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Answer :
The violin string is off-tune by 3 Hz from the piano's middle C.
When tuning a violin string to match a middle note C played on a piano, we use the concept of beat frequency to measure how off-tune the violin string is. In this case, when the middle note C is played on both the violin and the piano, nine beats are heard in three seconds.
The beat frequency is given as the number of beats per second, which can be calculated as follows:
[tex]\text{Beat frequency} = \frac{\text{Number of beats}}{\text{Time (in seconds)}}[/tex]
Given:
Number of beats = 9
Time = 3 seconds
[tex]\text{Beat frequency} = \frac{9}{3} = 3 \text{ Hz}[/tex]
This beat frequency tells us that the difference between the frequency of the piano's middle C note and the frequency of the violin string's note is 3 Hz.
Assuming the middle C on the piano is tuned to 261 Hz, the frequency of the violin string could either be:
(1) 3 Hz higher than 261 Hz, which is 264 Hz
(2) 3 Hz lower than 261 Hz, which is 258 Hz
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The violin is off tune by 3 Hz when compared to the piano's middle C pitch of 261.63 Hz, as determined by the beat frequency caused by the interference of the two slightly different frequencies.
The concept of beat frequency, which is covered in physics, particularly in the study of sound waves and musical acoustics. When tuning a violin string to the middle C pitch of a piano, beats are heard when two similar frequencies are close but not identical. This difference causes the sound waves to interfere, creating a pulsing effect known as beats.
The pitch of the middle C note on a piano is 261.63 Hz. When the student mentions that nine beats are heard in three seconds while tuning a violin string to this pitch, it indicates that the violin string's frequency is either slightly higher or slightly lower than the piano's middle C frequency. The number of beats per second, known as the beat frequency, is given by the absolute value of the difference between the two frequencies. Therefore, if nine beats are heard in three seconds, the beat frequency is 9 beats / 3 seconds = 3 beats per second (3 Hz).
Since the pitch of middle C on a piano is 261.63 Hz, the violin string's frequency must be either 261.63 Hz + 3 Hz or 261.63 Hz - 3 Hz, which equals 264.63 Hz or 258.63 Hz. Therefore, the violin is off by 3 Hz from the correct tuning pitch of middle C.
