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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 data:

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

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

The frequency of the first wave can be written as:

\({f_1} = \frac{v}{{2{l_1}}}\) 鈥� (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:

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

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:

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

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:

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

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:

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

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:

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

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 \(52.7\;{\rm{cm}}\).