Several times, I have read of someone who kept a pet deer mouse (Peromyscus maniculatus) who reported that it 'sang' to itself, with a beautiful musical sound.
The spring of 2000 was unusually early - I had bees at my crocuses 23 March, a week before my prior earliest record. Then, 7 April, we had a quick reversion to winter. A young deer mouse from the colony that lives under my shed found its way into the house, and made it under a door into a closet an inch ahead of my Puss. I rescued it, and installed it in an old aquarium with food and water until warm weather returned and I could safely release it. (A photo of an adult member of the colony is on my mammals page.)
It promptly began to sing to itself! And, once it got accustomed to a microphone placed close to its favourite resting spot, I got a recording.
As you can hear, it's as musical as described. It can be heard about 3 m away in a quiet house. It must be analogous in use to a cat's purr, to bond to its nestmates - it would clearly be counterproductive out in the open.
other notes on nature studies
I apologize for the brevity of the recording, but it was made before MP3 existed and when I only had 1 MB of web space. It actually sang like this for up to half an hour at a time.