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Umlaut marks are usually used to show that an inflection has occurred, that is the umlaut is a synchronic change. But some words with an umlauted character, such as Ärger, Hälfte, Käse, Lärm, März, Rätsel, Träne are not inflected forms of something else, so why do they have umlaut spellings?

Duden says that Bär was originally spelt with an e:

mittelhochdeutsch ber, althochdeutsch bero, eigentlich = der Braune, verhüllend Bezeichnung

but I don't think the "eigentlich = der Braune" explains the ä.

Ärger comes from ärgern

mittelhochdeutsch ergern, argern, althochdeutsch argerōn, gebildet zum Komparativ von arg und eigentlich = ärger, schlechter machen

Here the ä can be traced back to an inflected form (in this case a comparative) but it not clear when the spelling was changed from e.

So, even when umlauts are not indicating inflections in Modern German, it is clear that they do not all have the same explanation. So my question is:

Looking at umlauts that are not simply inflections, what are the various explanations, and when did these changes occur?

In most of your examples the vowel ä is an actual Umlaut, i.e. a changed vowel a or ā:

1. 
   Ärger relates to arg, https://www.dwds.de/wb/Ärger
   


2. 
   Hälfte relates to halb, https://www.dwds.de/wb/Hälfte
   


3. 
   Träne stems from Middle High German trān, Germanic *trahnu-, https://www.dwds.de/wb/Träne
   


4. 
   Käse stems from Latin cāseus, https://www.dwds.de/wb/Käse
   


5. 
   Lärm stems from Alarm, Italian all’arme ("to the weapons"), https://www.dwds.de/wb/Lärm
   

So your hypothesis that the vowel ä indicates inflection, is just a part of the truth. It seems to indicate that the word stem has been a in earlier stages of the word.

The exception Bär remains though. A hypothesis: Maybe the Umlaut indicates that the vowel has been long in Middle High German and is still long?

At least, other words with the morpheme ber- seem to have the short vowel nowadays:

• Berkelium


• Berlin


• Bermuda


• Berlocke


• Berme


• Bern


• Berserker


• bersten


• Bertha

I seriously doubt your premise of Umlaut marks. These two dots have been strokes before, which is simply Kurrent handwriting of the letter e. Older printed texts have an actual tiny e printed on top of a, u, o. And this is common practice even today: if no Umlaute are available on your computer, you should write ae, oe, ue instead.

These are just five additional vowels German has (short ä is identical to short e). Or diphthongs, whatever you like.

The sound changes in inflection and conjugation aren't special to the Umlaute, it's a general pattern and called Ablaut. This was already present in Proto-Indo-European.

You can see this in German where no Umlaute are involved:

rufen, er ruft, er rief

saufen, er säuft, er soff

bieten, er bietet, er bot

Umlaut marks are usually used to show that an inflection has occurred, that is the umlaut is a synchronic change.

Umlauted vowels are less common than unumlauted vowels, but I think it's going too far to say that they are "usually" used to show inflection. The front rounded vowel sounds /yː/, /ʏ/, /øː/, /œ/ are regularly written with ü and ö in German, and they don't only occur in inflected forms of words with a base form in u or o. For example, the prepositions über and für are not inflected forms, and the verb stören doesn't have o in any of its inflected forms. Historically, these sounds derive in most cases from the phonetic process called umlaut, which seems to have been active at some point before the "Middle High German" period. You can find more information about the evolution of German vowels over time on the following web page: German vowels over time (by Helmut Richter). Ablaut is a separate and much older type of vowel alternation.

In fact, all of your examples are with ä. The situation with ä is more complicated because, unlike ü and ö, it doesn't always represent a distinct sound of its own in the spelling of German words. It represents the same sound as the letter e in at least some cases. (The details vary depending on the word and the speaker.) So it is true in general to say that ä specifically is only used in modern spelling in words that are related in some way to words spelled with a. But the relationship does not have to be a matter of inflection: ä is also used in the spelling of some derived words. More details can be found in the answers to this Linguistics SE question: What was the original pronunciation of 'ä' in German?