Acoustics of the banjo: measurements and sound synthesis
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Acta Acustica 2021, 5, 15
Ó J. Woodhouse et al., Published by EDP Sciences, 2021
https://doi.org/10.1051/aacus/2021009
Available online at:
https://acta-acustica.edpsciences.org
AUDIO ARTICLE
Acoustics of the banjo: measurements and sound synthesis
Jim Woodhouse1,*, David Politzer2, and Hossein Mansour3
1
Cambridge University Engineering Department, Trumpington Street, CB2 1PZ Cambridge, UK
2
452-48 Caltech, Pasadena, 91125 CA, USA
3
Dassault Systèmes – SIMULIA, 5005 Wateridge Vista Dr., San Diego, 92121 CA, USA
Received 17 July 2020, Accepted 17 February 2021
Abstract – Measurements of vibrational response of an American 5-string banjo and of the sounds of played
notes on the instrument are presented, and contrasted with corresponding results for a steel-string guitar.
A synthesis model, fine-tuned using information from the measurements, has been used to investigate what
acoustical features are necessary to produce recognisable banjo-like sound, and to explore the perceptual
salience of a wide range of design modifications. Recognisable banjo sound seems to depend on the pattern
of decay rates of “string modes”, the loudness magnitude and profile, and a transient contribution to each played
note from the “body modes”. A formant-like feature, peaking around 500–800 Hz on the banjo tested, is found to
play a key role. At higher frequencies the dynamic behaviour of the bridge produces additional formant-like
features, reminiscent of the “bridge hill” of the violin, and these also produce clear perceptual effects.
Keywords: Banjo, Acoustics, Vibration, Synthesis
1 Introduction resonance frequencies, usually with fairly low damping
(measured Q-factors typically fall in the approximate range
Players of plucked-string musical instruments generally 30–100, at least for wooden instruments: see for example
have a clear impression that different instruments have Fig. 13 of [1]). As a result, the frequency response function
distinctive “voices”. This is most obviously true between characterising the transmission path between the string
instruments of different types but also between different vibration and the radiated sound pressure will inevitably
instruments of the same general type. A classical guitar is have large peaks and dips. A consequence of the peaky
different from a harp, a harpsichord or a banjo, but also response of an acoustic instrument is that the distribution
different guitars or banjos are different from each other. of sound energy between the overtones of a given plucked
Furthermore, players commonly maintain that the voice of note will vary enormously from one note to another on
an instrument can be discerned whatever music is played the instrument. Can it really be the case that a particular
on it. This seems familiar and obvious, but from the perspec- instrument has a characteristic “voice” which is recognisable
tive of acoustics the final statement is somewhat surprising. regardless of what music is played?
Every acoustical stringed instrument has some kind of To begin to address such a broad question, it makes
“soundboard”, a resonant structure that allows the energy sense to look at extreme cases. The topic of this study is
from the vibrating strings to couple effectively to sound to investigate what makes the sound of a banjo distinctively
waves in the surrounding air. The soundboard is most often different from other plucked-string instruments like the gui-
a wooden structure, as in a guitar or a harpsichord, but many tar. Even when strings, scale lengths, and pitches are chosen
other materials may be used: the banjo uses a stretched to be virtually identical, most listeners would agree that the
membrane similar to a drum skin, while around the world sounds of banjo and guitar can be distinguished from just a
stringed instruments can be found that employ gourds, tor- few plucked notes. Measurements will be presented here of
toise shells and many other natural or man-made materials. the vibration behaviour of banjo and guitar bodies and of
All these structures have one thing in common: they are the sounds produced during normal playing. These mea-
vibrating systems with normal modes and natural surements will confirm that there are indeed large differ-
ences of vibrational behaviour between a banjo and an
*Corresponding author: jw12@cam.ac.uk
The display of Audio files embedded in this PDF depends on the
acoustical guitar: far bigger than the differences between
software used (PDF reader, video player, installed codec, direct guitars of a similar type, or between different banjos.
display in the browser, etc.). Please see the Adobe Acrobat page Once the driving-point frequency response of an instru-
for more explanation. You can also find the audio files at ment body has been measured at the bridge notch for a par-
https://euphonics.org/ Chapter 5, Section 5.5. ticular string, it is straightforward to combine this with
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
2 J. Woodhouse et al.: Acta Acustica 2021, 5, 15
known properties of the string itself, and synthesise fully- compare with real instruments, so there is relatively little
coupled transient response to plucking the string. Such syn- overlap with the work to be described here and in the com-
thesised responses will be used for two purposes here. They panion paper.
can be compared to measurements of the distribution of loss Furthermore, the existing literature on the guitar turns
factors in frequency, and thus give a detailed interpretation out to give rather little information about the particular
of those measurements. They can also be turned into audio issues to be raised here. Many authors seem to agree that
files, to find out whether listeners are convinced that they the most important aspects of guitar acoustics concern a
“sound like a banjo” or “sound like a guitar”. small number of low-frequency modes: their natural
This project to explore the banjo has been driven frequencies, mode shapes and sound radiation behaviour.
throughout by listening to synthesised sounds based on a There is some literature on numerical modelling that
range of theoretical and numerical models. The models have extends to higher frequency [7], but much of the literature
been iterated until, in the opinion of the authors, they do is concentrated on measurements of mode shapes and sound
indeed produce sounds that are convincingly banjo-like fields, and increasingly elaborate numerical models: see for
while accurately reflecting relevant aspects of the underly- example [8–10]. The recent review article [11] gives many
ing physics. This background of interplay between theoret- other literature references. Guitar makers can manipulate
ical understanding, model development and sounds makes these low modes through constructional details, especially
the final story rather complicated to present. This paper the chosen material, mass and bracing pattern for the sound-
will describe experimental results and synthesis methods, board [12, 13]. There is some literature on perceptual assess-
then introduce a set of sound demonstrations. A companion ment of differences between guitars, but again it is largely
paper [2] will fill in the gaps in this description, with focussed on the influence of individual modes [14–17].
detailed analysis and modelling of the underlying physics. The individual body modes of a banjo are closely related
These models will be validated against the measurements to the well-studied case of drums (see for example [18–20]).
presented in this paper, and they then allow additional The accompanying web site [3] contains a discussion of
synthesised sounds to be created. Some synthesised sounds drums, and also (in section 5.3 of [3]) some measured exam-
are included in this paper, and a much larger collection of ples of vibration modes of the banjo to be studied here.
sounds has been provided via an accompanying web site However, it will be suggested in the present work that indi-
[3]: the reader is encouraged to consult this web site along- vidual low modes are less important for the banjo than in
side reading this paper. the case of the guitar: instead, several formant-like features
Beyond the basic question of recognising a banjo among will be revealed, and the sound examples provided on the
other instruments lie many details of interest to makers and accompanying web site [3] seem to suggest that these for-
players of instruments. Banjo enthusiasts, just like gui- mants have higher perceptual salience than individual
tarists or violinists, are sensitive to many nuances of sound modes. The available evidence [1, 21] suggests that there
and playing behaviour of their instrument. Arguments rage are no comparable formant-like features in guitars, at least
about the merits of different constructional details and in those with glued bridges relevant to this study. However,
adjustments. Synthesis based on physical models can be a comparable feature, usually called the “bridge hill”, has
used to explore the perceptual effects of changing model been extensively investigated in the violin [22–24]. This
parameters, such as the tension in the banjo head feature can be manipulated by violin makers, and is an
membrane or the “break angle” of the strings over the important resource for tonal adjustment. The same is
bridge. There will be some discussion here, and more exten- almost certainly true of the banjo formants.
sive discussion is given on the web site [3] which hosts the
main set of sound files.
It should be emphasised that this paper and its compan- 2 Banjo-guitar comparisons
ion do not include any formal psychoacoustical investiga- 2.1 The structure of the banjo
tions. On the web site and in the two papers, opinions
will be advanced about these sounds based on the authors’ The banjo is unusual among stringed instruments.
own listening, but readers can and should listen for Many of the elements responsible for the acoustical beha-
themselves; they may or may not agree with those opinions. viour can be adjusted by the player. The tension of the head
Of course, any such opinions need to be tested by careful can be altered, and indeed the head membrane can be
perceptual studies. This paper and its companion provide replaced; the bridge can be altered. No equivalent adjust-
some clear hypotheses, to be tested in future work: con- ments can be made on a fixed-bridge guitar: the player
straints of length do not allow such investigations to be has to accept what the guitar maker gives them. Also in
included here. strong contrast to instruments like the guitar or the violin
Although there is extensive literature on the acoustics of is the fact that the geometry of the resonant structure of
the guitar, the violin, and other stringed instruments, there a banjo is quite simple: a membrane is stretched across a
is rather little published work on the banjo [4–6]. These circular frame (the “pot”), tensioned by a mechanism similar
works shed some useful light on the physics of the instru- to that used in drums. Some banjos, including the main one
ment, but none of them examine all the consequences of tested here, have a metal ring (the “tone ring”) between the
using a membrane in place of a wooden box, or create a pot and the membrane to control the boundary conditions
physical model capable of detailed synthesis of sounds to at the edge of the membrane. As with most drums, the
J. Woodhouse et al.: Acta Acustica 2021, 5, 15 3
Table 2. Parameter values for the strings of the Deering banjo.
String 1 2 3 4 5
Diameter (mm) 0.25 0.28 0.33 0.53 0.24
Length (m) 0.67 0.67 0.67 0.67 0.50
Frequency (Hz) 293.7 246.9 196.0 146.8 392.0
Tension (kg) 6.19 5.29 4.66 5.76 5.56
Impedance (Ns/m) 0.154 0.157 0.174 0.287 0.139
“resonator” back, which among other things modifies the
sound radiation characteristics and provides a Helmholtz
resonance at low frequency. Figure 1 includes an approxi-
mate scale drawing of a cross-section through the body,
showing the configuration of the pot and resonator back.
Some results will also be shown for two other banjos.
Detailed comparisons will be made with a guitar by Martin
Woodhouse [26]. This guitar, somewhat unusually for a
steel-strung instrument, embodies a version of Torres-like
fan bracing.
2.2 Spectrograms and loudness profiles
Since the aim is to understand the characteristic sound
of a played banjo, it is useful to begin with some simple
experimental comparisons between notes on the banjo
and the guitar, based on normal playing. Conveniently,
the top strings of the Deering banjo and the Woodhouse
Figure 1. The Deering banjo, a schematic cross-section of the guitar are extremely similar: both are plain steel strings
body with main components drawn approximately to scale, and with the same diameter, and they are under very similar
a close-up of the bridge. The white patches on the bridge are tensions. This allows a rather clean comparison between
reflective tape for the laser vibrometer measurements. plucked notes on the two instruments. First, the note G4
(392 Hz) was plucked on both instruments: in standard
tuning this corresponds to the 3rd fret on the guitar and
Table 1. Parameter values for the Deering banjo. Head tension the 5th fret on the banjo. The notes were played in the
is estimated from computer modelling: see the companion paper normal way, and efforts were made to make them as similar
[2]. as possible. The same plectrum was used, and the plucking
Head diameter d 268 mm
distance from the bridge was 130 mm in both cases. The
Head thickness h 0.18 mm player sat in the same position with both instruments,
Mass per unit area r 0.30 kg/m2 and tried to use identical plucking gestures. The notes were
Tension T 5.33 kN/m recorded in a medium-sized domestic room suitable for
Bridge mass mb 2.2 g chamber music, using a microphone (GRAS 46AR) placed
Bridge footprint 78 5 mm on a stand above a carpeted floor, approximately 1 m away
from the player.
Spectrograms of the two resulting sounds are shown in
membrane was traditionally made of natural skin, but Figure 2, with identical scales including the colour scale
nowadays is usually made of a synthetic polymer: biaxi- indicating level. Both plots show the expected set of narrow
ally-oriented polyethylene terephthalate, more commonly vertical bands, associated with the near-harmonic overtones
known by trade names such as Mylar. The strings pass over of the string in both cases. As can be seen particularly
a small “floating” bridge which rests on the membrane. The clearly for the guitar note, some overtones have significantly
vibration of a circular stretched membrane is, of course, a higher amplitude than others. This is in part because of a
well-understood problem going back at least as far as familiar effect arising from the chosen plucking point: it
Rayleigh [18]. was roughly 1/5 of the way along the string, so overtones
The main 5-string banjo used for measurements in this 5 and 10 are relatively weak.
study is a Deering Eagle II, in the condition in which it was Careful inspection reveals some significant differences
received from the manufacturer [25]. Figure 1 shows the between the two plots. The banjo achieves higher levels
banjo, and a close-up of its bridge. Some parameter values than the guitar: the colour scale indicates decibels relative
for this banjo and the strings with which it is fitted are to the peak value for the banjo, and on this scale the peak
given in Tables 1 and 2. The banjo has a removable value for the guitar was 10.2 dB. The decay times for the
4 J. Woodhouse et al.: Acta Acustica 2021, 5, 15
1.5 0 4.5
4
-10
3.5
1 -20 3
Loudness (sone)
Time (s)
2.5
-30
2
0.5 -40 1.5
1
-50
0.5
0 -60 0
0 2 4 6 8 10 0 100 200 300 400 500 600
Frequency (kHz) Time (ms)
1.5 0
Figure 3. Short-term time-varying loudness computed from
plectrum-plucked notes from the first octave on the top string of
-10 the banjo (solid lines) and the guitar (dashed lines).
1 -20
A subjective observation can be mentioned at this stage.
Time (s)
-30 During the measurement of bridge admittance, to be
described in Section 2.4, it was noted that the “clonk” noises
0.5 -40 of the miniature impulse hammer hitting the bridge (with
the strings damped) were quite different for the banjo and
the guitar. The guitar made a sound familiar from many
-50
other tests on wooden instruments: in general terms,
guitars, lutes, violins, cellos and so on all make a rather
0 -60 similar sound during this testing. The banjo made a very
0 2 4 6 8 10
Frequency (kHz) different sound, reminiscent of a drum. This sound, arising
from the transient excitation of modes of the body, is closely
Figure 2. Spectrograms of plucked notes G4 (392 Hz) on the related to the feature just described in the spectrogram of a
top string of (a) the banjo; (b) the guitar. The colour scale plucked note.
corresponds to identical levels (in dB) in both plots.
The spectrograms already give an indication, from the
peak levels, that the banjo is louder than the guitar, a
overtones of the guitar string are generally significantly conclusion that does not come as a surprise. Nevertheless,
longer than for the banjo: a detailed quantitative analysis it is worthwhile to analyse loudness of played notes on
of decay times will be shown in Sections 2.3 and 3.2. For the two instruments in a quantitative way. With the same
some overtones of the banjo string (for example the 1st playing and recording setup just described, plectrum-played
and 3rd overtones), the colour can be seen to modulate in notes were recorded at every semitone from the open string
time, indicating beating between closely-spaced frequencies. up to the 12th fret, on both top strings. These were then
This is probably caused by a frequency split between the processed by the procedure described by Moore et al.
two polarisations of string vibration, because they are [27, 28], to calculate the short-term time-varying loudness
differently affected by coupling to the banjo body. Both in sones. The results are plotted in Figure 3: solid lines for
polarisations will normally be excited to some degree by a the banjo and dashed lines for the guitar.
pluck. There are variations between individual notes on each
Finally, there is a clear difference of appearance of the instrument, partly caused by physical differences and
two spectrograms within the first 0.2 s or so after the pluck. partly, no doubt, by the inevitable variability of a human
The banjo shows a bright patch indicating a significant player. Nevertheless, the two groups of plots are quite dis-
broad-band spread of radiated sound between the string tinct. The banjo notes are without exception louder than
overtones, extending up into the kHz region. For the guitar the guitar notes at early times following the pluck. They
the corresponding activity between the string overtones is then show more rapid decay than the guitar notes, and
less conspicuous, and largely confined to lower frequencies. the two clusters of curves merge into the same range after
As will be explored in detail in Section 3.2, these broad- about 400 ms. If the plot had been extended for a longer
band signals arise from transient excitation of modes of time, it would show the guitar notes eventually tending
the coupled string-body system which have energy mainly to be louder than the banjo notes in the tail of the decay.
in the body rather than in the string. The differences revealed in this plot give the first clear
J. Woodhouse et al.: Acta Acustica 2021, 5, 15 5
evidence that banjo notes will be perceptually distinct from 10-1
guitar notes (see for example Moore [29]).
Another subjective observation can be recorded, relat-
ing to these played notes. Although, as already noted, the 10
-2
two steel strings were in fact very similar, the impression
to the player was that the guitar string felt “tighter” and less
Loss factor
responsive. This impression is perhaps associated with the
-3
loudness difference: for a given plucking force applied by 10
the player, the banjo made a louder sound than the guitar.
If the player wanted to make sounds of similar loudness,
more force would be needed on the guitar. Perhaps this 10-4
leads to a subjective impression of unresponsiveness.
2.3 Distribution of loss factors
200 500 1000 2000 6000
Loudness is, of course, a very crude measure of differ- Frequency (Hz)
ence between musical sounds. For a more fine-grained view,
Figure 4. Loss factor versus frequency for modes excited by
the chromatic scales on the top strings of the banjo and plucking the top string of the guitar (black squares) and the
guitar can be analysed in a different way. The detailed test banjo (red circles). The analysis method cannot detect modes
procedure has been described in earlier papers [15, 30], and significantly above the magenta line. Green dashed lines indicate
a related test methodology has been used by Fréour et al. decay time constants (xg)1 = 50 ms (top line), 100 ms, 200 ms,
[31]. Time-frequency analysis can be used to deter- 300 ms and 400 ms (lowest line).
mine the best-fitted frequency and decay rate of every
spectral peak satisfying a set of criteria to ensure data
quality. The results can be plotted as a “cloud” of points associated with the fundamentals and second overtones of
to reveal patterns in the distribution of loss factors with the played notes.
frequency.
These measured results are plotted in Figure 4: guitar in 2.4 Bridge admittance measurements
black, banjo in red. Some areas of the plot have both red
and black points, but elsewhere they tend to occupy rather To compare the structural vibration behaviour of the
different regions. The magenta line indicates the limit of banjo and the Woodhouse guitar, the natural first step is
applicability of the analysis method. Above this line the to measure their input admittances at the bridge: the
decay time becomes too short to be resolved, and absence velocity response per unit force as a function of frequency.
of points in this region does not imply that no such For any stringed instrument, this admittance characterises
combinations of frequency and loss factor exist in the real the dominant linear coupling between string and body,
instruments. Lines to indicate the decay time constant are including the energy transfer rate from a string of given
plotted in green dashes: details are given in the caption. wave impedance. Measurements were made by the proce-
Detailed analysis of these results is deferred until dure described in earlier work [1]. Both instruments were
Section 3.2, when comparisons with corresponding synthe- tested with strings fitted and at normal playing tension.
sised signals allow more illuminating interpretation. The strings were damped by weaving thin paper through
However, some simple observations can be made immedi- them, without contacting the body or fingerboard. The
ately. The black and red points lying a little below the instruments were supported in a vertical position with soft
magenta line at frequencies up to about 1 kHz correspond foam at the tailpiece end and a foam-lined clamp on the
to “body modes”, already mentioned in connection with neck. Controlled force was applied to the bridge close to
Figure 2. The “string modes”, with an approximately the notch for the first string, using a miniature impulse
harmonic sequence of frequencies based the fundamental hammer (PCB 086D80) held in a pendulum fixture to
of each played note, show lower loss factors. For these, ensure that the same point was struck every time. The
the majority of the red points lie clearly higher than the resulting velocity response adjacent to the excitation point
black points, signalling slower decay for string modes of was measured with a laser-Doppler vibrometer (Polytec
the guitar and faster decay for the banjo. The relevant OFV-056). The signals were digitised into a PC at a
decay time constants can be deduced by comparison with sampling rate of 40 kHz, and processed with software
the green dashed lines: up to about 1 kHz, the banjo modes written in Matlab. Each measurement involved at least
tend to have decay times in the range 100–200 ms, while the 10 repeats, giving an averaged result for the frequency
guitar modes lie in the range 300–400 ms. These values tie response function together with the associated coherence
up well with the early decay times seen in the loudness function (see for example McConnell and Varoto [32]).
profiles shown in Figure 3. The range of fundamental Results are shown in Figure 5. The thick red line shows
frequencies for the banjo notes was 294–588 Hz, while the admittance of the Deering banjo, while the thick blue
that for the guitar notes was 330–660 Hz, so the pattern line shows the Woodhouse guitar. The dashed red line
revealed in the frequency range up to about 1 kHz is mainly shows the corresponding admittance of an inexpensive6 J. Woodhouse et al.: Acta Acustica 2021, 5, 15
20
10
Bridge admittance (dB re 1m/s/N)
0
-10
-20
-30
-40
-50
100 200 500 1000 2000 5000
Frequency (Hz)
Figure 5. Measured drive-point admittance functions, all close Figure 6. Admittance functions for the Deering banjo. Red: as
to the bridge notch for the first string. Heavy red line: Deering red curve in Figure 5; black dashes: modal fit to the red curve;
banjo; dashed red line: Ozark banjo; heavy blue line: Woodhouse blue: corresponding admittance with the resonator back
steel-string guitar; black lines: six other guitars of various types removed; black solid: bare head of the banjo with bridge, strings
(see text). and resonator removed.
10
banjo of Ozark brand. The cloud of thin black lines shows
the admittances of a representative selection of six other
0
guitars that are broadly comparable to the Woodhouse.
The selection includes classical, flamenco and steel-string
Bridge admittance (dB re 1m/s/N)
-10
guitars; makers include Fleta, Ramirez, Smallman, O’Leary
and Fylde; and the range includes fan-braced, lattice-braced
-20
and X-braced instruments.
The two banjos follow a very similar pattern, although
the individual peaks for the Ozark are a little lower in fre- -30
quency because it had a lower head tension than the
Deering. The guitars also follow a very consistent pattern. -40
Below about 300 Hz each guitar has a small number of
strong peaks, and the varieties of construction lead to signif- -50
icant variations in the individual modal frequencies [12].
However, above 300 Hz all the guitars occupy a very similar -60
100 200 500 1000 2000 5000
range. Strikingly, throughout the range above 300 Hz the Frequency (Hz)
admittance magnitude of the banjos is typically 20–30 dB
higher than that of all the guitars. This difference will be Figure 7. Admittance functions for the Deering banjo with the
shown in Section 3.2 to account for the results in Figure 4. resonator back removed. Red: as blue curve in Figure 6; blue:
measured near bridge notch for third string; black: measured on
Figure 5 suggests that the particular choice of guitar makes
side of bridge saddle, parallel to membrane.
little difference to the contrast with the banjo, especially in
respect of the results in Sections 2.2 and 2.3 since they only
involved playing on the top strings with all string overtone
frequencies above 294 Hz. They immediately reveal some important features, and they
At first sight, the banjo results shown here conflict with will play a role in the companion paper [2] by illuminating
the only previously published measurement of a banjo different aspects of the underlying physics and also giving a
admittance, in Figure 11 of Stephey and Moore [5]. How- wider range of validation cases for comparison with numer-
ever, that earlier measurement was made with undamped ical modelling. The blue curve in Figure 6 shows the effect
strings, giving a different appearance, and furthermore of removing the resonator back. The main difference from
the authors have confirmed that there was a calibration the red curve is at low frequency. As is familiar from earlier
error by a factor 10 (20 dB). Allowing for these two effects, literature on the guitar (see for example [9]), and also on
the measurement seems quite consistent with the one loudspeaker design [33], coupling to the internal air of the
presented here. resonator produces an additional degree of freedom, so that
Figures 6 and 7 show additional measurements that give the original measurement (red curve) has two peaks
a more complete characterisation of the Deering banjo. whereas without the resonator (blue curve) there is a singleJ. Woodhouse et al.: Acta Acustica 2021, 5, 15 7
peak near 300 Hz, lying between the pair. Of course, there
are other more subtle changes associated with removal of 350
the resonator: coupling to other internal acoustic reso-
300
nances in the cavity, and modifying the sound radiation
behaviour. The companion paper [2] gives detailed discus-
250
sion of the sound radiation question.
Level (dB)
The solid black curve shows the input admittance on 200
the banjo head with the strings and bridge removed, at a
position close to that used for the other tests. For this 150
measurement, the resonator was also removed to give a
system as close as possible to the ideal circular membrane. 100
The measurements on the bare membrane proved quite
challenging, for reasons to be explored in the companion 50
paper [2], but for all three measurements excellent coher-
ence was obtained up to at least 10 kHz. It is immediately 0
100 200 500 1000 2000
clear that removing the strings and bridge from the banjo to Frequency (Hz)
leave the bare membrane has a very profound effect on the
behaviour. Figure 8. Results for variations of head tension: waterfall plot
The final ingredient of Figure 6 is the dashed black of the frequency response deduced from microphone recordings
curve. This shows a fitted version of the red curve, based of head taps with eight different settings of the head tension in a
banjo. Tensions are indicated by colour: blue for the lowest
on modal decomposition (see for example Ewins [34]). It
tension, going through the rainbow sequence to red for the
shows an excellent level of agreement between the fitted highest tension. Curves are spaced by successive intervals of
curve and the original measurement. It will play a role when 40 dB for clarity.
synthesis methods are discussed in Section 3.
Figure 7 shows admittance measured at two other
positions on the banjo bridge, compared to the standard the geometry of the break angle of the strings over the
position near string 1. All three of these curves were bridge. These effects will be explored in detail in the
obtained with the resonator back removed. The red curve companion paper [2].
shows the admittance at the position of string 1, and is
the same as the blue curve in Figure 6. The blue curve 2.5 Varying the head tension
shows the admittance measured at the bridge centre, the
position of string 3. It follows the red curve closely at the It is useful to show one more set of measured results, to
lowest frequencies, but after that the two curves diverge. illustrate the effect of varying membrane tension. A banjo
Over most of the range 1–4 kHz the blue curve is signifi- with easily adjustable head tension (a Nechville Moonshine)
cantly below the red curve, but around 5 kHz it rises to a was set to a number of different tensions, covering the full
conspicuous peak. The black curve shows a rather similar feasible range from a tension that would be too low for
peak, centred around 3 kHz. This curve was obtained by any modern banjo with a Mylar head (although within
tapping and measuring on the side of the bridge saddle, the range used for some drums), up to a high tension on
parallel to the head membrane. It is the admittance rele- the brink of failure of the head. In each case the sound of
vant to excitation of the head by the polarisation of string tapping the head with a pencil was recorded. A set of
vibration lying in the plane parallel to the head. frequency spectra of these sounds is shown in Figure 8, with
Figure 7 gives a first glimpse of a phenomenon that will the lowest tension at the bottom and the highest at the top.
prove to be important: banjo admittance is influenced by Increasing the tension will naturally increase the
three strong formant-like features. The first of these is a frequencies of each individual vibration mode, and a
broad feature occupying the approximate frequency range pattern of steadily rising frequencies is immediately visible.
200–2000 Hz, dominating the trend of the red curve. The But the tension also influences the sound radiation and the
other two, around 3 kHz and 5 kHz, are brought out most associated radiation damping, as will be explained in detail
clearly by the black and blue curves respectively. It will be in the companion paper [2]: a higher tension means a higher
shown in the companion paper [2] that the two formants at wave speed, so that the speed in the membrane comes closer
higher frequency both have their origin in the dynamic to the speed of sound in air. Figure 8 clearly shows band-
behaviour of the banjo bridge, in conjunction with the width increasing with tension, most obviously for the lowest
membrane on which it sits. They are both closely analogous mode.
to the “bridge hill” of the violin: see for example [22–24]. The Figure 9 shows the frequency and loss factor of that
low-frequency formant has a quite different origin. The lowest mode for each case, plotted using the same colour
frequency and bandwidth of this formant are mainly deter- code as Figure 8. Included in the plot are results from
mined by the bridge mass, and two sources of stiffness: one two theoretical models that will be presented in the
from the transverse compliance of the membrane, and the companion paper [2]. Black symbols represent frequen-
other from a combination of effects of string tension and cies and radiation damping computed using a detailed
axial stiffness of the strings and membrane, acting through Finite Element/Boundary Element (FE/BE) model, while8 J. Woodhouse et al.: Acta Acustica 2021, 5, 15
-1
10
10-1
Loss factor
Loss factor
10-2
-2
10
-3
10
0 100 200 300 400 500 600 100 200 500 1000 2000 4000
Frequency (Hz) Frequency (Hz)
Figure 9. Results for variations of head tension: loss factor Figure 10. Loss factor against frequency for fitted modes of
against frequency for the fundamental mode of the membrane selected examples of the variable tension set. Tensions are
with the eight tension settings (circles, tensions indicated by indicated by colour as in Figure 8. Circles and stars show results
colour as in Fig. 8), compared to computed values of frequency from separate tests.
and radiation damping from the FE/BE model (black plus
symbols) and the rectangular membrane model (magenta
squares). radiation damping. The conclusion for the sound of the
banjo is that increasing the head tension will increase the
radiation efficiency of all modes, and also change the tonal
magenta squares were obtained from a simple model based balance across the frequency spectrum by shifting individ-
on rectangular geometry. These two lines of points follow ual resonances and perhaps by changing the general trend
the general rising trend of the measured values, but deviate of response with frequency.
in detail. A major difference between the two models is that
the FE/BE calculations were for an unbaffled structure,
whereas the rectangular model assumes a baffle. This 3 Sound synthesis based on measured
highlights an issue that will become important when sound admittance
demonstrations are discussed: results will prove to be 3.1 Types of synthesis model
sensitive to details of damping, and vibration damping is
notoriously hard to model theoretically. Measurements like Measured input admittance can be used to synthesise
these and the loss factor results seen in Figure 4 will be used plucked notes. Such synthesis can be used to understand
to fine-tune damping models. the results shown in Section 2.3, and also to generate sound
The theoretical results extend to far higher tensions files for listening purposes. As has been explored in some
than could safely be achieved on a real banjo. It may be detail in previous work [30, 35] there are several possible
noted that for the membrane wave speed to match the approaches to such synthesis. Two of the methods pre-
sound speed in air, the fundamental frequency would need sented in that work will be used here. The first method
to be at least doubled relative to the highest achieved in works in the frequency domain, and uses an inverse FFT
these measurements. The conclusion is that a Mylar to obtain the time history of the plucked note. This method
membrane, whether on a banjo or a drum, is always can use a measured bridge admittance with no additional
subsonic. processing. Alternatively, it can use a bridge admittance
Finally, the recordings were processed in the same way computed from a theoretical model.
used to generate Figure 4, giving the results plotted in The second method works by modal superposition. For
Figure 10. For clarity, only the extreme cases plus one inter- admittance derived from theoretical modelling, the modes
mediate tension are shown. For each tension (indicated by may be directly available from the model. For a measured
colour) two different taps were analysed and plotted with admittance like the ones shown here, a modal decomposi-
circle and star symbols respectively to give an indication tion must be carried out. This can be done using signal
of repeatability. As before, the magenta line indicates the processing methods derived from the fields of system identi-
approximate limit of applicability of the method: points fication or experimental modal analysis (see for example
lying much beyond this line will not be reliably detected. [34]). The result of such a modal fit to a measured banjo
The results tell a clear story. For each tension there is a admittance was shown in Figure 6.
cloud of points representing a range of possible loss factors The main motivation to make some use of this modal-
at any given frequency. These clouds overlap, but there is a based synthesis method is to quantify the distinction
systematic movement upwards in the diagram as tension between “string modes” and “body modes”. The simpler,
increases. This rise in damping is caused by increasing faster and more robust frequency-domain approach cannotJ. Woodhouse et al.: Acta Acustica 2021, 5, 15 9
give this information. The modal-based approach uses sep-
arate degrees of freedom for the string and body, so for any
given coupled mode of the combined system it is easy to
compute the fraction of total potential energy associated
with the string degrees of freedom. If that fraction is bigger
than 0.5 the mode is labelled a string mode, otherwise it is
labelled a body mode. In almost all cases the energy fraction
is very close to zero or 1, making the distinction uncontro-
versial. The synthesised pluck sound is created by a linear
superposition of all the modes, so if desired the string modes
and the body modes can be separately summed before being
combined to give the final sound. Sound examples will be
discussed in Section 3.3.
There is a further subdivision of synthesis methods,
depending on whether string vibration is confined to a
single polarisation plane, or whether both polarisations
are included. There is no doubt that a 2-polarisation model
captures more of the physics of the instrument. Examples of
the effect of including the second polarisation are included
among the sound demonstrations (see Sect. 3.3), but
otherwise for the purposes of this initial study only a
single polarisation will be considered, normal to the
“soundboard” of the instrument. This simplifies matters
because only a single input admittance is required, as
opposed to the 2 2 matrix of admittances required in
the more general case. The results to be presented here
and on the web site [3] give a strong indication that
recognisable banjo sound can be produced by this simpler
modelling option.
For all approaches to synthesis, properties of the strings
are needed: relevant properties of guitar strings have been
published previously [15], while properties of the banjo
strings are listed in Table 2. The main missing ingredient
is a model for the intrinsic damping of the banjo strings. Figure 11. Loss factor versus frequency for modes excited by
A suitable model was fine-tuned by comparing the results plucking the top string of (a) the guitar; (b) the banjo. Red
of synthesis with measurements of real plucked notes: the circles and stars: measured values; blue stars: values from
details will be presented in Section 3.2. synthesised notes; black dots: predicted loss factor for energy
Exploration of these varieties of synthesis model flow into the body alone, calculated from the measured
requires two stages: the first based on measurements, the admittances shown in Figure 5. The analysis method cannot
second based on listening to the resulting sounds. It is a detect modes significantly above the magenta line. Green dashed
common experience in musical acoustics that features which lines indicate decay time constants (xg)1 = 50 ms (top line),
show up very clearly in acoustical measurements do not 100 ms, 200 ms, 300 ms and 400 ms (lowest line). Blue dashed
line in (a) shows the internal loss factor of the string.
necessarily produce large audible effects, while conversely
some effects that are perceptually important can be remark-
ably hard to pin down by measurements.
the two sets are plotted as red circles and stars, and a
3.2 Model validation: distribution of loss factors reassuring correspondence can be seen between the two over
the important parts of both plots.
It has been seen in Figure 5 that the bridge admittances The blue and black points in these plots show theoreti-
of guitars and banjos are significantly different. To see how cal estimates of two different kinds. The black points show
this translates into different behaviour when the instru- estimates of the contribution to the loss factor arising only
ments are played in the normal way, the distribution of loss from energy flow from the string into the instrument body,
factors shown in Figure 4, deduced from playing chromatic calculated from the measured bridge admittance. The
scales on the top strings of the banjo and guitar, can be reflection coefficient, R(x), for a transverse wave on the
compared with estimates derived from modelling. The mea- string incident on the bridge can be expressed in terms of
sured clouds are plotted again in Figure 11 for the guitar the admittance Yb(x):
and the banjo separately: two sets of plucked notes were
Y b ðx Þ Y 0
recorded for each instrument, and processed independently RðxÞ ¼ ; ð1Þ
to give an indication of consistency of the measurements: Y b ðx Þ þ Y 010 J. Woodhouse et al.: Acta Acustica 2021, 5, 15
where Y0 is the characteristic admittance for the string, that the plotted points are spread out along the frequency
the inverse of the impedance listed in Table 1 (see for axis. The body modes have Q-factors (inverse of loss
example Cremer [36]). The loss factor gb,n associated with factors) around 100 or lower, while the string modes have
dissipation during this incomplete reflection for the nth Q-factors of a few thousand and their decay times deter-
overtone of string vibration depends only on the real part mine the duration of each played note. It is very striking
of R2: it can be written in Figure 11a that the line of the string modes is fairly
featureless, and mostly lies significantly above the black
1 ReðR2 Þ points. This confirms something reported earlier for guitars
gb;n ; ð2Þ
2pn (it is true for both steel-string and nylon-string guitars) [37]:
rather unexpectedly, the decay rate of string modes in these
and provided the ratio Y/Y0 is small in magnitude this
instruments is dominated by the damping of the string
reduces to
itself, and loss into the body of the instrument is usually
2ReðY Þ only a small perturbation. There are exceptions where the
gb;n : ð3Þ guitar body has a strong resonance, but it should be noted
pnY 0
that this particular plot is confined to the frequency range
Using the measured bridge admittance of the guitar and the relevant to the top string, and the strongest body
banjo, respectively, this loss factor has been calculated at resonances of the guitar lie lower in frequency.
intervals of 1/30 semitone for all overtones falling within The blue points for string modes follow the red points
the frequency range, and plotted as the black symbols in closely, but this is no coincidence: this data was used to
Figure 11. fine-tune the damping model for the string used in the
The blue points in Figure 11 are the result of repeating synthesis model. Earlier work investigating a range of
the chromatic scale experiment using the frequency-domain polymer-based musical strings demonstrated that a rather
synthesis model, and processing the results with the identi- simple model for the intrinsic damping of the strings gave
cal analysis code used for the experimental observations (so excellent agreement with measurements [38]. A version of
they are again limited to the region below the magenta that damping model has been successfully adapted for the
lines). These points can be directly compared to the steel strings played here: the loss factor resulting from this
measured red points, and they also serve to give an internal model is shown as the dashed blue line in Figure 11a. The
check both on the synthesis code and on the signal process- total modal loss factor is estimated by
ing method, by comparing with the black points. Both blue
gn gbackground þ gair þ gbend ; ð4Þ
and black points are computed, by very different
approaches, using the same bridge admittance, so the blue
in which a constant “background loss factor” gbackground =
points should always lie above the black points because the
3 105 is added to a term associated with damping from
synthesis model allows for additional energy loss in the
viscosity in the surrounding air, and a term associated
string. But this additional loss is quite small, so when
with bending stiffness of the string. The air damping term
the black points predict relatively high loss factor, the blue
is given by Fletcher and Rossing [19] in the form
points would be expected to follow them. These expecta- pffiffiffi
tions are reassuringly confirmed in the plots, most clearly qa 2 2M þ 1
in the banjo case. gair ; ð5Þ
q M2
The plots tell an interesting story. It is simplest to
explain the guitar case first, Figure 11a. The majority of where qa is the density of air, and
the red and blue points mark out two lines, one with loss sffiffiffiffiffiffiffiffiffi
factors of the order of 102 and the other of the order of d 2pfn
103. As already pointed out, these indicate “body modes” M¼ ; ð6Þ
4 ga
and “string modes” respectively. The body modes often
show as clusters of many points, because in principle these where ga is the kinematic viscosity of air. Textbook values
modes are excited by the transient nature of every plucked are used: qa = 1.2 kg/m3 and ga = 1.5 105 m2/s.
note, regardless of the played pitch, so that many estimates Energy loss from viscoelasticity in the string arises from
of these modes are obtained from the chromatic scale. the influence of bending stiffness. If the Young’s modulus
Strictly, each body mode is not exactly the same for every of the string has the complex value E(1 + igE), an argu-
played note, because it is perturbed by coupling to the ment based on Rayleigh’s principle [39] can be used to
string. However, except for special cases where a string yield an expression for the associated loss factor of the
overtone falls very close to an unperturbed body mode, nth string mode:
the shift is small. The blue and red points agree quite well
for these body mode clusters. Probably the line of points Ep2 d 2 n2 gE
gbend ; ð7Þ
would continue approximately horizontal beyond the 64qL4 f12 þ Ep2 d 2 n2
magenta line but for the limitations of the analysis
technique. where the string has vibrating length L, diameter d,
The string modes consist of an approximately harmonic density q and fundamental frequency f1. This bending
series based on the fundamental of each successive note so term turns out to have only a minor influence for plainJ. Woodhouse et al.: Acta Acustica 2021, 5, 15 11
steel strings over the frequency range examined here: the probably other important factors, not made apparent by
value gE = 103 was used, but the exact value does not this particular analysis. Physical measurements are never
matter very much. enough to settle perceptual questions: it is necessary to
The plot for the banjo, Figure 11b, is strikingly different listen to the sounds from the synthesis model and find out
from the guitar case. The black points lie considerably if they do in fact strike listeners as convincingly banjo-like.
higher over much of the frequency range, as a direct result
of the higher input admittance of the banjo. There is still a 3.3 Synthesised sound examples
trace of two lines showing string modes and body modes,
but whenever the black curve crosses above the position To accompany this discussion, the reader is referred to
where the line of string modes occurred for the guitar, it Section 5.5 of the web site [3]. All the sound examples use
carries the actual loss factors up with it. Energy dissipation the same short musical passage, the first few measures of
arising from different physical mechanisms is additive, so in a banjo arrangement of the tune “The Arkansas traveler”.
theory the total loss factor cannot be lower than the black The first example appears here and as Sound B.1 on
curve. the web site. It makes use of the measured admittance in
In the main this expectation is borne out by the data. the red curve of Figure 5, and it is the crucial first test of
There are a few red points lying below the black curve, whether the synthesis method can give plausible banjo
especially in the frequency range around 1 kHz, but these sounds. This can be compared with sounds based on other
may be associated with an aspect of the physics not taken measured admittances from this banjo, in Sounds C.1–C.8
into account in this simple description: each string mode on the web site. These include cases with and without the
can occur in two different polarisations. The description resonator back, cases measured at different points on the
here, and the basis for the calculation of the black points, bridge, and cases in which the regular bridge was substi-
considers only the string polarisation normal to the head. tuted for a rigid circular bridge. (This circular bridge is pic-
Vibration in the plane parallel to the head is likely to couple tured on the web site; it will be used in the companion
much less strongly to the head, and thus exhibit lower loss paper [2] as a validation case for modelling.)
factors. The real plucks during the test procedure will In the opinion of the authors, these cases all “sound like
involve a mixture of both polarisations, this being the expla- banjos”, although they sound different from each other in
nation of the beating behaviour noted in Figure 2b. Perhaps detail. This contrasts with Sounds C.9 and C.10, which
a few peaks associated with the second polarisation have use the measured admittances of two different guitars,
been caught by the analysis. Simple experiments using and Sound C.11, which uses the admittance of a violin.
the two-polarisation synthesis method confirm that this The difference is striking. It seems unlikely that any listener
kind of effect can indeed occur, but it would require more would mistake any of these sounds for a banjo.
careful and systematic efforts to attempt a quantitative Sounds E.1–E.4 illustrate the use of a two-polarisation
validation comparison. This lies beyond the scope of the synthesis model, using a measured 2 2 admittance matrix.
present work. Three different values of the plucking angle h = 0°, 45°, 90°
There are several consequences for the behaviour of are illustrated. The last of these sounds the most distinctive,
notes played on the banjo. For frequencies up to about which is not surprising given that it will be dominated by
1 kHz, the string modes often have significantly higher the horizontal bridge admittance plotted in black in
damping than for the same string attached to a guitar. Figure 7. Sound E.4 also exhibits a phenomenon to be
The decay time will be faster, and at some frequencies it will discussed shortly: there is an unrealistic “zinginess” accom-
be so fast that the distinction between string modes and panying the sound.
body modes is lost: this is flagged in the plot by the black Sounds E.5–E.7 relate to the modal synthesis approach.
points reaching levels comparable with the line of body They give an opportunity to hear the separate sounds of the
modes. Any note with a fundamental or low overtone lying string and body modes making up the full syn-
in one of these frequency ranges might be perceived by a thesised sound. This comparison gives a useful insight. On
player as “falling flat”: it will not ring on as much as usual. a casual listening, the string-only sound is quite similar to
Secondly, over most of the frequency range plotted here the the full synthesis with all modes, which is indistinguishable
string modes follow the black points quite closely. This from the frequency-domain datum sound presented above.
means that most of the energy put into the string is lost However, the difference is clearly audible, if hard to put into
by being transferred to the body, whereas in a guitar most words.
of it is dissipated by other loss mechanisms. Furthermore, it Figure 12 shows spectrograms of the synthesised
will be shown in the companion paper [2] that radiation passage for the string modes and the body modes sepa-
damping for many modes of the banjo head dominates over rately. They reveal that it is hardly surprising that some dif-
structural damping, so a significant proportion of this ference can be heard when the body modes are included.
energy from the strings is radiated as sound. Body modes are strongly excited over a frequency range
The result of these two factors taken together is that a extended up to about 2 kHz, and although they generally
note played on the banjo with the same player gesture as a have much faster decay times than the string modes, typical
note on the guitar will sound louder, and decay faster. banjo music like the passage used here has notes that come
No player is likely to quarrel with that description, and it thick and fast, and the body modes ring on for long enough
is at least part of the essence of “banjoness”. But there are to bridge the gap to the next played note. They surely make12 J. Woodhouse et al.: Acta Acustica 2021, 5, 15
0 The model used is a deliberately crude one, in order to
2 explore the hypothesis that anything with modal density
1.8 -10 and sound radiation behaviour that is membrane-like rather
than plate-like can “sound like a banjo”. So rather than
1.6
-20
using the detailed FE/BE model mentioned earlier, these
1.4 sound examples are based on a rectangular membrane:
the details of the model development and parameter selec-
Time (s)
1.2
-30
1
tion are described in the companion paper [2]. The datum
case for this model appears here and as Sound B.2
0.8
-40 on the web site, to be compared directly with Sound B.1
0.6 described earlier, based on measured admittance.
0.4 -50 The various sound examples illustrate the effect of
0.2 varying the size and tension of the banjo head, and also
the mass of the bridge and the added stiffness due to axial
0 -60
0 0.5 1 1.5 2 2.5 3 effects in the strings, in-plane effects in the head and string
Frequency (kHz) tension. The details are described on the web site. An
0 important conclusion emerges, illustrated by the admit-
2 tance plots in Figure 13. Adjustments that affect individual
1.8 -10 resonances but which do not change the low-frequency for-
mant, like the examples shown in Figure 13a, have a rela-
1.6
-20
tively subtle effect on the sound, whereas adjustments
1.4 that change this formant, like those shown in Figure 13b,
have a more pronounced effect. Here are the sounds corre-
Time (s)
1.2
-30
1 sponding to the blue and black curves in Fig-
ure 13a, and here are the sounds corresponding to the
0.8
-40 red and black curves in Figure 13b. Examples
0.6
like these provide the basis of the claim made earlier that
0.4 -50 this banjo formant appears to have higher perceptual sal-
0.2 ience than individual modes. This hypothesis deserves to
0 -60
be tested in a formal psychoacoustical study.
0 0.5 1 1.5 2 2.5 3 The final set of demonstrations, Sounds H.1–H.5, illus-
Frequency (kHz)
trate the influence of the formant near 3.5 kHz, seen in
Figure 12. Spectrograms to illustrate modal synthesis: (a) Figure 7. Weighted mixtures of the measured admittance
string modes only; (b) body modes only. and the simulated admittance from the rectangular model
were computed, and the crossover frequency between them
was varied so that the formant could either be excluded or
a significant contribution to the “punctuation” at the start included.
of each note, and it seems likely that this effect forms a Certain of the sound examples exhibit a phenomenon
significant part of characteristic banjo sound. that strikes many listeners as unrealistic. The effect was
Note that this body mode contribution to the sound first noticed in the context of synthesis using the model
gives a possible mechanism for recognising a particular based on rectangular membrane geometry, but it also
instrument, to some extent independent of what music is occurs in some cases based on measured admittance:
played. The mix of body modes is similar for every note, Sounds C.5 and E.4 give clear examples. These cases are
and constitutes a kind of acoustical fingerprint of the instru- all characterised by a sense of “something ringing on too
ment. A similar mechanism has been suggested for distin- long” in an unrealistic way, and the effect apparently arises
guishing different violins by their transient sound [40]. from damping that is too low in the mid-kHz range. Details
Fréour et al. [31] have studied the audibility of comparable of the problem and the measures taken to minimise its
body sounds in guitar notes, with the conclusion that the effects by adjustments to damping models are given on
effect is confined to frequencies below about 1 kHz and is the web site [3].
strongest for the low modes below 300 Hz, in keeping There is an intriguing possibility that deserves to be
with the impression given by Figure 2b. For the banjo, mentioned here. The effect is very persistent, so perhaps
Figures 2a and 12 suggest that the effect may extend over it is a real phenomenon, but one that is not obvious from
a wider frequency range, but this has yet to be formally normal banjo playing. It should be recalled that what is
tested. computed by the synthesis algorithm is not the radiated
The next set of sound demonstrations, Sounds F.1–F.10 sound, but the motion of the body at the bridge. Different
and G.1–G.13, are computed on a different basis. Rather modes of the banjo head have very different levels of radia-
than using measured admittance, they use a theoretical tion damping (see for example Fig. 7 of the companion
model in order to give access to parametric variations. paper [2]). Modes with low radiation damping, and thusYou can also read
