Can black keys be arranged in 3?

Everyone knows that the keys on the keyboard are arranged in the groups of two and three. Therefore, the octave has 7 white keys and 5 black (or the other way around). But can the sharps be arranged in the groups of 3?

It turns out that it's exactly the case with certain old historical organs. Watch this video with the oldest organ of Holland in Oosthuizen and notice that the bottom octave (called the short octave) is different. It's not complete and some of the rarely used accidentals are missing. That's why the black keys are arranged in threes there.

By the way, the top octave of this instrument is also not complete. I had a privilege to play the music of Sweelinck on a similar organ from 1511 in Alkmaar, Holland. It's not easy at first. You have to apply early keyboard fingering and get used to different arrangement of keys in order to play successfully.

Such an organ in a way is a real time machine. By playing it you can feel that you traveled about 400 or more years into the past. To the time when every citizen had to carry a dagger, when the liturgy lasted 3 hours, when a book was a luxury item, and the organ bellows had to be pumped by foot (or by hand).

The yellowish keys in this video remember these times. The instrument is so old, that you can almost smell the history (everyone who has been in an old church at an old organ will know what I mean).