Mathematical Model For The Tuning Imperfections in A Guitar [PDF]

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Mathematical Model for the Tuning Imperfections in a Guitar Rafael Resener Typical guitars have their frets arranged in such a manner that matches with those found in classical mathematical models for simple string vibration, with only minor adjustments in the bridge positioning being the difference between the real guitars and the simple models. Although their accuracy is already acceptable in most cases, they leave a margin for improvement in terms of conformity with the 12 Tone Equal Temperament (12TET) tuning system. 12 Tone Equal Temperament Tuning System In order to evaluate the conformity of an instrument’s tuning with any tuning system, we must firstly understand the system itself. Musical notes are vibrations in the air pressure which happen at a specific frequency and are perceived by humans as sound. The building blocks of the 12TET tuning system are the base frequency (usually the note A4, with a frequency usually of 440Hz), the ratio between two adjacent notes, and the idea of the octave. The relationship between the notes in the 12TET are constructed in such a way that the ratio between two consecutive notes is always the same value. Moreover, the ratio between one note and the note 12 steps higher than it (said to be one octave higher) has to be 2. The first construction constraint can be mathematically written as 𝑓𝑖+1 𝑓𝑖



= 𝑅



Where fi+1 and fi are, consecutively, the (i+1)th and i-th note frequencies, and R is the constant note frequency ratio that is characteristic of this tuning system in particular. From the rearrangement of the equation, one obtains 2𝑓𝑖+1 = 𝑅 𝑓𝑖 The ratio between any two notes 2 steps apart can be derived from the last equation 2



𝑓𝑖+2 = 𝑅 𝑓𝑖+1 = 𝑅 (𝑅 𝑓𝑖) = 𝑅 𝑓𝑖 And, in general, simple mathematical induction yields 𝑘



𝑓𝑖+𝑘 = 𝑅 𝑓𝑖 If the ratio between one note and the note 12 steps higher than it has to be 2, that means the second building block of the tuning system can be expressed in terms of equations in the following manner: 𝑓𝑖+12 = 2 𝑓𝑖 On the other hand, we can write fi+12 as



12



𝑓𝑖+12 = 𝑅



𝑓𝑖



Such that 12



𝑓𝑖+12 = 𝑅



𝑓𝑖 = 2 𝑓𝑖



Dividing the middle and right sides of the equation by fi generates 𝑅



12



= 2



And because R has to be a real number, the only solution for such an equation is (1/12)



𝑅 =2



=



12



2 ≃ 1. 05946309436



For the sake of analysis of the frequencies within the range of the guitar, it is convenient to call the base frequency, A1 = 55 Hz, to be the frequency f0. We can, therefore, infer that the general equation for the k-th note frequency has to follow the pattern (𝑘/12)



𝑓𝑘 = 55 × 2



[Hz]



Conversely, for a given frequency, its respective note can be calculated as



( ) [adimensional] 𝑓𝑘



𝑘 = 12 𝑙𝑜𝑔2



55



Classical Modelling and Engineering of Guitar Frets From [citation needed], it is known that for a string of length L [m], mass m [kg] and under a tension T [N], the frequency of vibration, f [Hz], of the string upon mechanical excitement is 𝑓=



1 2



𝑇 𝑚𝐿



[Hz]



It is intuitive that the mass of a string will be proportional to its length. A new variable, linear density (μ [kg/m]), is then introduced. This variable fulfills the role of being the proportionality constant between a string’s length and its mass. Knowing that μ = 𝑚/𝐿[kg/m] We can write the frequency equation as 𝑓=



1 2𝐿



𝑇 μ



[Hz]



Consider the setup in which we put a string of known linear density μ, between two fixtures, represented in the following figure by the points A and C, under a certain known tension force T. The distance between the fixtures A and C is known and represented by the symbol L0. If the string is initially mechanically



excited and then left to vibrate freely, its frequency of vibration can be determined using the equation derived just before.



Figure 1: Schematic of a string in vibration If the tension force T remains constant throughout the experiment, one can add a moveable constraint, depicted in the figure 1 by a triangle which constrains the region of vibration of the string between the points A and B (assuming the mechanical excitation happens between A and B). Because μ and T remain constant during the experiment, the frequency of vibration is solely a function of the distance L. By applying a certain tension such that the free vibration frequency in the region A to C is some musical note of frequency fi, found within the 12TET, we can, therefore, set constraints carefully placed such that for every constraint a different note is produced, including the case where no constraint other than the ones in the point C and A is present. That is the principle of a guitar, whose constraint points are its frets. By applying a small downward force to the strings, one is able to shorten it to the length set by any one of the frets and, with that, produce any of the musical notes available on the fretboard. The process is illustrated in figure 2.



Figure 2: Schematics of a guitar string and fretboard The classical approach of modeling the position of the frets assumes that the vertical force F applied to the strings has very little impact on its length, tension and linear density. Those assumptions are good enough when the analysis and manufacturing processes are taken into consideration, which become very simplified and have valuable practical purposes. There are, however, a few compromises made along the way, one being the fact that the aforementioned effects have small, but not entirely negligible impact on the instrument’s tuning.



The excess tension produced by the pressing force not only lengthens the string, effectively diminishing its linear density, but also introduces excess tension as a consequence of the string’s elastic properties, so that none of the independent variables come entirely untouched. We start by coming from the standard approach. Knowing that for the k-th note, its corresponding frequency is (𝑘/12)



[Hz]



𝑓𝑘 = 55 × 2



And that the frequency of vibration of any string with known tension, linear density and length is 𝑓=



1 2𝐿



𝑇 μ



[Hz]



It becomes obvious that we have to relate the inherently mathematical and physical equations to produce a musical instrument. In a practical scenario in which linear density and length of a string can be measured with a precision scale and any length measuring device, and by carefully controlling the tension by winding the string onto a shaft whose angle of rotation can be precisely administered, we can assume control over the frequency of vibration of the string. Such a setup is depicted in figure 3:



Figure 3: The setup for the building of a guitar. The tension controlling device is commonly known as a tuner, and is present in the vast majority of modern guitars, both electric and acoustic. It is easy to understand where the name comes from. In figure 3, the device is shown on the right side, in which the rotation of the horizontal peg, controlled by the user, determines the rotation of the vertical shaft. The open string frequency can be called fi0, with the subscript i concerning the fact that this is the i-th string (of, in general, but not limited to, a total of six), and the subscript “0” addressing that the string’s length is being constrained by the “zero”-th fret, also known as the nut. By using fi0 as the reference frequency, the other notes k steps above the base frequency are written as 𝑘



𝑓𝑖𝑘 = 𝑅 𝑓𝑖0



(𝑘/12)



⇒ 𝑓𝑖𝑘 = 2



𝑓𝑖0



We’ve already concluded that if the parameters μ and T remain unchanged (which they practically do in the scenario where we introduce an intermediate length constraining element), the frequency is only a function of the length. For the case of an open string vibrating, 𝑇 μ



𝑓𝑖0 = 1/(2𝐿0) And (𝑘/12)



𝑓𝑖𝑘 = 2



(𝑘/12)



𝑓𝑖0 = 2



/(2𝐿0)



𝑇 μ



The idea now is to find the appropriate string length that will produce the frequency of the k-th note on the string. We call this length Lk, and its subscript doesn’t contain information about which string it belongs to because it does not depend on either tension or linear density (the only varying parameters in a typical guitar string, or any other string for that matter), and only L0. Since T and μ are constant, 𝑓𝑖𝑘 = 1/(2𝐿𝑘)



𝑇 μ



𝐿𝑘 = 1/(2𝑓𝑖𝑘)



𝑇 μ



The length Lk can be isolated.



Using the expression for fik found in the equation before the last and substituting it yields



((



(𝑘/12)



𝐿𝑘 = 1/ 2 2



/(2𝐿0)



𝑇 μ



))



𝑇 μ



Which can be simplified to (−𝑘/12)



𝐿𝑘 = 2



𝐿0



As mentioned before, the length of the k-th note present in the string doesn’t depend on any physical property other than the open string length (L0) and the note itself (k). Plotting the position of the first 22 frets for a typical open string length of L0 = 650 mm and neck width of 50 mm, we get



Figure 4: Fret positions of a normal guitar Which is what will be observed on the smashingly vast majority of fretted stringed instruments. The black dots match the positions where they would be found in real instruments, and serve as a guide to the



instrumentalist. It is important to reiterate that the fundamental assumption of this approach is that T and μ are constant no matter what fret is pressed, and this assumption generates small errors in tuning. The New Approach Although still simple in its premise and results, this approach yields results slightly more compatible with the desired tuning compatibility.