Question

Caroline, a piano tuner, suspects that a piano's A4 key is out of tune. Normally, she would play the key along with her A4 tuning fork and tune the piano to match, but her A4 tuning fork is missing. Instead, she plays the errant key along with her G4 tuning fork (which has a frequency of 392.0 Hz), displays the resulting waveform on a handheld oscilloscope, and measures a beat frequency of 86.4 Hz. Then, she plays the errant key along with her B4 tuning fork (which has a frequency of 493.9 Hz) and measures a beat frequency of 15.5 Hz. What frequency is being played by the out-of-tune key? 509.4 Hz 305.6 Hz 478.4 Hz 407.5 Hz 38.4 Hz If the A4 key is supposed to produce a frequency of 440.0 Hz, is the frequency of the key lower than it should be ("flat") or higher than it should be ("sharp")? flat sharp

Answer 1

If beat frequency generated when A4 key and G4 tuning fork is 86.4 Hz,

then frequency of A4 key = 392 $\pm$ 86.4 = 478.4 Hz or 305.6 Hz

If beat frequency generated when A4 key and B4 tuning fork is 15.5 Hz,

then frequency of A4 key = 493.9 $\pm$ 15.5 = 478.4 Hz or 509.4 Hz

Hence frequency of A4 key = 478.4 Hz.

Frequency of out-of-tune A4 key is higher ( sharp ) than its normal frequency ( 440 Hz )