Classical guitar intonation and compensation: The well-tempered guitar

First, the string is elevated above the frets by the saddle and nut, so the fretted string is slightly elongated relative to the free string, and the resulting frequency is flattened in pitch. In principle, this effect could be accommodated by minute changes in the positions of the frets, but there are additional practical complications. For example, the string's tension is increased by the change in length, causing the frequency to sharpen by an amount that significantly exceeds the reduction caused by the increase in the resonant string length. In addition, the string is by no means ideal, and its intrinsic stiffness results in an additional increase in pitch that depends on its mechanical characteristics. These guitar intonation difficulties seem to preclude successful temperament, but remarkably, the instrument can be compensated by moving the positions of the saddle and the nut by small distances during the manufacturing process (Byers, 1995, 1996; Varieschi and Gower, 2010). This compensation process helps temper the guitar so that it is playable. It must also be tunable, so that the guitar strings can be brought into compliance with 12-TET quickly and accurately. This requirement places significant constraints on the physical properties of manufactured strings. Our goal in this work is to build an intuitive understanding of these effects to aid in the compensation and subsequent tuning of the classical guitar, with an accessible approach to making measurements of string properties and deducing the corresponding effects of playability (Dostal, 2020; Erkut , 2000; Noll, 2014).

We present the basics of our model of classical guitar strings in Sec. II, following the pioneering work of G. Byers (Byers, 1995, 1996, 2020). We offer empirical reasons to doubt the need for a complicated model of string fretting that incorporates either the depth of fretting or the shape of the finger in  Appendix A. Then, in  Appendix B, we derive a new expression for the allowed vibration frequencies of a stiff string [although more complicated dynamical models are available (Ducceschi and Bilbao, 2016)], derived under the assumption that the boundary conditions at the saddle and the nut are not symmetric. Based on this result, we then discuss in detail the four contributions to frequency shifts and errors of strings pressed behind a fret: the change in the resonant length of the vibrating string, a decrease in the linear mass density of the entire string, an increase in the tension of the entire string, and an increase in the mechanical stiffness of the resonating string. Our goal is to simplify the decoupled equations describing these effects through Taylor series expansions to allow an intuitive picture of the string's behavior to emerge. Nylon strings (Blanc and Ravasoo, 1996; Lynch-Aird and Woodhouse, 2017, 2018) behave very differently than the metal strings used on acoustic and electric guitars (Grimes, 2014; Kemp, 2017, 2020). After restringing, they first stretch and deform inelastically and then require hours to “settle” and reach a linear elastic equilibrium, and it is unlikely that the uniform stiff rod approach used to develop equations for the resonant frequencies (including ours) apply to either the monofilament treble strings or the wound bass strings until that equilibrium is reached. Nevertheless, we are able to develop a phenomenological model of the mechanical characteristics of these strings that is consistent with measurements of frequency deviations on uncompensated classical guitars.

In our analysis, we neglect two contributions to the sound produced by classical guitars. First, we do not account for the impact of the plucked string polarization on the boundary conditions we used to describe the vibrations of a string. A standard free stroke on a nylon classical guitar string results in an elliptically polarized response, with the major (horizontal) axis in the plane of the strings parallel to the fretboard. In principle, the vertical and horizontal oscillations could vibrate at different frequencies, but we see no evidence of peak splitting in spectrum analysis of the first half-dozen harmonics of fretted normal tension strings on the several classical guitars we tested. Second, given this result, we neglected all details of guitar body manufacturing such as the bridge admittance (Torres and Boullosa, 2009; Woodhouse, 2004) because they are unlikely to affect computations of setbacks in guitar compensation.

In Sec. II E, we collect all four of the effects mentioned above and develop a simple approximate expression for the total frequency shift of a fretted guitar string. We note that the two largest contributions to this deviation from 12-TET perfection are the increases in tension and bending stiffness due to fretting and that small changes in the positions of the nut and the saddle can largely compensate for these problems. In Sec. III, we demonstrate a simple, straightforward experiment to measure in situ the response of a string's fundamental frequency to a change in length (and therefore tension), and we test and rely on a subsequent phenomenological approach to determining the open-string bending stiffness. Then, in Sec. IV, we use these values for a normal-tension nylon string set (as well as other string sets in  Appendix D) to demonstrate a straightforward analytic approach to compensating for the tone errors in a guitar string. We check this method numerically by relying on a technique—described in  Appendix C—to minimize the root-mean-square (RMS) frequency deviation at each fret. With these results in hand, in Sec. V we discuss a collaboration of guitar manufacturer and musician to temper the guitar using harmonic tuning to optimize it for a particular piece. Finally, in Sec. VI, we summarize our results and suggest topics for future work.

This document—as well as the python computer code needed to reproduce the figures—is available at GitHub (Anderson and Beausoleil, 2024).

Our model is based on the schematic of the guitar shown in Fig. 1. The scale length of the guitar is X₀, but we allow the inside edges of both the saddle and the nut to be set back an additional distance, $Δ S$ and $Δ N$⁠, respectively. The location on the x axis of the center of the nth fret is X_n. In the y direction, y = 0 is taken as the surface of the fingerboard; the height of each fret is a, the height of the nut (i.e., the distance between the fingerboard and the bottom of the string) is a + b, and the height of the saddle is a + b + c. (For the moment, we are neglecting the art of relief practiced by expert luthiers that increases the effective height of a string as the fret number grows. We discuss this effect below in Sec. II C.) L_n is the resonant length of the string from the saddle to the center of fret n, and $L n ′$ is the length of the string from the fret to the nut. The total length of the string is defined as $L n ≡ L n + L n ′$⁠. As discussed in more detail in  Appendix A, we have chosen to include a line-segment intersection at a distance, d, behind fret n to represent the slight increase in the distance $L n ′$ caused by a finger. This differs from previous studies of guitar intonation and compensation (Byers, 1995, 1996; Varieschi and Gower, 2010), but our approach is consistent with our empirical observations for nylon strings.

The final form of Eq. (11) makes it clear that—for nylon guitar strings—there are four contributions to intonation:

1. Resonant length: The first term represents the error caused by the increase in the length of the fretted string L_n compared to the ideal length X_n, which would be obtained if $b = c = d = 0$ and $Δ S = Δ N = 0$⁠.

2. Linear mass density: The second term is the error caused by the reduction of the linear mass density of the fretted string. This effect will depend on the total length of the string: $L n = L n + L n ′$⁠.

3. Tension: The third term is the error caused by the increase of the tension in the string arising from the stress and strain applied to the string by fretting. This effect will also depend on the total length of the string $L n$⁠.

4. Bending stiffness: The fourth and final term is the error caused by the change in the bending stiffness coefficient arising from the decrease in the vibrating length of the string from L₀ to L_n.

Note that the properties of the logarithm function have decoupled these physical effects by converting multiplication into addition. We will discuss each of these sources of error in turn below.

In the discussion that follows, we will test our approximations for a prototypical instrument (hereafter referred to as the “Classical Guitar”) with the specifications listed in Table I. Refer to Fig. 1 for a graphical representation of these parameters. In addition, as we develop models of the physical effects discussed above, we will assume that the guitar string has the properties listed in Table II. The string constant κ and the open-string bending stiffness B₀ are introduced in Sec. II C and Sec. II D, respectively. The linear frequency shift parameter R is discussed in Sec. II E, and a method for determining both κ and B₀ in terms of R is discussed.

Counterintuitively, nylon classical guitar strings (Blanc and Ravasoo, 1996; Lynch-Aird and Woodhouse, 2017, 2018) have very different physical properties than those of steel strings (Grimes, 2014; Kemp, 2017, 2020), with completely different stress-strain curves. When fresh nylon strings are brought up to the required tension for the first time, they are stretched by a macroscopic distance, $Δ L$⁠, that varies from 7 cm (for the first E₄ string) to 2 cm (for the sixth E₂ string). After only a few minutes, the string must be re-tensioned because it has undergone nonlinear viscoelastic relaxation and has become flat by at least a half-step. [This stage of tensioning and strain is not well described by theories of nonlinear elasticity in soft materials (Mihai and Goriely, 2017).] This process continues for several hours until the strings begin to “settle” and remain properly tuned for longer periods; after about 10 h, they will respond at the correct frequencies for more than an hour provided that the temperature in the room does not change significantly (Blanc and Ravasoo, 1996; Lynch-Aird and Woodhouse, 2017). A string removed from the guitar after this stage has been reached will not relax back to its original “out-of-the-box” length—it has been permanently deformed. The frequency of most settled nylon strings string can be “dropped” one whole step and then returned to the initial value, but attempts to increase tension further will reach a nonlinear stage where the frequency increases much less quickly and will often result in a broken string.

In Fig. 2, we plot a comparison between the exact and approximate expressions for the frequency error resulting from the tension increase given by Eqs. (24) and (25) for two values of d. The normalized displacement Q_n is computed using Eq. (17) in the exact curves and Eq. (19) in the approximate curves. Here, the guitar has the specifications listed in Table I: the exact curves use $Δ S = 1.8$ mm and $Δ N = − 0.38$ mm, and the approximate curves ignore the setbacks entirely. The slight difference between the exact and approximate shifts for d = 10 mm at the first fret can be eliminated if we include a term quadratic in d in Eq. (19). As predicted above, we see that the dependence of Q_n—and therefore the tension shift—on the setback values is minimal.

In Fig. 4, we compare the total frequency shifts predicted by Eqs. (11) and (13) for the Classical Guitar specified by Table I with $Δ S = Δ N = 0$ mm and a string with the parameters listed in Table II at two different values of d. Note that the string is sharp at every fret, but even a large nonzero value of d is only important at the first fret. The bending stiffness is negligible at the first fret, but accounts for 65% of the shift at the 12th fret. The close agreement between the exact and approximate expressions for the frequency shifts gives us confidence that the equations we derive for the setbacks in Sec. IV will be useful.

It is relatively easy to measure in situ the value of R (and therefore infer κ and B₀) for any guitar string with the aid of a device that can measure frequency (Larsson, 2020), a simple ruler with fine markings (e.g., a string depth gauge), a magnifying glass or camera with a macro mode, and white correction fluid. For example, in Fig. 5 we show photographs of the nylon normal-tension first string on an Alhambra 8 P classical guitar. By depositing a small sample of correction fluid on the string, we can measure small displacements against a gauge marked in half-millimeter increments. Then we can pluck the open string and measure its vibration frequency. All of our measurements were made with strings that had settled into equilibrium after at least 10 h of use, and we completed each set of measurements of a string in less than 10 min so that it did not have time to relax further (Blanc and Ravasoo, 1996; Lynch-Aird and Woodhouse, 2017). We found that significantly stretching a string that had settled into equilibrium resulted in a nonlinear frequency shift, $Δ f$⁠, as a function of $Δ L$ (and occasionally broke the string). Therefore, prior to our measurements we tuned each string down one whole step by turning the tuning machine down five half-turns, stretching the string vertically to pull it through the nut, and then re-tensioning the string with two half-turns. The string stretches uniformly along its length, so at any position x the relative displacement $Δ x / x$ should be invariant. For convenience, we therefore chose to work near the first fret as a visual marker, which is located 614 mm from the saddle on a guitar with a 650 mm scale length. We made seven measurements of displacement over a 3 mm range (100 times the stretch that results from normal fretting), as well as the corresponding frequencies.

For example, we began with a normal-tension nylon classical string set (D'Addario, 2020; Santos, 2020) with the specifications listed in Table III using metric units.¹ In Fig. 6, we plot our measurements of $Δ f$ as a function of the displacement $Δ x$ relative to the frequency of the string when $Δ x = 0$⁠. The error bars (which arise primarily because of imperfect measurements of $Δ x$⁠) represent the standard deviation of ten independent measurements. We then performed a least-squares fit to a straight line (Bevington and Robinson, 2003) (also shown in Fig. 6), determined the derivative $Δ f / Δ L$⁠, and then computed R using Eq. (34), with L = 614 mm and f defined as the average frequency over the range. The results are shown in Table IV. Here, $κ = 2 R + 1$⁠, we compute the open-string bending stiffness B₀ using Eq. (39), and we also estimate an effective (differential) modulus of elasticity, $E eff$⁠, from Eq. (23), expressed in units of gigapascals (1 GPa = 10⁹ N/m²). Similar measurements and results for other string sets are provided in  Appendix D. Note—as predicted in Sec. II E and shown in Fig. 7 for all string sets except the light-tension set—the expectation that the guitar will be tunable results in R values of manufactured strings that are in the range $20 – 30$⁠. (It is unclear why the light strings appear to have much higher R values. Their volume densities are within a few percent of those of the normal-tension strings, so perhaps there is a significant difference in the corresponding manufacturing process.) The still more important requirement that the guitar be playable leads us to the discussion of compensation in Sec. IV.

Therefore, adopting the physical properties of the normal string set listed in Table IV and applying them to a computation of the frequency deviations for our standard classical guitar, we obtain the predictions shown in Fig. 8 using Eq. (11). Anticipating our treatment of exact compensation in Sec. IV and  Appendix C, we determine the RMS average of the frequency deviations for each string. This mean (over the first 12 frets) can be computed by squaring the frequency deviations shown in Fig. 8, averaging those values, and then taking the square root of the result.

In Sec. II, we noted that the bending stiffness and the increase in string tension due to fretting sharpen the pitch, but that we can flatten it with a positive saddle setback and negative nut setback. In  Appendix C, we develop an RMS least-squares fit method that numerically solves Eq. (11) for the values of $Δ S$ and $Δ N$ that minimize the RMS of the frequency errors of a string over a particular set of frets. In the case of our Classical Guitar with normal-tension nylon strings—shown in Fig. 8 for the case of zero setbacks—we use this method with d = 0 to obtain the nonzero setbacks listed in Table V. The corresponding frequency deviations are shown in Fig. 9(a) (assuming that all other aspects of the guitar remain unchanged). Of course, manufacturing a guitar with unique saddle and nut setbacks for each string (of a particular tension) can be challenging, so we also plot in Fig. 9(b) the shifts obtained by setting each of the values of $Δ S$ and $Δ N$ to the mean of the corresponding column in Table V. In both cases, the RMS error is significantly smaller than that of the uncompensated guitar shown in Fig. 8.

This purely numerical approach using Eq. (11) is accurate but not illuminating. Let us build an intuitive understanding of how guitar compensation works and then calculate approximate formulas for $Δ S$ and $Δ N$ that will help us appreciate the impact of particular choices of (for example) b, c, and d. As in Fig. 4, let us choose the guitar and string properties listed in Tables I and II, vary $Δ S$ and $Δ N$⁠, and then use Eq. (11) to determine the pitch of the string at each fret. Figure 10 shows that increasing the saddle setback tends to rotate the pitch curve clockwise, and increasing the magnitude of the negative nut setback displaces the pitch curve almost uniformly downward. It appears that we can compensate the guitar by finding a value of $Δ S$ that results in values of $Δ ν n$ that are equal for, say, frets with $n ≥ 3$⁠, and then calculating a value of $Δ N$ that sets $Δ ν 12 = 0$⁠.

This relationship remains true for $0 ≤ d ≤ 10$ mm. Therefore, we can either compute the mean of the saddle setbacks directly or using the average value of the string radii (⁠ $ρ ¯ = 0.43 ± 0.08$ mm). Either way, we obtain $Δ S ¯ = 1.8 ± 0.4$ mm. Similarly, in Fig. 11(b), we show a histogram of the values of the nut setbacks and compute the mean value $Δ N = − 0.38 ± 0.03$ mm, which we recall is proportional to the scale length X₀. Note that these results are remarkably similar to the values used in Fig. 9(b); if we plot the frequency deviations of those five string sets with these particular mean setback values, then we find that the maximum error always occurs at the 12th fret, and it is always less than 1 cent. In Sec. V, we discuss a method to temper the guitar to reduce these errors further.

“Temperament: A compromise between the acoustic purity of theoretically exact intervals, and the harmonic discrepancies arising from their practical employment.”—Dr. Theo. Baker (Baker, 1895).

In Fig. 9(b), a uniformly compensated classical guitar with normal tension strings tuned to 12-TET shows (of the treble strings) the third string has the greatest error in tuning across the fretboard. Tuning this guitar to 12-TET exacts a perfect-fifth in the third string while playing a C major chord in first position. This results in the third string being too sharp for the other common chords of E major (G#), A major, and D major (A), particularly when the guitar is played at a higher fret position. One way to reduce this error is by lowering the pitch of the third string below 12-TET with an electronic tuner. Another more comprehensive system is to tune all the strings harmonically to the fifth string, which lowers the third string by 7 cents as well as tempering the remaining strings.

In this particular case, the “harmonic tuning method” can be followed using these steps:

1. Begin by tuning the fifth string to A₂ = 110 Hz, resulting in a fifth-fret harmonic of A₄ = 440 Hz. (This can also be tuned by ear using an A₄ tuning fork.)

2. Tune that harmonic to the seventh fret harmonic of the fourth string, which is also A₄ = 440 Hz.

3. Tune the seventh fret harmonic on the fifth string (330 Hz, or 0.37 Hz sharper than 12-TET E₄) to the fifth-fret harmonic of the sixth string.

4. The seventh-fret harmonic on the fifth string can tune the remaining fretted strings: the ninth fret on the third string, the fifth fret on the second string, and the open first string.

We have summarized these steps in Table VI, and in Fig. 12 we show the same guitar tuned in this fashion. Although the RMS shift over all strings is similar to that obtained by 12-TET tuning, the reduction in errors by strings 2 and 3 on the second and higher frets is significant. Note that other tuning choices can be made depending on the piece being played. For example, the third string could also be tuned at the second fret to A₃ = 220 Hz using the fifth-string harmonic at the 12th fret, and/or the first string could be tuned at the fifth fret to A₄ using the fifth-fret harmonic of the fifth string. The flexibility of the harmonic tuning method—and its reliance on only an A₄ tuning fork—is a great asset for the classical guitarist. Of course, how the guitar string is plucked has an impact on the resulting tone, but we defer a discussion of this effect to the literature (Laurson , 2001; Migneco and Kim, 2011; Woodhouse, 2004).

In this work, we have constructed a model of classical guitar intonation that includes the effects of the resonant length of the fretted string, linear mass density, tension, and bending stiffness. We have described a simple experimental approach to estimating the increase in string tension arising from an increase in its length and then the corresponding mechanical stiffness. This allows us to determine the saddle and nut positions needed to compensate the guitar for a particular string, and we propose a simple approach to find averages of these positions to accommodate a variety of strings. This “mean” method benefits further from temperament techniques—such as harmonic tuning—that can enhance the intonation of the classical guitar for particular musical pieces.

Our calculations have relied on Eq. (6), which was derived by compromising for empirical reasons on symmetric boundary conditions and assuming that the string was pinned to the saddle rather than clamped. We then separated the contributions to the frequency deviations from ideal values caused by fretting by expressing these differences using the definition of logarithmic “cents” given by Eq. (2), resulting in the analytically exact expression for nonideal frequency shifts given by Eq. (11). We have used this equation to plot frequency errors at each of the first 12 frets for a prototypical Classical Guitar with a variety of compensation strategies based on an RMS fit method described in  Appendix C. Because the height of each string above the frets is small compared to the scale length, there are Taylor series approximations of the terms in Eq. (11) that we used to derive Eq. (31) to guide our understanding of the underlying principles of guitar compensation. This intuition led us to approximate estimates of the ideal values of the saddle and nut setbacks given quite accurately by Eq. (42). These setback estimates can be averaged across the string set to design compensated nuts and saddles that should be relatively easy to fabricate. Nevertheless, we understand that high-end (concert) guitars that are likely to rely on one or two string sets (and the appropriate value of d for one guitar player) will benefit from the full, more accurate treatment of individual string setbacks.

In the future, it could be worthwhile to study further the boundary conditions that result in the coefficient of the linear and quadratic B terms in Eq. (6), taking into consideration the polarization of the string's vibration. Although we saw no frequency difference between the horizontal and vertical eigenmodes, it is possible that asymmetric decay rates may change the elliptical polarization of the string's vibration and therefore the effective boundary conditions (particularly at the saddle.) We measured the frequency deviations of monofilament strings at the 12th fret of several guitars and were able to rule out a factor of 2 for the linear stiffness term in Eq. (6), but a more precise value would result in more accurate predictions of the saddle setback. (We speculate that this coefficient may also depend on the construction of the saddle.) Similarly, we measured the correct value for the radius of gyration of wound nylon strings to be $ρ / 2$ with a 30% standard deviation, which leaves some room for improvement. A numerical simulation using multiphysics software may be able to refine this value further. Finally, we are at a loss to understand the high R values of the light-tension string set that we measured; since they are not significantly different in volume mass density than the other sets we studied, we suspect that there is a different manufacturing process (such as chemical composition) at play.

We have placed the text of this manuscript (as well as the computational tools needed to reproduce our numerical results and all the graphs presented here) online (Anderson and Beausoleil, 2024) to invite comment and contributions from and collaboration with interested luthiers and musicians.

The authors have no conflicts to disclose.

The data that support the findings of this study are openly available in GitHub [see Anderson and Beausoleil (2024)].

Previous studies of guitar intonation and compensation (Byers, 1995, 1996; Varieschi and Gower, 2010) included a contribution to the incremental change in the length of the fretted string caused by both the depth and the shape of the string under the finger. As the string is initially pressed to the fret, the total length $L n$ increases and causes the tension in the string to increase. When the string is pressed further, does the additional deformation of the string increase its tension (throughout the resonant length L_n)? There are at least two purely empirical reasons to doubt this hypothesis. First, as shown in Fig. 13, we can mark a string (with a small deposit of white correction fluid) above a particular fret and then observe the mark with a magnifying glass. As the string is pressed flat on the fingerboard with two fingers, the mark does not move perceptibly—it has become clamped on the fret. Second, we can use either our ears or a simple tool to measure frequencies (Larsson, 2020) to listen for a shift as we apply different fingers and vary the fretted depth of a string. The apparent modulation is far less than would be obtained by classical vibrato (±15 cents)—which causes the mark on the string to move visibly—so we assume that once the string is minimally fretted the length(s) can be regarded as fixed. (If this were not the case, then fretting by different people or with different fingers, at a single string or with a barre, would cause additional, varying frequency shifts that would be audible and difficult to compensate.)

In Sec. II, we have included this concept in a simple way to determine the effect it will have on the frequency shift due to increased string tension. First, as shown in Fig. 14, as the string is pressed onto the fret, its shape is described quite well by two line segments intersecting behind the fret. Here, it is clear that the finger is shaped by the string more than the string is shaped by the finger. We have taken this observation into consideration in Fig. 1 by introducing such an intersection point at a distance d behind fret n to represent the slight increase in the distance $L n ′$ caused by a finger. The consequences of this choice are discussed in Sec. II B, and the impact it has on (for example) the tension is shown in Fig. 2.

We can further refine the predicted values of these setbacks to accommodate the small second-order terms in $Δ S$ and $Δ N$ neglected in the resonant length error approximation used in Eq. (C5). Relying on Eq. (11) as the exact expression for the frequency error $Δ ν n$⁠, we can use Eq. (C7) to provide initial values for a nonlinear minimization of $∑ n Δ ν n 2$ over the first 12 frets. We adopt the quasi-Newton algorithm of Broyden, Fletcher, Goldfarb, and Shanno (Nocedal and Wright, 2006), a second-order algorithm for numerical optimization. Typically, this additional step changes the setback values by only a small fraction of a percent. We will refer to this approach as the “RMS minimize” method, and we use it throughout this work to compute the setbacks for each string under study. Note that the approximate equations given by Eq. (42) also can be used to compute initial values for this final nonlinear minimization.