A piano tuner using a 264Hz tuning fork hears 4 beats per sec. There are 2 possible frequencies of vibration?

Answer 1

Yes, with a piano string vibrating at the same time as the tuning fork there are two different frequencies that the string can be at - 260 Hz and 268 Hz. The Tuner's ear would be able to hear the difference and know which way to go with the string - tighten or loosen.

Beats are caused by the constructive and destructive interference created by objects interacting at slightly different frequencies, and each beat per second corresponds to a difference of a single Hz, and is not specific as to which acoustic source is higher-pitched or lower.

Answer 2

The fork is not going to be resonating at more than one frequency at once... at least not until you get into the quantum mechanics of the system, by which point you are well beyond any frequencies/sizes that would have anything to do with the audible spectrum.

As far as sympathetic frequencies that could be stimulated by this fork, there is an additional one that would likely be 'heard'

264hz is close to middle c on a piano... 132 hz would be two octaves below it.

Additionally, a perfectly tensioned (sympathetic to the 264hz at 528 hz) C5 could be stimulated, if it were at least 4 wavelengths long.

Answer 3

Whenever you combine two waves you get four outputs. The two original waves plus the sum and differences. So, to get 4hz you could be combining the 264hz tuning fork wave with a 268hz or a 260hz original.