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My guess is that it was used to distinguish aspiration (as opposed to 't' in words of Latin/ Old French origin, which was not aspirated?). I'm pretty sure German lost its dental fricative to d pretty early on, so I know that's probably not it. Examples: thun, Thal, Thier etc.

Just to clarify, I do not mean words of Greek/English origin with 'th' (like ethisch, Thema or Thron).

user3482545
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German did lose its dental fricatives early on, but not quite early enough to avoid affecting the orthography.

The transition from /θ~ð/ to /t~d/ is usually considered part of the High German Consonant Shift. But this generalization can be misleading. This change did start in the same region as the other High German changes. But /θ/ shifted significantly later, around the 10th century, and spread significantly farther, affecting all of the West Germanic languages except English.

By this point, German had just begun to be written down. "Th" was chosen for /θ/, by analogy with Greek theta (transcribed as "th" in Latin, and now pronounced /θ/). So the cognate to English "think" was written thenken (modern German denken).

But then /θ/ disappeared from the language. With the distinction between /θ/ and /t/ lost, it was no longer clear which German words should be written with "th" and which should have "t". Thus "th" appeared in some words like Thal, which originally had /d/ (compare English "dale"). A similar process in English created the "th" in "Thomas" and "Thames".

Eventually this use of "th" fossilized. Words like Thal were permanently written with "th", while words like denken were permanently written with "d".

In 1901 the orthography reform finally removed this distinction, replacing Thal with Tal and Thier with Tier. "Th" now survives only in proper names (Goethe) and Greek-derived loanwords (Thron).

Source: I've been using Duden Online and Wright's Old High German Primer, which has good information on Old High German orthography and the consonant shift. However it was also published in 1888, so my information may be very out of date. Please correct me if so.

Draconis
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    I am afraid this answer is completely wrong. The "th" spellings occur only in New High German. There is no trace of them in Middle High German.The Germanic fricatives disappeared long before the emergence of NHG. This "th" is purely an orthographic fluke. – fdb Nov 08 '16 at 18:35
  • @fdb My understanding was that the spurious "th"s were an orthographic fluke, but that the "th" digraph existed in the first place because of the short-lived OHG fricatives setting precedent for loanwords. (Though again my source is memory + a single book from 1888, so this could be entirely off the mark.) – Draconis Nov 08 '16 at 20:25
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    No, you cannot jump straight from OHG to NHG. In MHG you have only tun, rat, tal. Thun, Rath, Thal come in at the time of Luther. – fdb Nov 08 '16 at 21:04
  • @Draconis Thanks for the answer, but as I said, I'm completely sure it's not the dental fricative. The phoneme turned into /d/ almost completely in Old High German and the words with 'th' in their orthography never were pronounced with a dental fricative to begin (just like you mentioned Tal goes back to Proto-Germanic *dalą). Proto-Germanic had /ð/ only as an allophone to /θ/ (whereas /t/ and /d/ were separate phonemes). And the /t/ in High German (which is the subject of my question) traces back to /d/ in Proto-Germanic. – user3482545 Nov 20 '16 at 13:18
  • @fdb Thank you for the additional information. I wouldn't be surprised to find out that it was just a fluke and had no real importance, but it would be nice to have it supported by sources, and I'd be interested to know why you think so. – user3482545 Nov 20 '16 at 13:24