What is the origin of traditionally German typefaces (for example, Fraktur? Why did these typefaces persist to such a degree in the German states long after they had declined in popularity elsewhere?
The typeface you gave above has its origin in the predominant book hand of high to late medieval Europe, Gothic.
First, reading. The definitive resource for Gothic scripts is the book by Derolez listed in our paleography booklist.
Second, terminology.
Gothic: The term "Gothic" was not used by contemporaries. Instead, it was coined by the late medieval Humanists, who saw it as a barbarism that had replaced what they thought as the true Roman script (actually Carolingian miniscule, whoops!).
Book Hand: We distinguish scripts by the context in which they were used, and medieval scribes were particular in using specific grades of script for different work. The more important the work, the more precise and stylized the script became.
Third, examples:
Here is an example of a middling grade Gothic book hand.
Here is a high grade Gothic book hand, used in a bible. Here is another example of the same sort of script, with more zoom.
Here is an example of a highly refined Gothic, writ large in order to be read easily - the book contains liturgical directions for a bishop.
Fourth, transmission:
When people first start printing books, they were concerned that they conserve as much of the manuscript aesthetic as possible, and tried to replicate the same appearance. If you look at the Gutenberg Bible, you should notice how similar it appears to the scripts above. This is pretty similar to the ways in which modern e-books try to replicate features of hard copy (pagination, etc).
As for why they persist in German states and not elsewhere? Sadly, my knowledge shuts off abruptly at the year 1500!
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I am sort of a conscious objector to changing default fonts. So I do not know anything about the specific history of typography. But I think that with a bit of context, the [Antiqua–Fraktur dispute] ( https://en.wikipedia.org/wiki/Antiqua%E2%80%93Fraktur_dispute) article on wikipedia provides an answer.
The Holy Roman Empire was closer related to modern super national institutions like the EU or the UN, than it was related to a modern state. And it dissolved in the wake of the French Revolution in 1806. In the first half of the 19th century modern Germany therefore consisted of hundreds of sovereign entities, which however all spoke essentially the same language. In this environment the German national movement developed.
In contrast with the French Revolution, which was about the organization of an already existing state, the German nationalists needed to argue for the founding of a state. And they needed a list whom to include. The deciding criterion was the "Deutsches Volk" ( literally German people, but in this context German tribe would probably be a better translation). The argument was, that everybody who belongs to the German culture also should be governed by a German state. And therefore they needed a canon of German culture. In this environment the brothers Grimm collected their fairy tales, and did also write a German dictionary. But with this comes the need to distinguish German from non-German culture. A preference of some kind of German font seems to fit well into this context.
As for sources, I do not have any in English. But some further reading in German would be the Der Spiegel series Die Erfindung der Deutschen ( The invention of Germans), which nicely is aviable in their archive, 4/2007 and following. Especially part IV, Hans-Ulrich Wehler's essay Gegen die Dynastien.