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?
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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.
Photograph-wise, it seems we may be out of luck. According to an article on the Classic FM website, the oldest surviving stringed keyboard instrument is a "Clavicytherium" ca. 1480, which clearly includes the standard arrangement of five "black" keys.
This instrument, from around 1480, was made in South Germany. It's an upright single-strung harpsichord in an outer case and is believed to be the earliest surviving stringed keyboard instrument.
The below images are from the Keyboard Instruments catalogue (p. 11) of the Royal College of Music Museum (London).