Coupled Modes and Time-Domain Simulations of a Twelve-String Guitar with
a Movable Bridge
Miguel Marques ?, José Antunes ?, Vincent Debut ?
? Applied Dynamics Laboratory, Campus Tecnológico e Nuclear, Instituto Superior Técnico
Universidade Técnica de Lisboa, Estrada Nacional 10, 2695-066 Bobadela, Portugal
and
Mechanical Engineering Department, Instituto Superior Técnico
Universidade Técnica de Lisboa, Av Rovisco Pais 1, 1049-001 Lisboa. Portugal
[email protected], [email protected], [email protected]
ABSTRACT
Coupling between the different vibrating sub-systems of a
musical instrument is an important feature in music acoustics. It is the reason why instruments of similar families
have such different and characteristic sounds. In this work,
we propose a model for a twelve strings (six pairs) guitar,
such that the strings are coupled with the instrument body
through the moving bridge, which is the relevant component for energy transmission from the strings to the guitar
body and back. In this preliminary study, the guitar body is
modelled as a simple plate, strings being assumed to display planar-only vertical motions. However, the coupled
equations thus obtained can be readily extended to cope
with real guitar body modes and orbital string motions. After obtaining the coupled modes of the instrument, we illustrate the instrument time-domain coupled dynamics, by
considering the characteristic modal frequencies typical of
a Portuguese guitar. In particular we show how, when only
one string alone is plucked, energy is transmitted to all
other strings, causing sympathetic vibrations, which contribute to give this guitar its own characteristic sound identity.
1. INTRODUCTION
The acoustic guitar has been the subject of many studies,
both on its theoretical description and on the experimental side. Yet, there still are limitations when it comes to
the description of real guitars with movable bridges, i.e.
when the bridge role is not simplified as a simple enforced
pinned boundary condition. Overcoming this limitation,
could surely help improving the guitar engineering, e.g. it
could allow us to optimise the bridge position in order to
achieve the best radiative acoustic response, without loss
of the archtop guitar sound characteristics.
Motivated by the wish to extend our understanding of
the acoustics of the Portuguese guitar, we set ourselves to
c
Copyright: 2013
Miguel Marques et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution 3.0 Unported License, which
permits unrestricted use, distribution, and reproduction in any medium, provided
the original author and source are credited.
model the dynamics of an archtop twelve strings acoustic guitar. For such purpose, we built a conceptual model
for a set of twelve (six pairs of) strings coupling, at some
point of the strings (where the bridge would be located),
to a body, through the bridge, which will be modelled as a
spring with a very large stiffness. In this paper, the body
will be simplified as a thin wood plate. Other coupling
techniques have already been proposed in the literature of
musical acoustics. Specifically for guitars, different models, which do not take into account the vibrations of the
dead side of the string, have already been proposed [1–4],
but to our knowledge, the sympathetic vibrations of a guitar with a movable bridge, displaying a dead side of the
string after the bridge, has never been addressed nor modelled. A model of coupling between one string and the
violin body, which has the same features of our model, has
been proposed by some of the present authors, in order to
address the problem of the wolf note and the nonlinear behaviour of the string/bow interaction [5].
The Portuguese guitar (Cithara lusitanica) is a pear-shaped
instrument with twelve metal strings (six courses), descendant from the renaissance european cittern. This instrument is widely used in Portuguese traditional music, mainly
in Fado, and more recently also started to play a considerable role among urban Portuguese musicians. Unlike most
common guitars, this guitar has a bent soundboard (arched
top) with a bridge somewhat similar, although smaller in
size, to the bridge of a violin, a neck typically with 22 fixed
metal frets and it is tuned by a fan-shaped tuning mechanism, consisting in twelve screws, acting as pegs, mounted
with small gliding pins where the strings are attached to
adjust its tension. It has the typical tuning of the European cittern tradition, and has kept an old plucking technique, described in sixteenth century music books. The
first courses are composed by plain steel strings and tuned
in unison, and the remaining are combinations of a plain
steel string and an overspun copper on steel string tuned
one octave bellow. There are two different models of the
Portuguese guitar: the Lisbon guitar and the Coimbra guitar, named after two Portuguese cities where the two most
important Fado styles emerged. They differ in some details, such as the body measurements, the string length (the
where j = 1, ..., J is related to the number of nodal lines
along the xx direction, l = 1, ..., L is the number of nodal
lines along the yy direction, mjl are the modal masses,
ωjl are the modal angular frequencies, ζjl are the modal
damping coefficients, and Fjl are the generalized forces.
To simplify the formulation, these equations can be organised in order to have a single modal index g, with a
total of G = J × L modes, by ordering the frequencies
in an ascendent form. The modal masses of the body are
L L
mg = ρ x4 y , with ρ being the wood surface density, and
Lx , Ly the plate dimensions. Each string will have its own
set of modal dynamical equations, given as
mn q̈ns (t) + 2mn ωn ζn q̇ns (t) + mn ωn2 qns (t) = Fns (t), (3)
Figure 1. The master musician Carlos Paredes, playing the
Coimbra Portuguese Guitar. Photo credits: Egidio Santos.
Coimbra guitar has a larger arm), and the Lisbon guitar is
tuned (each string) a whole tone above the Coimbra guitar.
Strings used in these two guitars are therefore different in
size and linear mass. More about the Portuguese guitar can
be read in [6, 7] and about experimental vibrational measurements of this instrument in [8]. In figure 1, a Coimbra
model of the Portuguese guitar is shown as an example.
2. GUITAR MODEL
Throughout this section, we will give a full description of
the computational model proposed in this paper. For simplicity, in this conceptual model, we will replace the guitar
body by a thin wood rectangular soundboard plate. Although it would be easy to implement in the model, we
will neglect any energy transfer occurring through the nut
or the fingerboard, as there are experimental studies revealing that this effect is not much relevant, and we will also
assume the energy transfer occurring at the tailpiece to be
negligible, due to its location on the Portuguese guitar geometry. Therefore, our fully coupled system will consist
in a rectangular plate with fixed ends, and twelve strings,
attached to the plate trough a set of springs, displayed in
six pairs. For simplicity, we will only consider string vibrations in the direction normal to the soundboard plate,
although it would be relatively straightforward to implement the other direction as well. We will decompose the
strings and the plate displacements respectively as
Ys (xs , t) =
∞
X
ψns (xs )qn (t),
n
Zb (xb , yb , t) =
∞
X
b
ψj,l
(xb , yb )qj,l (t),
(1)
j,l
with the script s = 1, ..., S, for a total of S strings; where
Ys is the displacement for each string, Zb is the displacement of the body, ψ stand for the modeshapes and q stand
for the modal amplitudes. In this modal formulation, the
plate dynamics is governed by
b
b
2 b
mjl q̈jl
(t) + 2mjl ωjl ζjl q̇jl
(t) + mjl ωjl
qjl (t) = Fjlb (t),
(2)
where n = 1, ..., Ns are the modes of each string s. The
modal masses of each string are mn = µ L2s , with µ being
the linear density of the string, and Ls the size of the string
(throughout this work, we will assume that all strings have
the same size, which is the case for the Portuguese guitar).
Furthermore, the string and the body will have modeshapes
respectively given by
nπxs
s
,
ψn (xs ) = sin
Ls
jπxb
lπyb
b
ψjl (xb , yb ) = sin
sin
,
(4)
Lx
Ly
and eigenfrequencies (assuming no inharmonicity) respectively given by
ωns
b
ωjl
= nω0s ,
s
= hb
Ey
12ρ 1 − vp2
"
jπ
Lx
2
+
lπ
Ly
2 #
, (5)
where ω0s is the s string fundamental frequency, hb is the
plate height, Ey is the Young bulk modulus, and vp is the
the Poisson coefficient. Throughout the paper, we will reb
b
(xb , yb ) as ψgb (r), assuming a
as ωgb , and to ψjl
fer to ωjl
proper indexation mapping g → (j, l), according to what
has been previously stated. Given a general force F (x, t),
acting on a surface S of a vibrating system with modeshapes ψm (x), the corresponding forces on the modal space
will be
Z
Fm (t) =
F (x, t)ψm (x)dS.
(6)
S
Let us define Fcs (xc , t) and Fcb (rsc , t) as being the force exerted on the bridge, as seen, respectively, from each string
s and the from body at each coupling point rsc (we are assuming that all strings will meet the bridge at the same distance from the nut, xc ). We will consider that the bridge
his thin enough such that it can be modelled as a spring
with very large stiffness. Therefore, these forces are
Fcs (xc , t) = Kc [δ(xs − xc )Ys (xc , t) + δ(r − rsc )Z(rsc , t)]
h
i
+Cc δ(xs − xc )Ẏs (xc , t) − δ(r − rsc )Ż(rsc , t)
8
9
Ns
<X
=
s
s
s
= δ(xs − xc )
ψm
(xc ) [Kc qm
(t) + Cc q̇m
(t)]
:
;
m=1
(
+δ(r − rsc )
G
X
h
i
b
b
b
ψm
(rsc ) Kc qm
(t) + Cc q̇m
(t)
)
m=1
(7)
where Kc , Cc are, respectively, the (very large) stiffness
constant and the damping constant of the bridge, and
Fcb (rsc , t) = −Fcs (xc , t), ∀s ∈ [1, S]
given the matrices
[Φsc ]
(8)
=
The modal projections forces (7) are given by
N
X
Fns (t) =
s−b Φc
=
s
s
s
[Kc qm
(t) + Cc q̇m
(t)] ψm
(xc )ψns (xc )
m=1
G
X
b s s
b
b
−
Kc qm
(t) + Cbr q̇m
(t) ψm
(rc )ψn (xc ),
s−b Φc
=
m=1
Fgb (t) =
S
X
s
−
G
X
b s b s
b
b
Kc qm
(t) + Cc q̇m
(t) ψm
(rc )ψg (rc )
Ns
S X
X
s
b
Φc =
m=1
s
s
s
[Kc qm
(t) + Cc q̇m
(t)] ψm
(xc )ψgb (rsc ).
m=1
(9)
We can rewrite this equations as the linear system of ODEs
The eigenvalues of (12) are related to the modal frequencies and modal damping coefficients as
λ̄k = −ω̄k ζ̄k ± iω̄k
[M]{Q̈(t)} + [C]{Q̇(t)} + [K]{Q(t)} = {F(t)}, (10)
with the matrices and vectors built from the model parameters from all strings and the body
=
diag(ms1 , . . . , msN , mb1 , . . . , mbG ),
[C]
=
s s
b b
2 · diag(ms1 ω1s ζ1s , . . . , msN ωN
ζN , mb1 ω1b ζ1b , . . . , mbG ωG
ζG ),
[K]
=
{Q(t)}
=
{F (t)}
=
s2
b2
diag(ms1 ω1s 2 , . . . , msN ωN
, mb1 ω1b 2 , . . . , mbG ωG
),
n
oT
s
s
b
b
q1 (t), . . . , qN (t), q1 (t), . . . , qG (t) ,
n
oT
s
b
F1s (t), . . . , FN
(t), F1b (t), . . . , FG
(t) .
(11)
Considering the eigensolutions qk (t) = q̄k eλ̄k t , we can
transform the system (10) in an eigenvalue problem, becoming
[I]
−[M]−1 [C̄]
–
«
− λ̄k [I]
q̄k
λ̄k q̄k
ff
= {0},
(12)
where [K̄] and [C̄] are the effective stiffness and damping
matrices, given by [K̄] = [K] + Kc [Φc ] and [C̄] = [C] +
Cc [Φc ], and [Φc ] is the coupling matrix,




[Φc ] = 



[Φs1
c ]
[0]
[0]
[Φs2
c ]
[0]
..
.
[0]
..
.
[0]
[0]
..
.
b−s1 Φc
[Φb−s2
] ...
c
(15)
ω̄d k = =(λ̄k ),
(16)
<(λ̄k )
ζ̄k = − λ̄ ,
k
(17)
ψ̄k (x)
=
Ns
S X
X
s
k
s
q̄m
Θ((s − 1)Ls < x < sLs )ψm
(x)
m=1
G
X
k
b
q̄m
Θ(x > SLs )ψm
(r),
(18)
m=1
2.1 Frequency Domain Analysis
[0]
−[M]−1 [K̄]
1 − ζ̄k2 .
and
+
„»
q
From this relationship, we can deduce the guitar modal
damped frequencies, modal damping coefficients and modeshapes to be
[M]
Notice that, each string will have its own, independent,
modal family Ns , and therefore, the dimension
PS of the matrices and vectors in (12) will be D = G + s Ns .
 s


 ψ1 (xc ) 
 ..
. ψ1s (xc ) . . . ψns (xc ) ,
.


 s

ψn (xc )
 s


 ψ1 (xc ) 
 ..
−
. ψ1b (rsc ) . . . ψgb (rsc ) ,
.


 s

ψn (xc )
 b s 

 ψ1 (rc ) 
 ..
−
. ψ1s (xc ) . . . ψns (xc ) ,
.

 b s 

ψg (rc )
 b s 
S 
 ψ1 (rc ) 
 X
..
. ψ1b (rsc ) . . . ψgb (rsc ) .
.



s=1 
ψgb (rsc )
(14)
[Φs1−b
]
c
[Φs2−b
]
c
..
.
..
..
.
.
...
[Φbc ]
...
...




,



(13)
where Θ(α) is an Heaviside-like step function, such that
it is equal to one when the argument α is true, and zero
otherwise (notice that no subsystems are being summed),
k
q̄m
is the mth term of the k th eigenvector, and x is defined
as
xs if Θ ((s − 1)Ls < x < sLs ) = 1
x=
(19)
r if Θ (x > SLs ) = 1
In short, the coupled system eigenfunctions (18) correspond
to the modeshapes of the S coupled strings and the modeshapes of the coupled soundboard, while (19) corresponds
to the coordinate of each string when dealing with the coupled strings modeshapes, and corresponds to the coordinates of the soundboard when dealing with the coupled
soundboard modeshapes. As for the original decoupled
modeshapes, we normalized (18) such that the maximum
amplitude of its absolute value is one.
2.2 Time Domain Analysis
3. RESULTS
2.2.1 Initial Conditions
Throughout this section, we will consider simulations using the typical values of a Lisbon Portuguese guitar. All
We will assume that the initially, all strings are static, havstrings have a length of 44 cm from the nut to the bridge,
ing zero initial displacements as well as zero initial velociand 17.5 cm from the bridge to the stop tailpiece (total
ties, and there will rather be an initial excitation force actlength is 61.5 cm). Table 1 contains the strings notes, the
ing on one or more strings (i.e. a finger or a nail plucking
corresponding frequencies according to the standard tunthe string, or a pick striking the string) during a short initial
ing of the Lisbon Portuguese guitar, and the correspondtime interval. We will model the excitation force in such a
ing linear masses. Notice that the frequencies shown in
way that its modal projection (over the excited string modal
table 1 represent the fundamental frequency at which the
space) will be
active part of the string, i.e. the length between the nut
n
h
io
ˆ
˜
s
(xf ), and the bridge, should be vibrating; the actual frequency at
Fnf (t) = Kf Zf (t) − Ys (xf , t) + Cf Żf (t) − Ẏs (xd , t) ψn
(20)
which the full-length string is tuned will be given by mul44
where Kf , Cf are respectively the finger/nail/pick stiff. Based
tiplying the presented value by the factor 44+17.5
ness constant (here taken to be very large) and damping
on the average results obtained in experimental identificonstant, Zf (t) is the finger/nail/pick displacement, and
cations, all strings will have equal damping coefficients
Ys (xf , t) is the string displacement at the point where it is
ζns = 0.05% ∀n ∈ Ns . Each string s will have a total numplucked/struck. This displacements are, respectively,
ber of degrees of freedom Ns such that the frequency of the
highest
mode will be fNs ∼ 10kHz. As for the body, we
Zmax tmax
Zf (t) = Żf t =
t|tmin ,
(21)
tmax
st
nd
rd
th
th
th
and
Ys (xf , t) =
Ns
X
s
s
qm
(t)ψm
(xf ).
(22)
m=1
This choice allow us to easily perform simulations in which
different strings are plucked/struck at different times.
2.2.2 Time-step integrating procedure
We start by reducing the ODEs system (10) to a system
of first order ODEs, based on the unconstrained subsystem
modal responses
Q(t)
Q̇(t)
[0]
[I]
=
−[M]−1 [K̄] −[M]−1 [C̄]
Q̇(t)
Q̈(t)
0
+
.
(23)
Ff (t)
We will refer to (23) as
ṗ(t) = A p(t) + Fe (t).
The analytical solution of (23) is
Z t
p(t) = p(t0 )eA(t−t0 ) + eAt
e−Aτ Fe (τ )dτ.
(24)
(25)
strings pair
notes
frequency
(Hz)
linear density
(10−4 kg/m)
number of
modes used
1
b4 b4
493.88
493.88
3.78
3.78
28
28
2
a4 a4
440
440
3.94
3.94
32
32
3
e4 e4
329.63
329.63
6.20
6.20
42
42
4
b4 b3
493.88
246.94
3.78
14.48
28
57
5
a4 a3
440
220
3.94
21.22
32
64
6
d4 d3
293.66
146.83
11.30
35.36
27
95
Table 1. Notes and corresponding frequencies of the Lisbon Portuguese guitar standard tuning.
will assume the soundboard plate to be squared with 30 cm
by side, surface density ρ = 0.5 kg/m2 , ζgb = 1% ∀g ∈
G, and we adjust the parameters Ey , vp and hb such that
the first body frequency will be f1b = 275 Hz (which is
one of the values measured in [8] for a Portuguese guitar
body); also, we have considered 36 modes, such that the
maximum body frequency is ∼ 10KHz. The total number of modes considered for the system is therefore 543.
The strings of each pair will be separated by a distance
of 4mm, and each pair will be separated by a distance of
8mm (these are the typical values chosen by portuguese
guitar luthiers).
t0
Assuming a very short time step ∆t = t − t0 → 0, it
becomes possible to approximate Fe (t) as being constant
during each ∆t. We can then discretize equation (25), obtaining the numerical solution
pti+1 = pti eA∆t + A−1 eA∆t − I Feti ,
(26)
where I is the identity matrix. Notice that at each step
t
ti , one must recompute the vector Fe i−1 , using the results
ti−1
p
, according to what as previously stated in (20). This
method is quite stable (for the considered external forces
Fe (t)) if given an accurate calculation of the state transition matrix eA∆t . As to accuracy, this method provide
highly accurate results up to the time step bound ∆t ≤
Tmin /10, where Tmin is the smallest period of the system
(the period of the highest considered mode).
3.1 Frequency Domain Results
In Figure 2, we show the modeshape of the first fully coupled system mode, which is dominated by the 11th string,
the lowest tuned frequency of the system. Except for the
11th string, all other strings will have qualitatively equivalent vibrations to that of the 1st string in this system mode,
as detailed in the figure. In Figure 3, we show the modeshape of the fifteenth system mode, which is dominated by
the 1st string. In this mode, we observe significant vibrations in all strings tuned to b4 or b3. We find that the first
pair of strings has a rather different phase than the b4 string
of the fourth pair, and on the other hand, the b3 string of
the fourth pair, which frequency corresponds to the second
mode of the b4 strings, share the same phase as the first
pair, and has an amplitude considerably smaller than that
Figure 2. First mode of the coupled system. The red dots
represent the bridge position. Upper plot, string modal responses; medium plot, detail of the 1st string modal response; lower plot, detail of the body modal response.
Figure 3. Fifteenth mode of the coupled system. The red
dots represent the bridge position. Upper plot, string modal
responses; medium plot, detail of the 11th string modal
response; lower plot, detail of the body modal response.
of the b4 strings. The eleventh string (which is the d3 string
of the sixth pair), is interestingly vibrating in its fifth mode
with a very low amplitude; the fourth mode of this string
has a frequency of about 525Hz, which is almost 30Hz
higher than the fifteenth system mode. We also notice that
there are small discrepancies between the frequency of the
system modes and the frequency of the dominating strings.
This suggests that the coupling of the different subsystems,
as well as the dead side of the strings, adds some inhamornicity to the system, which would account for the audible
differences between different musical instruments.
Although typically twelve-string guitar players would pluck
a string pair rather than a single string, we will focus in
analysing the most simple possible scenarios. The string
excitation lasts for 0.01s, and the simulation will last for
3s; however, we find that in the first scenario, almost all
energy of the system has been damped after 1.5 seconds,
which is remarkably similar to what we typically hear when
plucking a portuguese guitar string. In Figures 5 and 6, we
show the plots of the energy evolution in each string, in the
body, and the total energy of the system, respectively for
the scenarios (i) and (ii). In Figure 4, we show the average energy of each subsystem, relative to the average total
energy of the coupled system.
As expected, in the first scenario, the first string is the
subsystem which has the most significant amount of energy, but interestingly, its energy will practically vanish
after the first 0.6 seconds, while the coupled system still
3.2 Time Domain Results
We performed time domain simulations of the situations in
which: (i) the musician will only excite the first (b4) string,
(ii) the musician will only excite the eleventh (d3) string.
has a fair amount of energy. Shortly after 0.2 seconds have
passed, the energy of the second, seventh and eight strings
(all the strings tuned to the same fundamental frequency
b4, and the string tuned to the second harmonic b3) will be
comparable in magnitude with the energy of the first string,
while the total energy of the system will have a larger magnitude from this moment on. Particularly the second string,
at the time in which it achieves its maximum energy (when
close to 0.3 seconds), has almost the same energy as the
excited string, which is not surprising given that they are
tuned to the same frequency and they are very closely located at the bridge. The body displays an energy profile
which reveals that there is energy being transferred back
and forth between itself and the twelve strings. All the
strings of the system will display some amount of energy,
even if they have been tuned in frequencies rather lower
than that of the excited strings. This will give this guitar its
own distinguishable sound.
In the second scenario, there is not a very efficient transfer of energy from the excited string to the other subsystems (comparing with the first scenario), and the only subsystems which will receive a relevant amount of energy
are the soundboard, the d4 string (which is within the same
pair, and tuned to the second harmonic of the excited string),
and the strings of the ”neighbour” pair(a3, a4). Based in
our observations, the most efficient transfer of energy occurs when exciting strings with more than one subsystem
tuned to a multiple frequency, due to the fact that this subsystems will experience sympathetic vibrations.
5. REFERENCES
[1] Woodhouse, “On the Synthesis of Guitar Plucks”, in
Acta Acustica United With Acustica Vol. 90, 2004, pp.
928–944.
[2] G. Derveaux, A. Chaigne, P. Joly, and E. Bécache,
“Time-domain simulation of a guitar: Model and
method”, in J. Acoust. Soc. Am. Vol. 114, No.6, 2003,
pp. 3368–3383.
[3] E. Bécache, G. Derveaux, A. Chaigne, and P. Joly,
“Numerical Simulation of a Guitar”, in Computers
Structures Vol. 83, 2005, pp. 107–126.
[4] A. Nackaerts, B. Moor, and R. Lauwereins, “Coupled
string guitar models”, in Proceedings of the WSES International Conference on Acoustics and Music: Theory and Applications 2001, Skiathos, 2001, p.119.
[5] O. Inácio, J. Antunes, and M.C.M. Wright, “Computational modelling of string-body interaction for the violin family and simulation of wolf notes”, in J. Of Sound
And Vibration Vol. 310, 2008, pp. 260–286.
[6] P. C. Cabral, A Guitarra Portuguesa. Ediclube, 1999.
[7] ——, The History of The Guitarra Portuguesa, (chapter in The Lute In Europe 2). Menziken, 2011.
[8] O. Inácio, F. Santiago, and P. C. Cabral, “The
portuguese guitar acoustics: Part 1 - vibroacoustic
measurements”, in Proc. of the 4th ibero-American
Congress Acústica, Guimarães, 2004, pp. 14–17.
4. CONCLUSIONS
We have developed a conceptual model to accurately perform, in both frequency and time domain, analysis of twelvestring guitars as a fully coupled system. The formulation
considered is sufficiently versatile to be also applicable to
model any other plucked string instrument, regardless of
the number of strings or the geometry in which they are
disposed on the instrument body. In this preliminary analysis, we have found results which corroborate that the body,
through the bridge, will account for a significant part of the
energy transmission across the multiple subsystems. We
also show the relevance of the ”dead side” of the string
(i.e. the continuation of the string after the bridge) for the
wave propagation across the string. We stress that this results might be a significant contribution for works on the
optimisation of guitar characteristics and radiation.
Acknowledgments
The authors warmly thank Professors Pedro Serrão, António
Relógio Ribeiro and Octávio Inácio for interesting discussions on modelling issues, as well as Pedro Caldeira Cabral
for very important discussions and valuable information on
guitar building and performance issues. This work was
supported by FCT - Fundação para Ciência e Técnologia
- Portugal under the project PTDC/FIS/103306/2008.
Figure 4. Average Energy of each subsystem; upper plot:
excitation of the first (b4) string, lower plot: excitation of
the eleventh (d3) string.
Figure 5. Energy plots for an excitation of the first (b4)
string.
Figure 6. Energy plots for an excitation of the eleventh
(d3) string.