How are vowel sounds changed by the diaeresis/trema?

Question

When learning about plurals, I picked up on some vowels changing their pronunciation — I say "picked up on" here because there was no formal explanation or lesson about how the diaeresis/trema works in changing sounds.

I noticed that the letter a alone (alone here meaning not followed by another vowel) such as in Mann or Apfel appears to generally read as ah, whereas ä such as in Männer or Äpfel appears to change the sound to be read like the e in the English word men.

I also noticed that au such as in Maus and Haus appears to read as ou in the English word house, whereas äu such as in Mäuse and Häuser appears to change to something like oi in the English word moist.

And finally I also noticed that the o in Vogel appears to have more of an English oh sound, whereas the ö in Vögel and Österreich appears to turn into more of an English oo or uh sound.

There were other examples that do not immediately come to mind now, but that lead me to pick up on what I think is a pattern.

So basically my question here is: are these sound changes constant? As in, are they actually a rule or just something that happened to be the case in all the examples I listed and learned?
And if they are a rule, how are all other vowel sounds affected by the diaeresis/trema?

Answer 1

About umlauts, trema and diaeresis

In case of German language, the dots you see in ä, ü and ü are neither called diaeresis nor trema, and they also don't fall into the category of diacritics. Those dots are just called "dots" (»Punkte« in German).

Umlauts are distinct letters

The letters ä, ö, ü and ß are four additional letters, that German has (compared to English or Latin). So German has 30 lowercase letters and 29 uppercase letters (there is no official uppercase version of ß).

Trema can be added to letters

In German texts trema can appear in rare words, but they all are foreign words (mainly imported from French), like

• Zaïre
• Citroën

Im German texts, trema appear only on i and e, i.e. on vowels, to which no umlaut exist.

The letters ä, ö and ü are called umlauts (»Umlaute« in german). The dots on them are just called "dots", they have no special name. Do not think of umlauts as vowels with something added to them. Think of them as distinct letters! (Well, the truth is, historically they did develop from the dotless versions, but now they are separate letters.)

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Pronunciation of German vowels

In written texts, German has 9 different vowels (monophthongs) and 8 different diphthongs:

a, e, i, o, u, y, ä, ö, ü
au, ei, ai, ey, ay, eu, äu, ui

In spoken language, German has much more vowels. Depending if you treat stressed and unstressed vowels as equal or different, you count 23 or 30 different spoken vowels (monophthongs) in German language, and four standard-diphthongs (there are almost a dozen non-standard-diphthongs in German dialects).

To keep it simple, I do not separate stressed and unstressed vowels. But I separate long and short vowels, because there are minimal pairs where only the length of the vowel makes a difference in the meaning, like in this pairs:

• Stahl [ʃtaːl] - Stall [ʃtal] (steel - hutch)
• du wägst [vɛːkst] - du wächst [vɛkst] (you weigh, you grow)

But how do 9 written vowels match with more than 20 spoken vowels? And how to 8 written diphthongs match with only 4 spoken diphthongs?

Diphthongs

Let's start with the diphthongs, they are easier. There is only one diphthong containing an umlaut, it is äu, item #3 in this list:

1. au = [aʊ̯]
   You find this diphthong in English in "mouse" = [maʊ̯s], "loud" = [laʊ̯d]
   German examples:

   • Baum = [baʊ̯m] (tree)
   • Maus = [maʊ̯s] (mouse)
   • Auge = [ˈaʊ̯ɡə] (eye)

2. ei, ai, ey, ay = [aɪ̯]
   You find this diphthong in English in "I" = [aɪ̯], "eye" = [aɪ̯], "buy" = [baɪ̯], "bye" = [baɪ̯], "bicycle" = [ˈbaɪ̯sɪkəl]
   German examples:

   • Leim = [laɪ̯m] (glue, made from bones)
   • Mais = [maɪ̯s] (corn)
   • Speyer [ˈʃpaɪ̯ɐ] (name of a German town)
   • Mayer = [ˈmaɪ̯ɐ] (extinct profession, and one of the most common German surnames)

3. eu, äu = [ɔɪ̯]
   You find this diphthong in English in "boy" = [bɔɪ̯], "oyster" = [ˈɔɪ̯stə]
   German examples:

   • Heu = [hɔɪ̯] (hay)
   • Läufer = [ˈlɔɪ̯fɐ] (runner)

4. ui = [ʊɪ̯]
   This one is rare in standard German (but common in many German dialects), and some experts say, that it even isn't a standard diphthong (but I think it is).
   I think, there is no English word containing this diphthong, but ui in "St. Luis" = [seɪnt ˈluːɪs], and ew in "Jerry Lee Lewis" = [d͡ʒɛri liː ˈluːɪs] are very close.
   German examples:

   • pfui! = [pfʊɪ̯] (tut! or ugh!)

Monophthongs (singel vowels)

Some German written vowels appear together with additional consonants or vowels which remain silent. This additional letters are length-markers, making the vowel to be spoken longer. I will add this length-markers if they are used.

You find umlauts in this list here:

• ä at number 6 and 7
• ü at number 18, 19 and 20
• ö at number 21, 22 and 23

Here is the full list of all German vowels:

1. [i]
   short close front unrounded vowel
   In English: happy ['hæpi]
   Written in German:

   • i - direkt [diˈʀɛkt] (direct)

2. [iː]
   long close front unrounded vowel
   In English: see [siː], thief [θiːf], be [biː], beat [biːt]
   Written in German:

   • i - Mine [ˈmiːnɘ] (mine)
   • ie - Biene [ˈbiːnɘ] (bee)
   • ih - ihn [iːn] (him)

3. [ɪ]
   short near-close near-front unrounded vowel
   In English: bit [bɪt]
   Written in German:

   • i - Mitte [ˈmɪtə] (the middle)

4. [e]
   short close-mid front unrounded vowel
   This vowel is not used in British received pronunciation or US-pronunciation.
   In English: bed [bed] (only Australian pronunciation)
   Written in German:

   • e - Debatte (first e) [deˈbatə] (debate)

5. [e:]
   long close-mid front unrounded vowel
   This vowel is not used in British received pronunciation or US-pronunciation.
   In English: play [pleː] (only Indian pronunciation)
   Written in German:

   • e - beten (first e) [ˈbeːtn̩] (to pray)
   • ee - Schnee [ʃneː] (snow)
   • eh - Lehrer (first e) [ˈleːʀɐ] (teacher)

6. [ɛ]
   short open-mid front unrounded vowel
   in English: let [lɛt], men [mɛn]
   Written in German:

   • e - nett [nɛt] (nice)
   • ä - Hälfte [ˈhɛlftʰə] (the half)

7. [ɛ:]
   long open-mid front unrounded vowel
   In English: bed [bɛːd] (only US-pronunciation)
   Written in German:

   • ä - Käse [ˈkʰɛːzə] (cheese)

8. [a], [ä]
   short open central unrounded vowel
   ^{Note, that the International Phonetic Alphabet officially has no dedicated letter for this sound between front [a] and back [ɑ]. Most often the letters [a] or [ä] are used. But be aware! If you find [ä] in a phonetic notation, it never is how the German letter »ä« sounds! For this reason, I use here [a].}
   This vowel is not used in British received pronunciation. In US-pronunciation it appears only in some geographic regions.
   In English: cot [kʰat] (pronunciation of southern Michigan)
   Written in German:

   • a - Wall [val] (rampart); Katze [ˈkaʦə] (cat)

9. [a:], [ä:]
   long open central unrounded vowel
   ^{The note for #8 applies here too.}
   This vowel is not used in British received pronunciation or US-pronunciation.
   In English: car [kʰaː] (Australian pronunciation and Cultivated South African pronunciation)
   Written in German:

   • a - Wal [vaːl] (whale); (er) kam [kaːm] ((he) came)
   • aa - Waal [vaːl] (irrigation canal); Saal [zaːl] (hall)
   • ah - Wahl [vaːl] (election); wahr [vaːɐ̯] (true)

10. [ə]
    short mid central vowel (also known as "schwa" or "shwa")
    This vowel is always unstressed and appears most often in reduction syllables at the end of words, both, in English and German.
    In English: broken [ˈbɹəʊkən]
    Written in German:

    • e - viele (last e) [ˈfiːlə] (many)

11. [ɐ]
    short near-open central vowel
    This vowel is always unstressed and appears in German in reduction syllables at the end of words.
    In English: nut [nɐt] (British Received Pronunciation and pronunciation of California and New Zealand)
    Written in German:

    • er - oder [ˈoːdɐ] (or)

12. [o]
    short close-mid back rounded vowel
    I found this vowel in english words only as part of a diphthong.
    In English: go [ɡoʊ] (only US-pronunciation)
    Written in German:

    • o - Rosine [ʀoˈziːnə] (raisin)
    • ow (rare!) - Pankow [ˈpaŋko] (a district of Berlin)

13. [o:]
    long close-mid back rounded vowel
    This vowel is not used in British received pronunciation or US-pronunciation.
    In English: yawn [joːn] (Australian pronunciation)
    Written in German:

    • o - Ton [toːn] (tone, clay)
    • oo - Boot [boːt] (boat)
    • oh - bohren [ˈboːʀən] (to drill)

14. [ɔ]
    short open-mid back rounded vowel
    The long version of this vowel is the standard-O-vowel in American English. You find it in "thought" [θɔːt]. But its hard to find an English word where this vowel is short.
    In English: not [nɔt] (only old Received Pronunciation)
    Written in German:

    • o - Sonne [ˈzɔnə] (sun)

15. [u]
    short close back rounded vowel
    In English: book [buk] (only Australian Pronunciation)
    Written in German:

    • u - kulant [kuˈlant] (obliging, fair)

16. [u:]
    long close back rounded vowel
    In English: boot [buːt]
    Written in German:

    • u - Fuß [fuːs] (foot)
    • uh - Kuhle [ˈkuːlə] (hollow, scrape)

17. [ʊ]
    short near-close near-back rounded vowel
    In English: hook [hʊk]
    Written in German:

    • u - Butter ['bʊtɐ] (butter)

18. [y]
    short close front rounded vowel
    This vowel is not used in British received pronunciation or US-pronunciation.
    This vowel, and the next both (#19 and #20 in this list) are the vowels, that cause the most problems to English native speakers learning German. It is very important not to pronounce it as [ʊ] or [u]! [y] is a sound in the phonetic neighborhood of [i]. You can train it if you say »cheese« with a very long [i:] (like while waiting for the photographer to take a picture), but while you do so, round your lips, without changing anything inside your mouth! The [i:] will turn into the sound we need. You will get the sound #19 in this list. This here, #18 is just the short form of it.
    Written in German:

    • ü - Büro [byˈʀoː] (office)
    • y - Physik [fyˈzɪk] (physics)

19. [y:]
    long close front rounded vowel
    This vowel is not used in British received pronunciation or US-pronunciation.
    Read the comments in #18!
    In English: few [fjyː] (only young females in South Africa and in multicultural parts of London)
    Written in German:

    • ü - üben [ˈyːbn̩] (to practice)
    • üh - kühl [kyːl] (cool)
    • y - Typ [tyːp] (type, fellow)

20. [ʏ]
    short near-close near-front rounded vowel
    This vowel is not used in British received pronunciation or US-pronunciation.
    Read the comments in #18! This vowel is the rounded twin of [ɪ]. Say "bit", "kiss" or "Tim" with rounded lips, and you will get exactly the vowel [ʏ].
    In English: foot [fʏt] (only multicultural London and rural whites in southern regions of USA)
    Written in German:

    • ü - Mücke [ˈmʏkə] (mosquito)

21. [ø]
    short close-mid front rounded vowel
    This vowel is not used in British received pronunciation or US-pronunciation.
    Written in German:

    • ö - Ökonomie [ˌøkonoˈmiː] (economy)

22. [ø:]
    long close-mid front rounded vowel
    This vowel is not used in British received pronunciation or US-pronunciation.
    In English: bird [bø̠ːd] (only South African Pronunciation)
    Written in German:

    • ö - schön [ʃøːn] (beautiful)
    • öh - gewöhnlich [ɡəˈvøːnlɪç] (common)

23. [œ]
    short open-mid front rounded vowel
    This vowel is not used in British received pronunciation or US-pronunciation. Only the long version of this vowel exists in some regions, i.e. In the British dialect cockney and in New Zealand: "bird" [bœ̠ːd], but we are talking here about the short version of this vowel.
    Written in German:

    • ö - löschen [ˈlœʃn̩] (delete)

You can group this list to get a simplified list:

a = 8 [a], 9 [a:]
e = 4 [e], 5 [e:], 6 [ɛ], 10 [ə], 11 [ɐ]
i = 1 [i], 2 [iː], 3 [ɪ]
o = 12 [o], 13 [o:], 14 [ɔ]
u = 15 [u], 16 [u:], 17 [ʊ]
ä = 6 [ɛ], 7 [ɛ:]
ö = 21 [ø], 22 [ø:], 23 [œ]
ü = 18 [y], 19 [y:], 20 [ʏ]
y = 18 [y], 19 [y:]

Answer 2

There is no rule, the Umlaute are vowels on their own. For Germans speakers, an umlaut isn't ¨ but one of ä, ö, ü.

The two dots on top of a, u, o originate in the lowercase letter e, which was written on top of those three vowels to indicate a sound shift “including e”. This e was written " in handwriting and eventually replaced by it also in typography. Today, ä can also be written as ae, ö as oe, und ü as ue. But it was always a writing thing, in speech the Umlaute are three additional vowels.

So you have to learn these sounds separately. There are even short and long variants of ä, ö, ü, which sound differently. And of course, all the diphtongs made up from Umlaute, too.

Fun fact: aeu is a diphtong in German, though it seem to have three vowels. In reality, it's only two: ä and u, and it sounds extremely similar (in most dialects identical) to the diphtong eu.