Question

Question: (III) When a player’s finger presses a guitar string down onto a fret, the length of the vibrating portion of the string is shortened, thereby increasing the string’s fundamental frequency (see Fig. 12–36). The string’s tension and mass per unit length remain unchanged. If the unfingered length of the string is l= 75.0 cm, determine the positions x of the first six frets, if each fret raises the pitch of the fundamental by one musical note compared to the neighboring fret. On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is 2^1/12.

Figure 12-36

Short answer

The positions x of the first six frets are \(4.3\;{\rm{cm}}\), \(8.3\;{\rm{cm}}\), \(12.1\;{\rm{cm}}\), \(15.7\;{\rm{cm}}\), \(19.1\;{\rm{cm}}\) and \(52.7\;{\rm{cm}}\).

Step-by-step solution

Determination of the length of the first fret

The length of the first fret can be obtained by taking the ratio of the fundamental frequency of the first fret to the standard fundamental frequency in the periodic form, which can be represented as the ratio of the speed to twice of the unfingered length.

Given information

Given data:

The unfingered length of the string is \(l = 75.0\;{\rm{cm}}\).

The ratio of frequencies of neighboring notes is \({2^{1/12}}\).

Evaluation of the length and position of the first fret

The frequency of the first wave can be written as:

\({f_1} = \frac{v}{{2{l_1}}}\) … (1)

After dividing the equation (1) by the standard equation of fundamental frequency, you get:

\(\begin{array}{c}\frac{{{f_1}}}{f} = \frac{{\left( {\frac{v}{{2{l_1}}}} \right)}}{{\left( {\frac{v}{{2l}}} \right)}}\\\frac{{{f_1}}}{f} = \frac{l}{{{l_1}}}\\{2^{1/12}} = \frac{{0.75\;{\rm{m}}}}{{{l_1}}}\\{l_1} = 0.707\;{\rm{m}}\end{array}\)

This is the length of the first fret.

The position of the first fret can be calculated as:

\(\begin{array}{c}{x_1} = l - {l_1}\\{x_1} = \left( {75\;{\rm{cm}}} \right) - \left( {0.707\;{\rm{m}} \times \frac{{100\;{\rm{cm}}}}{{1\;{\rm{m}}}}} \right)\\{x_1} = 4.3\;{\rm{cm}}\end{array}\)

Evaluation of the length and position of the second fret

The length of the second fret can be calculated as:

\(\begin{array}{c}\frac{{{f_2}}}{{{f_1}}} = \frac{{{l_1}}}{{{l_2}}}\\{2^{1/12}} = \frac{{0.707\;{\rm{m}}}}{{{l_2}}}\\{l_2} = 0.667\;{\rm{m}} \times \frac{{100\;{\rm{cm}}}}{{1\;{\rm{m}}}}\\{l_2} = 66.7\;{\rm{cm}}\end{array}\)

The position of the second fret can be calculated as:

\(\begin{array}{c}{x_2} = l - {l_2}\\{x_2} = \left( {75\;{\rm{cm}}} \right) - \left( {66.7\;{\rm{cm}}} \right)\\{x_2} = 8.3\;{\rm{cm}}\end{array}\)

Evaluation of the length and position of the third fret

The length of the third fret can be calculated as:

\(\begin{array}{c}\frac{{{f_3}}}{{{f_2}}} = \frac{{{l_2}}}{{{l_3}}}\\{2^{1/12}} = \frac{{66.7\;{\rm{cm}}}}{{{l_3}}}\\{l_3} = 62.9\;{\rm{cm}}\end{array}\)

The position of the third fret can be calculated as:

\(\begin{array}{c}{x_3} = l - {l_3}\\{x_3} = \left( {75\;{\rm{cm}}} \right) - \left( {62.9\;{\rm{cm}}} \right)\\{x_3} = 12.1\;{\rm{cm}}\end{array}\)

Evaluation of the length and position of the fourth fret

The length of the fourth fret can be calculated as:

\(\begin{array}{c}\frac{{{f_4}}}{{{f_3}}} = \frac{{{l_3}}}{{{l_4}}}\\{2^{1/12}} = \frac{{62.9\;{\rm{cm}}}}{{{l_4}}}\\{l_4} = 59.3\;{\rm{cm}}\end{array}\)

The position of the fourth fret can be calculated as:

\(\begin{array}{c}{x_4} = l - {l_4}\\{x_4} = \left( {75\;{\rm{cm}}} \right) - \left( {59.3\;{\rm{cm}}} \right)\\{x_4} = 15.7\;{\rm{cm}}\end{array}\)

Evaluation of the length and position of the fifth fret

The length of the fifth fret can be calculated as:

\(\begin{array}{c}\frac{{{f_5}}}{{{f_4}}} = \frac{{{l_4}}}{{{l_5}}}\\{2^{1/12}} = \frac{{59.3\;{\rm{cm}}}}{{{l_5}}}\\{l_5} = 55.9\;{\rm{cm}}\end{array}\)

The position of the fifth fret can be calculated as:

\(\begin{array}{c}{x_5} = l - {l_5}\\{x_5} = \left( {75\;{\rm{cm}}} \right) - \left( {55.9\;{\rm{cm}}} \right)\\{x_5} = 19.1\;{\rm{cm}}\end{array}\)

Evaluation of the length and position of the sixth fret

The length of the sixth fret can be calculated as:

\(\begin{array}{c}\frac{{{f_6}}}{{{f_5}}} = \frac{{{l_5}}}{{{l_6}}}\\{2^{1/12}} = \frac{{55.9\;{\rm{cm}}}}{{{l_6}}}\\{l_6} = 52.7\;{\rm{cm}}\end{array}\)

The position of the sixth fret can be calculated as:

\(\begin{array}{c}{x_6} = l - {l_6}\\{x_6} = \left( {75\;{\rm{cm}}} \right) - \left( {52.7\;{\rm{cm}}} \right)\\{x_6} = 22.3\;{\rm{cm}}\end{array}\)

Thus, the positions x of the first six frets are \(4.3\;{\rm{cm}}\), \(8.3\;{\rm{cm}}\), \(12.1\;{\rm{cm}}\), \(15.7\;{\rm{cm}}\), \(19.1\;{\rm{cm}}\) and \(52.7\;{\rm{cm}}\).