A Pitch Error Model for Fretted String Instruments John Lane1 and Takis Kasparis2 1
2
University of Central Florida, Orlando, Florida Cyprus University of Technology, Lemesos, Cyprus
ABSTRACT A mathematical model is presented for analyzing string and fingerboard parameters of a fretted string musical instrument in order to predict the tuning degradation associated with instruments that tune well and those that don’t. The proposed model is based on two fundamental string equations: the relationship between string tension, pitch (fundamental frequency), and string length; and the equation that relates the change in tension to the change in length using Young’s modulus. A third equation essential to this work relates string deflection perpendicular to the fretboard to the change in string length. This model quantifies the observation that poor fret design can lead to a large and unacceptable tuning error due strictly to geometrical considerations. The model discusses a means to analyze and optimize tuning of a string instrument based on variable instrument parameters, such as string density and bridge position, as well as fixed instrument parameters, such as fret height.
INTRODUCTION In practice, stringed instruments (especially guitars) are tuned to an open string and at the 12th fret. This tuning strategy is based on the assumption that errors can be evenly distributed over the fingerboard of the instrument. Adjustment of bridge position modifies the string length so that the pitch at the 12th fret is exactly twice that of the open string. In this case, the string tension to a good approximation is considered constant. The differential equation of motion for string of length L under constant tension T, fastened at both ends1, is: ,
(1)
where y is the displacement perpendicular to the string, x is the distance along the length of the string, and is string density per unit length. The solution to Equation (1) gives a harmonic series where the fundamental frequency is: /
/
/2
.
(2)
In the traditional tuning strategy, T is assumed to be a constant when fretting different positions on the neck. Ideally then, if L goes down by exactly a factor of two by fretting at the 12th fret, then f goes up by exactly a factor of two. The bridge position can be adjusted so that the 12th fret corresponds to exactly a factor of two. All other fret positions are treated similarly so that the new string length at each fret corresponds to a frequency (pitch) of the equal temperament scale, so that the product of the free string length at each fret and frequency (pitch) is equal to a constant, / / /2, from Equation (2). The fundamental frequencies of the equal temperament scale are based on the 12th root of 2: 2
/
Corresponding Author: John Lane,
[email protected], 321-867-6939.
,
(3)
where is the frequency of an open string corresponding to length . Fingering the kth fret position produces a string length of and using Equation (2) and (3) is: 2
/
,
(4)
is the string length from the bridge position to the fingered kth fret. In this notation, k = where 0 corresponds an open string and k = 1 corresponds to the first fret position closest to the neck nut, and k =12 corresponds to the octave fret position, halfway between the nut and the bridge. Figure 1 shows various fretted string instruments, all of which have fret spacing based on Equation (4). There are at least two problems with relying upon Equation (2) for modeling and predicting tuning and pitch at every fret position: 1. Equation (2) is valid only for a very small amplitude string displacement and a homogeneous string density1. 2. Equation (2) is the result of a one-dimensional analysis where the string thickness (or cross-section) is ignored. The finite cross section of a string leads to a string pitch characteristic referred to as inharmonicity2. Inharmonicity requires a modification to Equation (2) where the true frequency of a string of length L, under tension T, of modulus E, cross-sectional area A, and string density , is approximated by3: 1
2
,
(5)
For the purpose of the work, Equation (5) will be assumed to lead to only a small pitch error, on the same order of magnitude (less than a cent) as errors due to fret position inaccuracies, and errors due to non-homogeneous string density due to dirt or rust. The primary source of pitch error to be discussed in the remaining sections of this paper is caused by vertical string displacement due to fingering a note on the fretboard. In the reaming text, the term pitch will be synonymous with the fundamental frequency. In other words, the problems with higher harmonics due to in inharmonicity effects will not be considered. FRETTED STRING MODEL The basic components of the fretted string instrument system are depicted in Figure 2. Fret height is exaggerated in the figure for convenience of explaining the mathematical modeling details. Figure 3 shows the details of the string and fret position model parameters. In this ideal one dimensional model, there are two primary mechanisms that determine the final pitch of the vibrating string. The first mechanism has been described by Equation (1) and (2) where the string tension is constant. The second primary mechanism is due to the stress-strain influence on the string. . For In the case of the first primary mechanism, the characteristic string length is th the open string, and for the octave position at the 12 fret, /2 . The second
mechanism, the stress-strain relationship for a string of length L, under tension T, modulus E, cross-sectional area A, and string density , is given by: ∆
∆
,
(6)
where T = change in string tension (stress) and L = change in string length due to bending (strain). In the case of Equation (6), the string length of significance is . The length associated with Equation (2) is the vibrating string length between the kth fret position (or nut position corresponding to k = 0) and the bridge position. The density associated with Equation (2) includes the total mass per unit length and total cross-sectional area A of the string. The length associated with Equation (6) is the entire length of the string from the tuning key to the tail-piece, since neither the bridge nor the head nut clamp the string in place. If the string is a single wire, then the mass density and cross-section are the same in the case of both Equation (2) and Equation (6). However, in most cases, the string is not a single wire but consists of a single core and a wrapped outer section as shown in Figure 4 for the case of a guitar G-string. The effective diameter of a wrapped string applicable to Equation (2) is approximately 0.9 of the measured cross-section due to voids produced by the wrapping (see Appendix A for a derivation of this result). For a wrapped string, the outer sheath is approximately the same density as the core (mass density and modulus of steel). The total mass density per length of the string goes into Equation (2). However, in the stress-strain relationship of Equation (6), it is only the mass density per length and cross-section of the wire core that is significant. STRING FORCE MODEL The relationship between string tension T and a vertical displacement force Fk(y) above the kth fret position can be approximated for small displacements, as: ,
(7)
where y is the vertical distance toward the fretboard from the string’s normal position (see Figure 5). The rationale behind Equation (7) can be more easily seen in the special case of k = 12 where . This result follows by considering the vertical component of tension T as 2 sin
2 tan
2
/
. Equation (7) is a generalization of this result for the kth
fret position. Initial Force-Deflection: Region I Region I can be defined as the initial deflection from zero vertical displacement to the point where the string makes first contact with the kth fret (see Figure 6). The vertical distance y to initial contact of string to fret is ≡ /2 . As shown in Figure 5, hk is the distance to the fret top from the undisplaced string centerline and D is the full string diameter, including any wrapping. In this region, the force versus deflection curve is characterized by a relatively small
force constant. This in turn, as will be quantified in a later section, will result in only a small pitch error. Using the stress-strain relationship from Equation (6) and going back to Equation (7), the tension for region I is now a function of y : Δ Δ
,
(8)
where ≡ (see Figure 3) and where is the tension in the undeflected string. The change in string length Δ due to a deflection in y can be found by considering the right triangles formed on either side of the kth fret position, from the new string position, and the original string position: /
Δ
/
Approximating the square root terms as 1 as: Δ
/
1
.
(9)
, for u
