I have a vague memory of seeing a photo of an ancient keyboard instrument (a calvichord?) that had only some of the black keys - maybe just B flat and F sharp. Alas, I can't find anything similar now. Can anybody here help?
Yes, many old organs were built with only some of the 12 chromatic keys in some octaves (particularly the lowest one, since the biggest pipes are the most expensive ones). The reason is that the more remote chromatic tones were rarely used in compositions of the time, and so this saved a lot of money for only a little inconvenience.
The same was also done for other keyboard instruments. Here's a section from German wikipedia with pictures of a harpsichord and clavichord like this.