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Simplest method is probably a 556 timer IC or two 555 timers. The 556 is two 555s in a single package.
Trigger the first in monostable mode from the collector of the transistor and use it's output to reset the other half in astable mode.
The monostable half will determine how long the beep lasts and the astable half will determine the pitch of the beep. It should drive a small loudspeaker without additional amplification.