The motion of the floating bridge of the banjo, in conjunction with the break angle of the strings over that bridge, produces string tension modulation that is first order in the amplitude of the string motion.  This note refines a previous suggestion regarding the impact on the frequencies of the strings’ and bridge’s motion.  For a given mode frequency pair of string and bridge, the resulting tension modulation produces a new, additional motion characterized by the sum and difference of the original ones.Strictly speaking, this corresponds to canonical “frequency modulation” only in the limit of modulation slow compared to the string frequency.  The more general result is precisely an example of what is known as “parametric oscillation,” first analyzed by Rayleigh.  The qualitative impact of tension modulation on banjo timbre remains as suggested previously.  It is only the precise math and physics that warrants this correction

Politzer, David

English

arXiv

Caltech Authors - Main

HDP: 20 – 01
Banjo Break Angle Tension Modulation as Parametric Oscillation
David Politzer∗
(Dated: January 9, 2020)
The motion of the floating bridge of the banjo, in conjunction with the break angle
of the strings over that bridge, produces string tension modulation that is first order
in the amplitude of the string motion. This note refines a previous suggestion regard-
ing the impact on the frequencies of the strings’ and bridge’s motion. For a given
mode frequency pair of string and bridge, the resulting tension modulation produces
a new, additional motion characterized by the sum and difference of the original ones.
Strictly speaking, this corresponds to canonical “frequency modulation” only in the
limit of modulation slow compared to the string frequency. The more general result
is precisely an example of what is known as “parametric oscillation,” first analyzed
by Rayleigh. The qualitative impact of tension modulation on banjo timbre remains
as suggested previously. It is only the precise math and physics that warrants this
correction.
∗ politzer@theory.caltech.edu; http://www.its.caltech.edu/˜politzer; Pasadena CA 91125
2
Banjo Break Angle Tension Modulation as Parametric Oscillation
BACKGROUND
As they pass over the bridge, the strings of the banjo form a “break” angle — typically
somewhere between 3o and 16o — which is determined by the height of the bridge and the
geometry of the tailpiece, as illustrated in FIG. 1.
bridge
neck
string
rim
tailpiecehead
to nut
break angle
FIG. 1
It is long-known to players and builders that increasing that angle enhances the “ring”
and “clang” of the instrument. A direct demonstration of that phenomenon was presented
in ref. [1], which contains sound samples of a particular instrument where the break angle
can be either zero or 13o. Note that zero degrees requires a specially designed bridge to
allow the strings to function normally without buzzing, as shown on page 1. Links to the
recordings themselves are also given in this note’s appendix.
For a given length and linear density, the fundamental pitch frequency of a string is
proportional to the square-root of the tension. It was an obvious leap to imagine that
tension modulation produced frequency modulation. The sound of audio range frequency
modulation has a rich and wonderful history since its 1973 discovery by John Chowning,
the licensing to Yamaha to become Stanford University’s second highest earning patent,
the production of the Yamaha DX7, and all that followed.[2] Chowning described his first
examples as sounding metallic.
While the string stretch associated with small vibrations and fixed ends is second order in
the amplitude, a non-zero break angle produces stretching that is first order in the amplitude
when the bridge moves. A natural assumption was that this phenomenon produced the sound
of Chowning’s frequency modulation.[3]
But frequency modulation is usually understood to have a precise mathematical meaning.
An initially sinusoidal oscillation in time is modulated in frequency at some other sinusoidal
3
frequency and amplitude. When the modulation is weak in amplitude and much lower in
frequency, its principal effect is to add side bands at the sum and difference frequencies.
(This is the basis of FM radio transmission.)
An explicit analysis of the basic equations of motion given below shows that the physical
mechanism of tension modulation literally produces frequency modulation only in the limit
that the modulation frequency is much less than the initial string frequency. In the case
of the banjo, there are certainly frequency components of the tension modulations that are
comparable and higher than some of the initial string frequencies. And then a purely sinu-
soidal tension modulation does not translate into a purely sinusoidal frequency modulation.
What does in fact happen is an example of what has come to be called parametric oscil-
lation, which was first analyzed by Rayleigh.[4] For a given modal frequency pair of string
and small amplitude bridge motion, the result is new, additional string and bridge motions
characterized by the sum and difference frequencies of the original ones.
TENSION MODULATION IN THE STRING EQUATION OF MOTION
The zeroth order string motion can be expressed as a sum over the ideal normal modes.
It could be just one of them or plucked triangular traveling waves or any other possible free
motion of the string. The tension modulation (again, assumed to be simultaneous along the
length of the string) is a function of time. It need not have the periodicity of the string or
even be periodic. It is the net result of the forces of all modes of all the strings on the bridge
and head. In any case, it has a Fourier transform. Even if there is only a single string’s
motion determining the bridge motion, in Fourier space we generally have to consider terms
with string and bridge having different frequencies.
In the following, a dimensionless parameter b is the relative amplitude of the tension
modulation and is assumed to be small. As shown in ref. [3], b depends linearly on the
string amplitude, A. I work to lowest non-trivial order in b because keeping anything higher
than b1 would be inconsistent — such things having been ignored in the ideal string equation
in the first place.
So (as they say) without loss of generality, we can consider a single ideal string mode, with
frequency ωo and wave number k, as the zeroth order string motion, and a single frequency
of tension modulation ω. In particular, let the tension T (t) be To(1 + b cos ωt). Let ψ(z, t)
be the string transverse displacement. For small amplitudes, it satisfies
4
ρψ̈ = T (t)ψ′′ .
And let ψ = ψo + bφ , with both ψo and φ vanishing at both ends. Take
ψo = A cos ωot sin kz .
Even before trying to find φ, the zeroth order ψo exerts anO(b) vertical force on the bridge
of bTocos ωt
∂
∂z
ψo = bToAk cos ωt cos ωot =
1
2
bToAk(cos ω+t+ ω−t), where ω± ≡ ωo ± ω .
To find the equation for φ(z, t) to lowest order in b: Let vo ≡ (To/ρ)1/2 = ωo/k . Keeping
only the terms linear in b, you get
φ̈ = v2oφ
′′ + v2o cos ωt ψ
′′
o .
Let φ(z, t) = f(t) sin kz. Then
f̈(t) = −v2ok2f(t)− v2ok2A cos ωt cos ωot = −ω2of(t)− 12ω
2
oA(cos ω+t+ cos ω−t) .
So f(t) is simply a forced SHO. You add the particular solutions for the two forcing fre-
quencies. In particular, with
f(t) = f+(t) + f−(t) ,
f±(t) =
ω2oA
2(ω2o−ω2±)
cos ω±t =
ω2oA
2ω(∓2ωo−ω) cos ω±t .
The general solution for f(t) also includes oscillation at ωo, independent of ω. That part
should be absorbed in ψo(t). The pole at ω = 0 is just a permanent retuning of the
string. The pole in f−(t) at ω = 2ωo is the standard parametric oscillator resonance. In
particular, tension modulation at ω ≈ 2ωo drives the string mode with cos ω−t ≈ cos ωot,
i.e., approximately at the original mode frequency.
The resulting vertical force on the bridge due to φ(z, t) is bTokf(t). Adding this to
the O(b) force on the bridge due to ψo and T (t), the total O(b) force on the bridge is
1
2
TobA{(1− 1ω
ωo
(2+ ω
ωo
)
)cos(ωo + ω)t+ (1 +
1
ω
ωo
(2− ω
ωo
)
)cos(ωo − ω)t} .
DECAY TIMES & RESONANCE WIDTHS
A is the amplitude of a plucked string mode. So it decays in time more or less (or exactly)
like an exponential. The modulation amplitude b also decays similarly, although potentially
with a somewhat different decay time. (Strictly speaking, the decays can be the sum of
more than one exponential and also may exhibit beats.) The tension-modulation-induced
amplitude decays like bA, i.e., faster than either separately. Hence, the tension modulation
effect appears in the attack or early part of the pluck sound.
These decay times are also reflected in the behavior near the resonance in f±(t). The
actual behavior will have a finite maximum and a related resonance width. This could
5
be exhibited explicitly by including damping in the equations of motion. But it is also a
necessary consequence of the fact that the resonance amplitude can only build up to a finite
value in a finite amount of time.
THE PROBLEM WITH FREQUENCY MODULATION
The starting equation, ρψ̈ = T (t)ψ′′ = To(1 + b cos ωt)ψ
′′, suggests that the whole effect
might simply be viewed as a modulation of vo, i.e., vo → v(t) = vo(1 + 12b cos ωt) or a
modulation of the flow of time, e.g., t→ τ(t) = t− b
2ω
sin ωt. However, any such ansatz for
the wave ψ does not satisfy the basic equation of motion. The discrepancy is O( ω
ωo
b), i.e.,
it goes away if the modulation is slow compared to the original frequency.
In terms of traveling waves, tension modulation changes the speed of travel but not the
wave shapes in space only if ω  ωo. More generally, tension modulation changes the wave
shapes as well.
CONCLUSION
The implication of all this is that the general motion of the banjo floating bridge with
a non-zero break angle can introduce components into the early part of the sound that are
not the harmonics of the plucked string. Note that even for frequencies already present in
the zeroth order string motion, additional sounds at those same frequencies will alter the
timbre. I would welcome any suggestion of an alternative explanation for the differences in
sound that are presented in the Appendix, below. A proper analysis would require a model
of the total bridge motion and a double sum over string and bridge motion modes. The low
frequency string modes generally produce the largest bridge motion (b), so it’s expected that
significant contributions come from ω  ωo and this gives the sound of canonical frequency
modulation.
APPENDIX
Here are links to the sounds of the 0o and 13o break angle configurations. The banjo is a
year 2000 Deering Goodtime, played with a solid disk back spaced 1/4′′ off the rim. The head
was tensioned to 91 on a DrumDial. For two tunes, each played in the two configurations:
click here for sample A,
click here for sample B,
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click here for sample C, and
click here for sample D,
or go to http://www.its.caltech.edu/~politzer/zero-break/A.mp3 and then switch A to B,
C, and D. If you can’t hear which is which, this exercise is pretty bootless.
[1] D. Politzer, Banjo Ring from Stretching String: A Zero Break Angle Demo, the MARCH 2019
entry at Banjo Physics 411, i.e., http://www.its.caltech.edu/~politzer.
[2] J. Chowning, The Synthesis of Complex Audio Spectra by Means of Frequency Modulation, J.
Audio Eng. Soc. 21 (7), 526 (1973). And some Yamaha PR: history of FM synthesizers at
https://usa.yamaha.com/products/contents/music production/synth 40th/history/chapter02/index.html
[3] D. Politzer, Banjo timbre from string stretching and frequency modulation, Acta Acustica
(united with Acustica), AAuA 101(1) 1, January 2015. That’s a brief, formal published ver-
sion. Far easier to access and replete with sound samples and more complete explanations is the
APRIL 2014 entry at http://www.its.caltech.edu/~politzer — or directly as String Stretching,
Frequency Modulation, and Banjo Clang.
[4] Lord Rayleigh, On Maintained Vibrations, Phil.Mag.S.5, 15, No.94, 231 (1883).


Banjo Break Angle Tension Modulation as Parametric Oscillation

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