The sound of your beep will be strongly affected by the shape of the electric waveform sent to the speaker. For example, if you send a tone that is a pure sine wave, it will sound quite pure and perhaps "sweet" to you. If you have a square wave, it will sound much more "urgent". Some waveshapes are a blend between the two, like for example a triangle wave is somewhat raspy but not as urgent as the square wave. If you were to distort the square wave by adding overshoot and ringing then it would sound even more urgent and perhaps even a bit weird. So, there are many sounds you can get out of a single tone. To be even more complicated, you can have more than one tone at a time, and you can vary the relative amplitude of these two tones as well as the shape of the waveform of each to experiment with different kinds of beeps. This is what is done in small toy organs (like those little electric pianos that are made for babies to play with, or the kind that are put into children's books) and in fact, if you can find one that is being thrown away this might be a very good starting place for your application.