Are "ä", "ö" and "ü" considered as rhyming with other non-umlauted vowels?

Question

Which vowels can rhyme with the umlaut-vowels?

My guess:

ä rhymes with e (almost sure)
ö perhaps with e or o
ü perhaps with u or ie

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The origin of the question is Trakl's sonetto Traum des Bösen:

Verhallend eines Gongs braungoldne Klänge –
Ein Liebender erwacht in schwarzen Zimmern
Die Wang’ an Flammen, die im Fenster flimmern.
Am Strome blitzen Segel, Masten, Stränge.

Ein Mönch, ein schwangres Weib dort im Gedränge.
Guitarren klimpern, rote Kittel schimmern.
Kastanien schwül in goldnem Glanz verkümmern;
Schwarz ragt der Kirchen trauriges Gepränge.

Aus bleichen Masken schaut der Geist des Bösen.
Ein Platz verdämmert grauenvoll und düster;
Am Abend regt auf Inseln sich Geflüster.

Des Vogelfluges wirre Zeichen lesen
Aussätzige, die zur Nacht vielleicht verwesen.
Im Park erblicken zitternd sich Geschwister.

And I guess here schimmern and verkümmern are paired and, less probably perhaps, Bösen and lesen.

Answer 1

There are perfect rhymes and imperfect rhymes, with varying degrees of imperfection. (The German terms are reiner Reim and unreiner Reim, respectively.) In German, two words rhyme perfectly if they sound exactly the same from the last stressed vowel onward, but are not altogether identical. The spelling is not essential: For example, Reste/Paläste, Beute/läute, Mine/Biene are all perfect rhymes, whereas Tage/Courage do not rhyme at all.

What imperfections are acceptable is completely up to the poet and the audience. Rounded and unrounded vowels at the same articulation site are commonly considered sufficiently similar, as can be seen from your example (ö/e, ü/i). Long and short vowels of the same quality may be considered contrived (gezwungener Reim: Reim dich oder ich freß dich!), but your sonnet has an example of this, too: düster/Geflüster (long vowel in düster, short vowel in Geflüster and Geschwister). Other deviations, regarding the consonants as well, are possible.

There is also a dialectal or regional perspective to this. A poet from northern Germany might rhyme Säge/Gehege, or a Palatine Spaß/Haß, which are perfect rhymes in their respective varieties of German; a reader from a different region is going to interpret them as imperfect rhymes.

Answer 2

There is a historical reason for this. At the time of classical German literature, that is to say, in the late 18th century, the most prestigious pronunciations of standard German did not have rounded front vowels at all. They were Middle German pronunciations, especially Upper Saxon German. At the time, the city said to have the best pronunciation was Meissen, not Hannover.

Since these pronunciations did not have any rounded front vowels, pairs like the following were perfect rhymes:

• ziehen [ˈtsiː(n)] – blühen [ˈbliː(n)]
• wissen [ˈʋɪsə(n)] – müssen [ˈmɪsə(n)]
• gehn [ˈɡeː(n)] – schön [ˈʃeː(n)]
• Wetter [ˈʋɛdɐ] – Götter [ˈɡɛdɐ]
• Seite [ˈsaɪdə] – Leute [ˈlaɪdə]

The idea that these are “impure” rhymes is an explanation after the fact. During the 19th century, when Prussia became the dominating state within Germany, the Northern pronunciation of standard German became the most prestigious one – that is to say, the Prussian pronunciation. Since it has front rounded vowels, pairs like the ones mentioned before do not rhyme. However, these pairs are common in classical German poetry. Therefore, they were now explained as “impure” rhymes, ignoring the fact that they had originally been perfectly pure rhymes.

Answer 3

One observation. Poetry is not my thing, but:

Guitarren klimpern, rote Kittel schimmern.
Kastanien schwül in goldnem Glanz verkümmern;

As far as I'm concerned, the highlighted words make a rather poor rhyme. However, please note that there's something else going on:

Guitarren klimpern, rote Kittel schimmern.
Kastanien schwül in goldnem Glanz verkümmern;

The actual rhyme is more complex than trying to match an Umlaut to a vowel; somebody who actually knows poetry can probably explain this construct. I believe it's quite clever, though.

If you read it out loud, "im" rhymes/harmonizes with "im", "ü" with "ü", and "ern" with "ern". In other words, each line is a self-contained rhyme and both lines rhyme with each other because of the endings. If anything, an actual rhyme between "schimmern" and "verkümmern" would create an imbalance.