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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∗

Abstract

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

∗Email: gvarieschi@lmu.edu; cgower@lion.lmu.edu

1

Contents

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

I. INTRODUCTION

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

numbers.

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

2

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

3

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

above.

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.

II. GEOMETRICAL MODEL OF A FRETTED STRING

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]:

νn=n

2LsT

µ,(1)

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=pT/µ is the wave velocity.

In the equal-tempered musical scale an octave is divided into twelve semitones, mathe-

matically:

4

νi=ν02i

12 ≃ν0(1.05946)i,(2)

where ν0and νiare respectively the frequencies of the ﬁrst note in the octave and of the

i−th 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:

Li=L02−i

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

sub-lengths.

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 1.0−1.5mm above the ﬁngerboard level.

The positioning of the frets follows Eq. (3), which we rewrite in terms of our new variable

X:

Xi=X02−i

12 ≃X0(0.943874)i≃X017

18i

,(4)

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.

5

X

X0

X1

X2

X3

X12

X19 .

0...

... X5X7.

fret positions

fingerboard

bridge

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.

6

fingerboard

frets

ii-1

L0

Li

a)

b)

saddle

nut

c

b

fingerboard

XiXi-1

a

a

hi

di

figi

li1

li2 li3

li4

X

X0

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 i−1. In part b) we show the details of our deformation model, in terms of the four

diﬀerent sub-lengths li1−li4of 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 i−th fret, thus producing the i−th 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

7

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:

L0=qX2

0+c2.(5)

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 i−1, thus reducing

the vibrating portion of the string to the part between the saddle and the i−th 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 i−th 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:

Li=li1+li2+li3+li4,(6)

where these four sub-lengths can be evaluated from the geometrical parameters as follows:

8

li1=rX02−i

12 2+ (b+c)2(7)

li2=his1 + f2

i

4h2

i

+f2

i

4hi

ln 2hi

fi1 + q1 + f2

i/4h2

i

li3=his1 + g2

i

4h2

i

+g2

i

4hi

ln 2hi

gi1 + q1 + g2

i/4h2

i

li4=rX2

01−2−i−1

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:

di=fi+gi=Xi−1−Xi=X02−i

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:

Qi=Li−L0

L0

(9)

and they can also be computed for any fret iusing Eqs. (5) - (8).

III. COMPENSATION MODEL

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

9

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):

νn≃n

2LsT

ρS "1 + 2

LrESk2

T+4 + n2π2

2ESk2

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

π2

T L2

ES k2≈369; 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 L≃1

3L0≃21.5cm. The approximation in Eq. (10) is certainly valid for our

strings, for at least n.19.

10

νn≃n

2LsT

ρS 1 + 2 β

L+αn

β2

L2.(11)

We then consider just the fundamental tone (n= 1) as being the frequency of the sound

perceived by the human ear4:

ν1≃1

2LsT

ρS 1 + 2 β

L+αβ2

L2,(12)

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 i−th 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 i−th note can be expressed by combining together Eqs.

(2) and (12):

νi=ν02i

12 ≃1

2L0sT(L0)

ρS(L0)"1 + 2 β(L0)

L0

+α[β(L0)]2

L2

0#2i

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].

11

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:

νi≃1

2l′

i1sT(Li)

ρS(Li)"1 + 2 β(Li)

l′

i1

+α[β(Li)]2

l′2

i1#,(14)

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:

1

2L0sT(L0)

ρS(L0)"1 + 2 β(L0)

L0

+α[β(L0)]2

L2

0#2i

12 =1

2l′

i1sT(Li)

ρS(Li)"1 + 2 β(Li)

l′

i1

+α[β(Li)]2

l′2

i1#.

(15)

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′

i1:

l′

i1≃li1

1 + h1 + 2β(L0)

li1+α[β(L0)]2

l2

i1i−1

[1+Qi(1+R)] h1 + 2β(L0)

L0+α[β(L0)]2

L2

0i

h1 + 4β(L0)

li1+3α[β(L0)]2

l2

i1i

.(16)

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):

R=dν

dLL0

L0

ν0

,(17)

i.e., the frequency change dν 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

X′

i=pl′2

i1−(b+c)2≃l′

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.

12

li1(see Fig. 2) so that the same Eq. (16) can be used to determine the new fret positions

from the old ones:

X′

i≃Xi

1 + h1 + 2β(L0)

li1+α[β(L0)]2

l2

i1i−1

[1+Qi(1+R)] h1 + 2β(L0)

L0+α[β(L0)]2

L2

0i

h1 + 4β(L0)

li1+3α[β(L0)]2

l2

i1i

.(18)

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

{X′

i}i=1,2,... to a canonical fret position equation (similar to the original Eq. (4)) of the form:

X′

i=X′

02−i

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′

NU T =X′

0+ ∆Nand combining together the last two equations, we

obtain the deﬁnition of the “nut adjustment” ∆Nas:

∆N=X0−(X′

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.

13

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

follows:

∆Sresc =X0

X′

0

∆S(21)

∆Nresc =X0

X′

0

∆N.

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.

14

IV. EXPERIMENTAL MEASUREMENTS

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

change.

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.

15

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

16

# 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 .

17

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

instrument.

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.

18

String 1

Fret

number

Note

Perfect

intonation

Frequency

(Hz)

Without

compensation

Frequency

(Hz)

Without

compensation

Freq. deviation

(cents)

With

compensation

Frequency

(Hz)

With

compensation

Freq. deviation

(cents)

0C3130.813 130.813 0 130.813 0

1C#

3138.591 143.832 64.3 137.958 −7.9

2D3146.832 150.551 43.3 147.323 5.8

3D#

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

6F#

3184.997 188.754 34.8 184.763 −2.2

7G3195.998 200.386 38.3 195.878 −1.1

8G#

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.

19

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.

20

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.

strings.

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

21

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.

22

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.

V. CONCLUSIONS

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

23

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.

Acknowledgments

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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25

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26