I now realize that I judged Lord Frederick Leighton's painting "Music Lesson" too hastily. There ARE six tuning pegs on the instrument in the painting; you are viewing 3 of them end-on. They stick out from the front of the headstock, at right angles to the one the woman is tuning.

I figure this instrument is a saz or bağlama or something closely related to it. It's a traditional Turkish instrument. It's normal for a bağlama not to have a sound-hole, though I think some of them do. They come in different sizes—although the ones in different sizes might have different names.

I read that the frets are moveable.

I haven't seen one with a body shape quite like the one in the painting, though.

Go to YouTube and search for bağlama or saz and you will find lots of interesting videos.