Apparently there's a geneology that J.S. Bach wrote tracing back the musicians in his family, and in it he said his great-great-grandfather Veit Bach was a miller and an amateur musician who used to play an instrument called a cythringen while he was grinding meal. The cytheringen was a type of cittern ... it was sort of a simplified lute very similar to a modern Irish bouzouki.
So it wasn't precisely what we would call a zither today, because the fretboard is up on the neck of the instrument, but its name comes from the same word as "zither" (so does "guitar," I've learned), and apparently the Germans have used the word "zitter" and "zither" at different times for different types of instruments.
Main thing about it, I think, is that it was easy to play, and amateurs were attracted tp it. If I'm interpreting the drawing in Wikipedia right, it looks like it might have a diatonic fretboard
A municipal website put up by the city of Günthersleben-Wechmar in Thuringia quotes the passage from Bach's geneology that describes Veit Bach's playing at the mill at http://www.wechmar-bach.de/tourismus/veit_muehle.htm ... (information supplied to the city by Dagmar Schipanski, a German academic and politician who is also a member of the Kuratorium der Internationalen Martin Luther Stiftung ... and there's even a picture of the mill below. As follows:
From a municipal website put up by the city of Günthersleben-Wechmar in Thuringia.
Die „Wiege der Musikerfamilie Bach“
Ministerin Prof. Dr. Dagmar Schipanski eröffnete am Samstag, den 29. 11. 2003, 16 Uhr, die Veit-Bach-Obermühle Wechmar.
Johann Sebastian Bach schrieb 1735 über seinen Wechmarer Ururgroßvater Veit Bach „Er hat sein meistes Vergnügen an einem Cythringen gehabt, welches er auch mit in die Mühle genommen und unter währendem Mahlen darauf gespielet. Es muß doch hübsch zusammen geklungen haben, wiewohl er doch dabey den Takt sich hat imprimieren lernen und dieses ist gleichsam der Anfang zur Musik bei seinen Nachkommen gewesen …“.