap Figure 10: autocorrelation of a single tap

The waveform of a single tap is shown in Fig. 9. A rough periodicity can be
estimated from from the graph by counting 29 cycles in 0.121 s. That
corresponds to 240 Hz. Autocorrelation is a well-defined mathematical
calculation that can be performed on a signal. In cases of complicated
sources, the location of a single high peak of the autocorrelation generally
agrees with the brain’s attempt to identify a single pitch. Audacity’s
calculation of the autocorrelation of the tap from Fig. 9 is shown in Fig. 10.
In the Audacity Frequency Analysis Autocorrelation window, the entry called
“Peak:” locates the highest peak at

0.0042 s, corresponding to 238 Hz (an A#3).

## References

  * (1) Acoustics of the Banjo: measurements and sound synthesis & theoretical and numerical modelling, J. Woodhouse, D. Politzer, and H. Mansour, Acta Acustica, 5, 15 and 16 (2021) https://doi.org/10.1051/aacus/2021009 and https://doi.org/10.1051/aacus/2021008; included are links to synthesized sound samples based on measurements and modeling, which are posted at https://euphonics.org; specifically, the synthesized banjo sounds are all in §5.5; more generally, euphonics.orgeuphonics  is an extensive resource on the science of musical instruments, currently being assembled by author J.W.
  * (2) https://euphonics.org.
  * (3) For example, subtle but discernible linear physics effects include string polarizationeuphonics  and sympathetic coupling of harmonics between different strings, which is considerably stronger on banjo than guitar.sympathetic
  * (4) D. Politzer, String Sing-Along (Sympathetic Vibration) Is Not the Key to Banjo Sound, https://www.its.caltech.edu/~politzer/turkey/turkey.pdf
  * (5) Johnny Butten plays a rectangular banjo built by Mike Gregory: https://www.youtube.com/watch?v=1FxfZ9yZ0PY and https://www.youtube.com/watch?v=rYUoIewJpro
  * (6) E. Skudrzyk, The mean-value method of predicting the dynamic-response of complex vibrators, Journal of the Acoustical Society of America, vol. 67, no. 4, pp. 1105 - 1135, 1980.
  * (7) D. Politzer, High Frequency Formants from Banjo Bridge Design, HDP: 21 – 02, https://www.its.caltech.edu/~politzer/bridge-hills/bridge-hills.pdf
  * (8) Steve Huber’s YouTube on how to tap tune a banjo head: https://www.youtube.com/watch?v=zsLxHOa4RUY
  * (9) J. Strutt, The problem of the whispering gallery, Phil. Mag 20 1001 (1910) (Scientific Papers 5 617 (1912)) and Further applications of Bessel’s functions of high order to the Whispering Gallery and allied problems, Phil. Mag 27 100 (1914) (Scientific Papers 6 211 (1920)).