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A hammer in an out-of-tune piano.

This question aims to find the highest and lowest frequency of the string of an out-of-tune piano. The hammer of this piano hits two strings and produces a beat of 7 Hz and one of the strings of this piano is tuned to 131 Hz.

The number of waves passing through a fixed point in a fixed amount of time is called frequency. Frequency is the reciprocal of time. For example, if a wave takes 3 seconds to pass through a fixed point, then it means its frequency is 3 Hz. The unit of frequency is Hertz and it is represented as Hz.

The high peaks in the graph show high frequency and the small peaks show low frequency. When an object experiences periodic motion, it creates vibrations of certain frequencies.

The sound of a hammer hitting the strings of the piano tells whether a piano is perfect or out of tune. When the hammer of a piano hits more than one string tunes to the same pitch then the piano is perfectly fine.

If the piano sounds off- beat then the strings of the piano are out of tune. The sound of the piano tells a lot about the piano. Irritating loud sounds tell us that the piano needs tuning.

Expert Answer

The given values of frequencies are:

\[  Frequency 1 = 7 Hz  \]

\[  F_1 = 7 Hz  \]

\[  Frequency 2 = 131 Hz  \]

\[  F_2 = 131 Hz  \]

The string has the highest frequency as follows:

\[  Highest Frequency  =  F_1  +  F_2  \]

\[  Highest Frequency  =  131 + 7  \]

\[  Highest Frequency  =  138 Hz  \]

The string has the lowest frequency as follows:

\[  Lowest Frequency  = F_2 –  F_1  \]

\[  Lowest Frequency  = 131 – 7  \]

\[  Lowest Frequency  = 124 Hz  \]

Numerical Solution

The highest frequency of the string is $ 131 Hz $ and the lowest frequency of the string is $ 124 $ Hz.

Example

If the frequencies of the two strings of a guitar are 16 Hz and 23 Hz then find the highest and lowest frequencies.

The given values of frequencies are:

\[  Frequency 1 = 16 Hz  \]

\[  F_1 = 16 Hz  \]

\[  Frequency 2 = 23 Hz  \]

\[  F_2 = 23 Hz  \]

The string has the highest frequency as follows:

\[  Highest Frequency  =  F_1  +  F_2  \]

\[  Highest Frequency  =  16 + 23 Hz  \]

\[  Highest Frequency  =  39 Hz  \]

The string has the lowest frequency as follows:

\[  Lowest Frequency  =  F_2 –  F_1  \]

\[ Lowest Frequency  =  23 – 16 Hz \]

\[ Lowest Frequency  =  7 Hz \]

The highest frequency of the string is $ 39 Hz $ and the lowest frequency of the string is $ 7 Hz $.

If these strings produce the frequency of 12 Hz and 10 Hz then the highest frequency is $22 Hz$

Image/Mathematical drawings are created in Geogebra

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