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Intonation and Compensation of Fretted String Instruments


Abstract and Figures

In this paper we present mathematical and physical models to be used in the analysis of the problem of intonation of musical instruments such as guitars, mandolins and the like, i.e., we study how to improve the tuning on these instruments. This analysis begins by designing the placement of frets on the fingerboard according to mathematical rules and the assumption of an ideal string, but becomes more complicated when one includes the effects of deformation of the string and inharmonicity due to other string characteristics. As a consequence of these factors, perfect intonation of all the notes on the instrument can never be achieved, but complex compensation procedures are introduced and studied to minimize the problem. To test the validity of these compensation procedures, we have performed extensive measurements using standard monochord sonometers and other basic acoustical devices, confirming the correctness of our theoretical models. In particular, these experimental activities can be easily integrated into standard acoustics courses and labs, and can become a more advanced version of basic experiments with monochords and sonometers. Comment: Improved version, with minor changes. 25 pages, including 6 figures and 2 tables. Accepted for publication in the American Journal of Physics (AJP)
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arXiv:0906.0127v2 [physics.class-ph] 5 Sep 2009
Intonation and Compensation of Fretted String Instruments
Gabriele U. Varieschi and Christina M. Gower
Department of Physics, Loyola Marymount University - Los Angeles, CA 90045, USA
In this paper we present mathematical and physical models to be used in the analysis of the
problem of intonation of musical instruments such as guitars, mandolins and the like, i.e., we study
how to improve the tuning on these instruments.
This analysis begins by designing the placement of frets on the fingerboard according to math-
ematical rules and the assumption of an ideal string, but becomes more complicated when one
includes the effects of deformation of the string and inharmonicity due to other string characteris-
tics. As a consequence of these factors, perfect intonation of all the notes on the instrument can
never be achieved, but complex compensation procedures are introduced and studied to minimize
the problem.
To test the validity of these compensation procedures, we have performed extensive measure-
ments using standard monochord sonometers and other basic acoustical devices, confirming the
correctness of our theoretical models. In particular, these experimental activities can be easily
integrated into standard acoustics courses and labs, and can become a more advanced version of
basic experiments with monochords and sonometers.
PACS numbers: 01.55.+b; 01.50.Pa; 43.75.Bc; 43.75.Gh
Keywords: acoustics, musical acoustics, guitar, intonation, compensation
I. Introduction 2
II. Geometrical model of a fretted string 4
A. Fret placement on the fingerboard 4
B. Deformation model of a fretted string 7
III. Compensation model 9
A. Vibrations of a stiff string 9
B. Compensation at nut and saddle 11
IV. Experimental measurements 15
A. Description of the apparatus 15
B. String properties and experimental results 17
V. Conclusions 23
Acknowledgments 24
References 24
The physics of musical instruments is a very interesting sub-field of acoustics, which
connects the mathematical models of vibrations and waves to the world of art and musical
performance. This connection between science and music has always been present, since
the origin of art and civilization. Classic books on the field are for example [1], [2], [3],
[4]. In the 6th century B.C., the mathematician and philosopher Pythagoras was fascinated
by music and by the intervals between musical tones. He was probably the first to perform
experimental studies of the pitches of musical instruments and relate them to ratios of integer
This idea was the origin of the diatonic scale, which dominated much of western music,
and also of the so-called just intonation system which was used for many centuries to tune
musical instruments, based on perfect ratios of whole numbers. Eventually, this system was
abandoned in favor of a more mathematically refined method for intonation and tuning, the
well known equal temperament system, which was introduced by scholars such as Vincenzo
Galilei (Galileo’s father), Marin Mersenne and Simon Stevin, in the 16th and 17th centuries,
and also strongly advocated by musicians such as the great J. S. Bach. In the equal-tempered
scale, the interval of one octave is divided into 12 equal sub-intervals (semitones), achieving a
more uniform intonation of musical instruments, especially when using all the 24 major and
minor keys, as in Bach’s masterpiece, the “Well Tempered Clavier.” Historical discussion
and complete reviews of the different intonation systems can be found in Refs. [5], [6], [7].
Mathematically, the twelve-tone equal temperament system requires the use of irrational
numbers, since for example the ratio of the frequencies of two adjacent notes corresponds
to 12
2. On a fretted string instrument like a guitar, lute, mandolin, or similar, this into-
nation system is accomplished by placing the frets along the fingerboard according to these
mathematical ratios. Unfortunately, even with the most accurate fret placement, perfect
instrument tuning is never achieved. This is due mainly to the mechanical action of the
player’s fingers, which need to press the strings down on the fingerboard while playing, thus
altering their length, tension and ultimately changing the frequency of the sound being pro-
duced. Other causes of imperfect intonation include inharmonicity of the strings, due to
their intrinsic stiffness and other more subtle effects. An introduction to all these effects can
be found in Refs. [8] and [9].
Experienced luthiers and guitar manufacturers usually correct for this effect by intro-
ducing the so-called compensation, i.e., they slightly increase the string length in order to
compensate for the increased sound frequency, resulting from the effects described above (see
instrument building techniques in [10], [11], [12], [13], [14]). Other solutions are reported in
luthiers’ websites ([15], [16], [17], [18], [19]) or in commercially patented devices ([20], [21],
[22]). These empirical solutions can be improved by studying the problem in a more scien-
tific fashion, through proper modeling of the string deformation and other effects, therefore
leading to a new type of fret placement which is more effective for the proper intonation of
the instrument.
Some mathematical studies of the problem appeared in specialized journals for luthiers
and guitar builders ([23], [24]), but they were particularly targeted to luthiers and manu-
facturers of a specific instrument (typically classical guitar). We are not aware of similar
scientific studies being reported in physics or acoustics publications. For example, in general
physics journals we found only basic studies on guitar intonation and strings (see [25], [26],
[27], [28], [29], [30], [31], [32], [33]), without any detailed analysis of the problem outlined
Therefore, our objective is to review and improve the existing mathematical models
of compensation for fretted string instruments and to perform experimental measures to
test these models. In particular, the experimental activities described in this paper were
performed using standard lab equipment (sonometers and other basic acoustic devices) in
view of the pedagogical goal of this project. In fact, all the experimental activities detailed in
this work can be easily introduced in standard sound and waves lab courses, as an interesting
variation of experiments usually performed with classic sonometers.
In the next section we will start by describing the geometry of the problem in terms of
a simple string deformation model. In Sect. III we will examine the theoretical basis for
the compensation model being used, and in Sect. IV we will describe the outcomes of our
experimental activities.
In this section we will introduce the geometrical model of a guitar fingerboard, review
the practical laws for fret placement and study the deformations of a “fretted” string, i.e.,
when the string is pressed onto the fingerboard by the mechanical action of the fingers.
A. Fret placement on the fingerboard
We start our analysis by recalling Mersenne’s law which describes the frequency νof
sound produced by a vibrating string [8], [9]:
where n= 1 refers to the fundamental frequency, while n= 2,3... to the overtones. Lis
the string length, Tis the tension, µis the linear mass density of the string (mass per unit
length), and v=pTis the wave velocity.
In the equal-tempered musical scale an octave is divided into twelve semitones, mathe-
12 ν0(1.05946)i,(2)
where ν0and νiare respectively the frequencies of the first note in the octave and of the
ith note (i= 1,2, ..., 12). For i= 12, we obtain a frequency which is double that of the
first note, as expected. Since Mersenne’s law states that the fundamental frequency of the
vibrating string is inversely proportional to the string length L, we simply combine Eqs. (1)
and (2) to determine the correct string lengths for all the different notes (i= 1,2,3, ...) as a
function of the original string length L0(open string length, producing the first note of the
octave considered), assuming that the tension Tand the mass density µare kept constant:
12 L0(0.943874)i.(3)
This equation can be immediately used to determine the fret placement on a guitar or a
similar instrument, since the frets essentially subdivide the string length into the required
In Figure 1 we show a picture of a classical guitar as a reference. The string length is the
distance between the saddle1and the nut, while the frets are placed on the fingerboard at
appropriate distances. We prefer to use the coordinate X, as illustrated in the same figure,
to denote the position of the frets, measured from the saddle toward the nut position. X0
will denote the position of the nut (the “zero” fret), while Xi,i= 1,2, ..., are the positions
of the frets of the instrument. On a classical guitar there are usually up to 19 20 frets on
the fingerboard and they are realized by inserting thin pieces of a special metal wire in the
fingerboard, so that the frets will rise about 1.01.5mm above the fingerboard level.
The positioning of the frets follows Eq. (3), which we rewrite in terms of our new variable
12 X0(0.943874)iX017
1The saddle is the white piece of plastic or other material located near the bridge, on which the strings are
resting. The strings are usually attached to the bridge, which is located on the left of the saddle. On other
type of guitars, or other fretted instruments, the strings are attached directly to the bridge (without using
any saddle). In this case the string length would be the distance between the bridge and the nut. Our
analysis would not be different in this case: the bridge position would simply replace the saddle position.
X19 .
... X5X7.
fret positions
saddle nut
Classical Guitar
FIG. 1: Illustration of a classical guitar showing our coordinate system, from the saddle toward
the nut, used to measure the fret positions on the fingerboard (guitar by Michael Peters - photo
by Trilogy Guitars, reproduced with permission).
where the last approximation in the previous equation is the one historically employed by
luthiers to practically locate the fret positions. This is usually called the “rule of 18,” which
requires placing the first fret at a distance from the nut corresponding to 1
18 of the string
length (or 17
18 from the saddle); then place the second fret at a distance from the first fret
corresponding to 1
18 of the remaining length between the first fret and the saddle, and so
on. Since 17
18 = 0.944444 0.943874, this empirical method is usually accurate enough for
practical fret placement2, although modern luthiers use fret placement templates based on
the decimal expression in Eq. (4).
2Following Eq. (4), frets number 5, 7, 12, and 19, are particularly important since they (approximately)
correspond to vibrating string lengths which are respectively 3/4, 2/3, 1/2, and 1/3 of the full length, in
line with the Pythagorean original theory of monochords.
li2 li3
FIG. 2: Geometrical deformation model of a guitar string. In part a) we show the original string
in black (of length L0) and the deformed string in red (of length Li) when it is pressed between
frets iand i1. In part b) we show the details of our deformation model, in terms of the four
different sub-lengths li1li4of the deformed string.
B. Deformation model of a fretted string
Figure 2 illustrates the geometrical model of a fretted string, i.e., when a player’s finger
or other device is pressing the string down to the fingerboard, until the string is resting on
the desired ith fret, thus producing the ith note when the string is plucked. In this
figure we use a notation similar to the one developed in Refs. [23], [24], but we will introduce
a different deformation model.
Figure 2a shows the general geometrical variables for a guitar string. The distance X0
between the saddle and the nut is also called the scale-length of the guitar (typically between
640 660 mm for a modern classical guitar) but this is not exactly the same as the real
string length L0, because saddle and nut usually have slightly different heights above the
fingerboard surface. The connection between L0and X0is simply:
The metal frets rise above the fingerboard by a distance aas shown in Figure 2. The
heights of the nut and saddle above the top of the frets are labeled in Figure 2 as band c,
respectively. All these heights are greatly exaggerated; they are usually small compared to
the string length. The standard fret positions are again denoted by Xiand, in particular,
we show the situation where the string is pressed between frets iand i1, thus reducing
the vibrating portion of the string to the part between the saddle and the ith fret.
Figure 2b shows the details of the deformation caused by the action of a finger between
the two frets. Previous works ([23], [24]) modeled this shape simply as a sort of “knife-edge”
deformation which is not quite comparable to the action of a fingertip. We improved on this
point by assuming a more “rounded” deformation, considering a curved shape as in Figure
2b. The action of the finger depresses the string behind the ith fret by an amount hi
below the fret level (not necessarily corresponding to the full height a) and at a distance fi,
compared to the distance dibetween consecutive frets.
In Sect. IV we will describe how to set all these parameters to the desired values with
our experimental device and simulate all possible deformations of the string. It is necessary
for our compensation model, described in the next section, to compute exactly the length
of the deformed string for any fret value i. As shown in Fig. 2, the deformed length Liof
the entire string is the sum of the lengths of the four different parts:
where these four sub-lengths can be evaluated from the geometrical parameters as follows:
12 2+ (b+c)2(7)
li2=his1 + f2
ln 2hi
fi1 + q1 + f2
li3=his1 + g2
ln 2hi
gi1 + q1 + g2
12 2+b2.
In Eq. (7) the sub-lengths li2and li3were obtained by using a simple parabolic shape
for the “rounded” deformation shown in Fig. 2b, due to the action of the player’s fingertip.
They were computed by integrating the length of the two parabolic arcs shown in Fig. 2b,
in terms of the distances fi,giand hi.
The distances between consecutive frets are calculated as:
12 21
12 1(8)
so that, given the values of X0,a,b,c,hiand fi, we can compute for any fret number ithe
values of all the other quantities and the deformed length Li. We will see in the next section
that the fundamental geometrical quantities of the compensation model are defined as:
and they can also be computed for any fret iusing Eqs. (5) - (8).
In this section we will describe the model used to compensate for the string deformation
and for the inharmonicity of a vibrating string, basing our analysis on the work done by G.
Byers ([16], [24]).
A. Vibrations of a stiff string
Strings used in musical instruments are not perfectly elastic, but possess a certain amount
of “stiffness” or inharmonicity which affects the frequency of the sound produced. Mersenne’s
law in Eq. (1) needs to be modified to include this property of real strings, yielding the
following result (see Ref. [34], chapter 4, section 16):
ρS "1 + 2
T+4 + n2π2
T L2#,(10)
where we have rewritten the linear mass density of the string as µ=ρS (ρis the string
density and Sthe cross section area). The correction terms inside the square brackets are
due to the string stiffness and related to the modulus of elasticity (or Young’s modulus)
Eand to the radius of gyration k(equal to the string radius divided by two, for a simple
unwound steel or nylon string). Following Ref. [34], we will use c.g.s. units in the rest of
the paper and in all computations, except when quoting some geometrical parameters for
which it will be more convenient to use millimeters.
The previous equation is an approximation valid for E Sk2
T L2<1
n2π2, a condition which
is usually met in practical situations3. When the stiffness factor ES k2
T L2is negligible, Eq.
(10) reduces to the original Eq. (1). On the contrary, when this factor increases and
becomes important, the allowed frequencies also increase, following the last equation, and
the overtones (n= 2,3, ...) increase in frequency more rapidly than the fundamental tone
(n= 1). The sound produced is no longer “harmonic” since the overtone frequencies are
no longer simple multiples of the fundamental one, as seen from Eq. (10). In addition,
the deformation of the fretted string, described in the previous section, will alter the string
length Land, as a consequence of this effect, will also change the tension Tand the cross
section Sin the last equation. These are the main causes of the intonation problem being
studied. Additional causes that we cannot address in this work are the imperfections of the
strings (non uniform cross section or density), the motion of the end supports (especially
the saddle and the bridge, transmitting the vibrations to the rest of the instrument) which
also changes the string length, the effects of friction, and others.
Following Byers [24] we define αn=4 + n2π2
2and β=qES k2
T, so that we can simplify
Eq. (10):
3The condition is equivalent to n2<1
T L2
ES k2369; 803; 5052, where the numerical values are related
to the three steel strings we will use in Sect. IV (see string properties in Table I) and for the shortest
possible vibrating length L1
3L021.5cm. The approximation in Eq. (10) is certainly valid for our
strings, for at least n.19.
ρS 1 + 2 β
We then consider just the fundamental tone (n= 1) as being the frequency of the sound
perceived by the human ear4:
ρS 1 + 2 β
where α=α1=4 + π2
2and βis defined as above. In Eq. (12) Lrepresents the vibrating
length of the string, which in our case is the length li1when the string is pressed down onto
the ith fret. To further complicate the problem, the quantities T,Sand βin Eq. (12)
depend on the actual total length of the string Li, as computed in Eq. (6). In other words,
we tune the open string, of original length L0, at the appropriate tension T, but when the
string is “fretted” its length is changed from L0to Li, thus slightly altering the tension, the
cross section, and also βwhich is a function of the previous two quantities. This is the origin
of the lack of intonation, common to all fretted instruments, which calls for an appropriate
compensation mechanism, which will be analyzed in the next section.
B. Compensation at nut and saddle
The proposed solution [24] to the intonation problem is to adjust the fret positions to
accommodate for the frequency changes described in the previous equation. The vibrating
lengths li1are recomputed as l
i1=li1+ ∆li1, where ∆li1represents a small adjustment in
the placement of the frets, so that the fundamental frequency from Eq. (12) will match the
ideal frequency of Eq. (2) and the fretted note will be in tune.
The ideal frequency νiof the ith note can be expressed by combining together Eqs.
(2) and (12):
12 1
ρS(L0)"1 + 2 β(L0)
12 ,(13)
4This statement is also an approximation since the pitch (or perceived frequency) is affected by the presence
of the overtones. See for example the discussion of the psychological characteristics of music in Olson [4].
where all the quantities on the right-hand side of the previous equation are related to the
open string length L0, since ν0is the frequency of the open string note. On the other hand,
we can write the same frequency νiusing Eq. (12) directly for the fretted note:
ρS(Li)"1 + 2 β(Li)
where now we use the “adjusted” vibrating length l
i1for the fretted note and all the other
quantities on the right-hand side of Eq. (14) depend on the fretted string length Li. By
comparing Eqs. (13) and (14) we obtain the master equation for our compensation model:
ρS(L0)"1 + 2 β(L0)
12 =1
ρS(Li)"1 + 2 β(Li)
We obtained an approximate solution5of the previous equation by Taylor expanding the
right-hand side in terms of ∆li1and by solving the resulting expression for the new vibrating
lengths l
1 + h1 + 2β(L0)
[1+Qi(1+R)] h1 + 2β(L0)
h1 + 4β(L0)
In this equation the quantities Qiare derived from Eq. (9) and from our new deformation
model described in Sect. II B, while an additional experimental quantity Ris introduced in
the previous equation and defined as (see Ref. [24] for details):
i.e., the frequency change relative to the original frequency ν0, induced by an infinitesimal
string length change dL, relative to the original string length L0. This quantity will be
measured in Sect. IV for the strings we used in this project.
The new vibrating lengths l
i1from Eq. (16) correspond to new fret positions X
i, since
i1for (b+c)l
i1. A similar relation also holds between Xiand
5Our solution in Eq. (16) differs from the similar solution obtained by Byers et al. (Eq. 17 in Ref. [24]).
We believe that this is due to a minor error in their computation, which yields only minimal changes in
the numerical results. Therefore, the compensation procedure used by G. Byers in his guitars is essentially
correct and practically very effective in improving the intonation of his instruments.
li1(see Fig. 2) so that the same Eq. (16) can be used to determine the new fret positions
from the old ones:
1 + h1 + 2β(L0)
[1+Qi(1+R)] h1 + 2β(L0)
h1 + 4β(L0)
At this point a luthier should position the frets on the fingerboard according to Eq. (18)
which is not anymore in the canonical form of the original Eq. (4). Moreover, each string
would get slightly different fret positions, since the physical properties such as tension, cross
section, etc., are different for the various strings of a musical instrument. Therefore, this
compensation solution would be very difficult to be implemented practically and would also
affect the playability of the instrument6.
An appropriate compromise, also introduced by Byers [24], is to fit the new fret positions
i}i=1,2,... to a canonical fret position equation (similar to the original Eq. (4)) of the form:
12 + ∆S(19)
where X
0is a new scale length for the string and ∆Sis the “saddle setback,” i.e., the
distance by which the saddle position should be shifted from its original position (usually
S > 0 and the saddle is moved away from the nut). The nut position is also shifted, but we
require to keep the string scale at the original value X0, therefore we need X
NU T +S=X0,
where X
NU T is the new nut position in the primed coordinates. Introducing the shift in the
nut position ∆Nas X
0+ ∆Nand combining together the last two equations, we
obtain the definition of the “nut adjustment” ∆Nas:
0+ ∆S).(20)
This is typically a negative quantity, indicating that the nut has to be moved slightly forward
toward the saddle.
Finally, instead of adopting a new scale length X
0, the luthier might want to keep the
same original scale length X0and keep the fret positions according to the original Eq. (4).
6Nevertheless some luthiers actually construct guitars where the individual frets under each string are
adjustable in position by moving them slightly along the fingerboard. Each note of the guitar is then indi-
vidually fine-tuned to achieve the desired intonation, requiring a very time consuming tuning procedure.
Since the corrections and the effects we described above are essentially all linear with respect
to the scale length adopted, it will be sufficient to rescale the nut and saddle adjustment as
Sresc =X0
Nresc =X0
This final rescaling is also practically needed on a guitar or other fretted instrument,
because the compensation procedure described above has to be carried out independently
on each string of the instrument, i.e., all the quantities in the equations of this sections
should be rewritten adding a string index j= 1,2, ..., 6, for the six guitar strings. Each
string would get a particular saddle and nut correction, but once these corrections are all
rescaled according to Eq. (21) the luthier can still set the frets according to the original
Eq. (4). The saddle and nut will be shaped in a way to incorporate all the saddle-nut
compensation adjustments for each string of the instrument (see [24], [16] for practical
illustrations of these techniques).
In practice, this compensation procedure does not change the original fret placement
and the scale length of the guitar, but requires very precise nut and saddle adjustments
for each of the strings of the instrument, using Eq. (21). Again, this is just a convenient
approximation of the full compensation procedure, which would require repositioning all
frets according to Eq. (18), but this would not be a very practical solution.
In the next section we will detail the experimental measures we performed following the
deformation and compensation models outlined above. Since all our measurements will be
carried out using a monochord apparatus, we will work essentially with a single string and
not a whole set of six strings, as in a real guitar. Therefore, we will use all the equations
outlined above without adding the additional string index j. However, it would be easy to
modify our discussion in order to extend the deformation-compensation model to the case
of a multi-string apparatus.
In Figure 3 we show the experimental setup we used for our measurements. Since our goal
was simply to test the physics involved in the intonation problem and not to build musical
instruments or improve their construction techniques, we used standard lab equipment, as
illustrated in the figure.
A. Description of the apparatus
A standard PASCO sonometer WA-9613 [35] was used as the main device for the exper-
iment. This apparatus includes a set of steel strings of known linear density and diameter,
and two adjustable bridges which can be used to simulate nut and saddle of a guitar. The
string tension can be measured by using the sonometer tensioning lever, or adjusted directly
with the string tensioning screw (on the left of the sonometer, as seen in Fig. 3). In par-
ticular, this adjustment allowed the direct measurement for each string of the Rparameter
in Eq. (17), by slightly stretching the string and measuring the corresponding frequency
On top of the sonometer we placed a piece of a classical guitar fingerboard with scale
length X0= 645 mm (visible in Fig. 3 as a thin black object with twenty metallic frets, glued
to a wooden board to raise it almost to the level of the string). The geometrical parameters
in Fig. 2 were set as follows: a= 1.3mm (fret thickness), b= 1.5mm,c= 0.0mm (since
we used two identical sonometer bridges as nut and saddle). This arrangement ensured that
the metal strings produced a good quality sound, without “buzzing” or producing undesired
noise, when the sonometer was “played” like a guitar by gently plucking the string. Also,
since we set c= 0, the open string length is equal to the scale length: L0=X0= 645 mm.
The mechanical action of the player’s finger pressing on the string was produced by using
a spring loaded device (also shown in Fig. 3, pressing between the sixth and seventh fret)
with a rounded end to obtain the deformation model illustrated in Fig 2b. Although we
tried different possible ways of pressing on the strings, for the measurements described in
this section we were always pressing halfway between the frets (fi=gi=1
2di) and all the
way down on the fingerboard (hi=a= 1.3mm). In this way, all the geometrical parameters
of Fig. 2 were defined and the fundamental quantities Qiof Eq. (9) could be computed.
FIG. 3: Our experimental apparatus is composed of a standard sonometer to which we added a
classical guitar fingerboard. Also shown is a mechanical device used to press the string on the
fingerboard and several different instruments used to measure sound frequencies.
The sound produced by the plucked string (which was easily audible, due to the resonant
body of the sonometer) was analyzed with different devices, in order to accurately measure
its frequency. At first we used the sonometer detector coil or a microphone, connected to
a digital oscilloscope, or alternatively to a computer through a digital signal interface, as
shown also in Fig. 3. All these devices could measure frequencies in an accurate way, but we
decided to use for most of our measurements a professional digital tuner [36], which could
discriminate frequencies at the level of ±0.1cents 7. This device is shown near the center
7The cent is a logarithmic unit of measure used for musical intervals. The octave is divided into twelve
semitones, each of which is subdivided in 100 cents, thus the octave is divided into 1200 cents. Since
# String 1 String 2 String 3
Open string note C3F3C4
Open string frequency (H z) 130.813 174.614 261.626
Radius (cm) 0.0254 0.0216 0.0127
Linear density µ(g/cm) 0.0150 0.0112 0.0039
Tension (dyne) 5.16 ×1065.88 ×1064.41 ×106
Young’s modulus E(dyne/cm2) 2.00 ×1012 2.00 ×1012 2.00 ×1012
R parameter 130 199 78.7
Rescaled saddle setback ∆Sresc (cm) 0.733 0.998 0.518
Rescaled nut adjustment ∆Nresc (cm)2.31 2.41 1.35
TABLE I: The physical characteristics and the compensation parameters for the three steel strings
used in our experimental tests are summarized here.
of Fig. 3, just behind the sonometer.
B. String properties and experimental results
For our experimental tests we chose three of the six steel guitar strings included with
the PASCO sonometer. Their physical characteristics and the compensation parameters are
described in Table I.
The open string notes and related frequencies were chosen so that the sound produced
using all the twenty frets of our fingerboard would span over 2-3 octaves, and the tensions
were set accordingly. We used a value for Young’s modulus which is typical of steel strings
and we measured the Rparameter in Eq. (17) as explained in the previous section. The
rescaled saddle setback ∆Sresc and the rescaled nut adjustment ∆Nresc from Eq. (21) were
computed for each string, using the procedure outlined in Sect. III and the geometrical and
physical parameters described above.
an octave corresponds to a frequency ratio of 2 : 1, one cent is precisely equal to an interval of 2 1
1200 .
Given two frequencies aand bof two different notes, the number nof cents between the notes is n=
1200 log2(a/b)3986 log10(a/b). Alternatively, given a note band the number nof cents in the interval,
the second note aof the interval is a=b×2n
1200 .
We then carefully measured the frequency of the sounds produced by pressing each string
onto the twenty frets of the fingerboard in the two possible modes: without any compensa-
tion, i.e., setting the frets according to Eq. (4), and with compensation, i.e., after shifting
the position of saddle and nut by the amounts specified in Table I and retuning the open
string to the original note.
Table II illustrates the frequency values for String 1, obtained in the two different modes
and compared to the theoretical values of the same notes for the case of a “perfect intonation”
of the instrument. The measurements were repeated several times and the quantities in Table
II represent average values.
In this table, fret number zero represents the open string being plucked, so there is no
difference in frequency for the three cases. On the contrary, for all the other frets, the
frequencies without compensation are considerably higher than the theoretical values for a
perfectly intonated instrument. This results in the pitch8of these notes to be perceived
being higher (or sharper) than the correct pitch. In fact, when we “played” our mono-
chord sonometer in this first situation, it sounded definitely out of tune. The frequency
values obtained instead by using our compensation correction appear to be much closer to
the theoretical values, thus effectively improving the overall intonation of our monochord
In Table II we also show the frequency deviation of each note from the theoretical value
of perfect intonation for both cases: with and without compensation. The frequency shifts
are expressed in cents (see definition in note at the end of Sect. IV A) rather than in Hertz,
since the former unit is more suitable to measure how the human ear perceives different
sounds to be in tune or out of tune. The frequency deviation values illustrate more clearly
the effectiveness of the compensation procedure: without compensation the deviation from
perfect intonation ranges between 14.3 and 64.3cents, with compensation this range is
reduced to values between 7.9 and +7.3cents.
In view of the previous discussion, we prefer to plot our results for String 1 in terms
of the frequency deviation of each note from the theoretical value of perfect intonation.
8We note that the frequency of the sound produced is the physical quantity we measured in our experiments.
The pitch is defined as a sensory characteristic arising out of frequency, but also affected by other subjective
factors which depend upon the individual. It is beyond the scope of this paper to consider these additional
subjective factors.
String 1
Freq. deviation
Freq. deviation
0C3130.813 130.813 0 130.813 0
3138.591 143.832 64.3 137.958 7.9
2D3146.832 150.551 43.3 147.323 5.8
3155.563 159.126 39.2 155.363 2.2
4E3164.814 168.407 37.3 164.070 7.8
5F3174.614 178.348 36.6 173.933 6.8
3184.997 188.754 34.8 184.763 2.2
7G3195.998 200.386 38.3 195.878 1.1
3207.652 212.105 36.7 207.632 0.2
9A3220.000 224.644 36.2 220.081 0.6
10 A#
3233.082 237.495 32.5 233.136 0.4
11 B3246.942 252.345 37.5 247.123 1.3
12 C4261.626 266.338 30.9 261.505 0.8
13 C#
4277.183 281.958 29.6 277.076 0.7
14 D4293.665 298.545 28.5 293.688 0.1
15 D#
4311.127 315.276 22.9 311.463 1.9
16 E4329.628 334.822 27.1 329.787 0.8
17 F4349.228 353.408 20.6 348.785 2.2
18 F#
4369.994 373.545 16.5 370.330 1.6
19 G4391.996 396.597 20.2 393.335 5.9
20 G#
4415.305 418.742 14.3 417.068 7.3
TABLE II: Frequency values of the different notes obtained with String 1: theoretical perfect
intonation values are compared to the experimental values with and without compensation. Also
shown are the frequency deviations (in cents) from the theoretical values, for both cases.
Fig. 4 shows these frequency deviations for each fret number (corresponding to the different
musical notes of Table II) in the two cases: without compensation (red circles) and with
compensation (blue triangles). Error bars are also included, coming from the computed
standard deviations of the measured frequency values.
We also show in the same figure the so-called pitch discrimination range (region between
green dashed lines), i.e., the difference in pitch which an individual can effectively detect
when hearing two different notes in rapid succession. In other words, notes within this range
will not be perceived as different in pitch by human ears. It can be easily seen in Fig. 4 that
all the (red) values without compensation are well outside the pitch discrimination range,
thus will be perceived as out of tune (in particular as sharper sounds). On the contrary, the
(blue) values with compensation are in general within the green dashed discrimination range
of about ±10 cents9. The compensation procedure has virtually made them equivalent to
the perfect intonation values (corresponding to the zero cent deviation - perfect intonation
level, black dotted line in Figure 4). We note again that fret number zero simply corresponds
to playing the open string note which is always perfectly tuned, therefore the experimental
points for this fret do not show any frequency deviation.
We repeated the same type of measurements also for String 2 and 3, which were tuned
at higher frequencies as open strings (respectively as F3and C4, see Table I). In this way
we obtained experimental sets of measured frequencies, with and without compensation, for
these two other strings, similar to those presented in Table I. For brevity, we will omit to
report all these numerical values, but we present in Figs. 5 and 6 the frequency deviation
plots, as we have done for String 1 in Fig. 4.
The results in Figs. 5 and 6 are very similar to those in the previous figure: the fre-
quencies without compensation are much higher than the perfect intonation level, while
the compensation procedure is able to reduce almost all the frequency values to the region
within the green dashed curves (the pitch discrimination range). Using again the procedure
outlined in Ref. [4], the discrimination ranges in Figs. 5 and 6 were computed respectively
as ±8.6cents and ±5.2cents, due to the different frequencies produced by these two other
9This discrimination range was estimated, for the frequenqies of String 1, according to the discussion in
Ref. [4], pages 248-252. In general, this range varies from about ±5cents -±10 cents for frequencies
between 1000 2000 Hz, to even larger values of ±40 cents -±50 cents at lower frequencies, between
60 120 Hz.
FIG. 4: Frequency deviation from perfect intonation level (black dotted line) for notes obtained
with String 1. Red circles denote results without compensation, while blue triangles denote results
with compensation. Also shown (region between green dashed lines) is the approximate pitch
discrimination range for frequencies related to this string.
For the three cases we analyzed, we can conclude that the compensation procedure de-
scribed in this paper is very effective in improving the intonation of each of the strings we
used. Although more work on the subject is needed (in particular we need to test also
nylon strings, which are more commonly used in classical guitars), we have proven that the
intonation problem of fretted string instruments can be analyzed and solved using physical
FIG. 5: Frequency deviation from perfect intonation level (black dotted line) for notes obtained
with String 2. Red circles denote results without compensation, while blue triangles denote results
with compensation. Also shown (region between green dashed lines) is the approximate pitch
discrimination range for frequencies related to this string.
and mathematical models, which are more reliable than the empirical methods developed
by luthiers during the historical development of these instruments.
FIG. 6: Frequency deviation from perfect intonation level (black dotted line) for notes obtained
with String 3. Red circles denote results without compensation, while blue triangles denote results
with compensation. Also shown (region between green dashed lines) is the approximate pitch
discrimination range for frequencies related to this string.
In this work we studied the mathematical models and the physics related to the problem
of intonation and compensation of fretted string instruments. While this problem is usually
solved in an empirical way by luthiers and instrument makers, we have shown that it is
possible to find a mathematical solution, improving the original work by G. Byers and
others, and that this procedure can be effectively implemented in practical situations.
We have demonstrated how to use simple lab equipment, such as standard sonometers
and frequency measurement devices, to study the sounds produced by plucked strings, when
they are pressed onto a guitar-like fingerboard, thus confirming the mathematical models
for intonation and compensation. These activities can also be easily presented in standard
musical acoustics courses, or used in sound and waves labs as an interesting variation of
experiments usually performed with classic sonometers.
This work was supported by a grant from the Frank R. Seaver College of Science and
Engineering, Loyola Marymount University. The authors would like to acknowledge useful
discussions with John Silva of Trilogy Guitars and with luthier Michael Peters, who also
helped us with the guitar fingerboard. We thank Jeff Cady for his technical support and
help with our experimental apparatus. We are also very thankful to luthier Dr. Gregory
Byers for sharing with us important details of his original study on the subject and other
suggestions. Finally, the authors gratefully acknowledge the anonymous reviewers for their
useful comments and suggestions.
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