The ventilation fan in my bathroom had an epic hum for a while that produced two pitches an octave apart. The lowest pitch was in the bass range and the upper pitch was in the tenor range. I couldn't help but reflexively hum a tune above the hum of the fan whenever I heard it.
I thought this would be a great improvisational exercise not only for those who are learning how to improvise, but for those who already know how to improvise and want to put their skills to the test in a fun way.
The associated video demonstrates this exercise where the hum of the fan is represented by middle C. Your improvisation can be performed on any musical instrument, provided you have a steady pitch to improvise around. If you're playing a keyboard or string instrument, you can use any pitch to represent the fan hum. If you're playing the organ, you could hold down a pedal throughout your entire improvisation. If you're playing bagpipes, the "hum" is already built-in!
The sheet music containing my improvisation is provided below merely as an example. However, like with all of my improvisational exercises, it's important for you create your own improvisation without referencing mine.