The glyph ä, known as U+00E4 LATIN SMALL LETTER A WITH DIAERESIS in Unicode and ä in HTML, denotes at least three different sounds and is found in a number of primarily European scripts.

The most common use of "ä" is in German and Swedish as a representation for umlaut. To briefly summarize, in some transformations (esp. plurals) in these languages the vowel sound moves from back to front, as in German Mann becoming Männer, where the "ä" is pronounced the same as the "e" in English "men". However, since the sound was originally written by adding a small e atop the letter a, many Germanic languages (including Danish, Norwegian and Icelandic) have opted to use the AE ligature æ instead; some other Germanic languages, like English and Dutch, just use a simple e. (Yes, English man/men is the same case of umlaut, spelled a little differently.)

But when the Swedes hopped across the Gulf of Finland and conquered the illiterate peasants on the other side, things started to get a bit hairy. Finnish is not a Germanic language, and it differentiates not only between back vowels and front vowels, but open vowels and closed vowels as well. This distinction was not found in the Swedish script, so at first "e" and "ä" were used rather indiscriminately. Eventually this evolved into using "e" for the closed vowel (English "men", IPA e) and "ä" for the open variant (English "bad", IPA æ1). This selection was intuitive, since instead of umlaut Finnish has vowel harmony: whereas in German "a" tends to change into "ä", in Finnish conjugations change so that "a" and "ä" are never found in the same word.

Finnish orthography finally being settled, it was borrowed into most other languages in the Finnish branch of the Finno-Ugrian family, including Estonian and Sámi. Hungarian makes do without though, since the sound denoted by "ä" is not found in the language, and they worked out their own orthography anyway. The letter "ä" is also employed in some Turkic languages, although Azeri has recently (and inexplicably) decided to start using the IPA schwa sign instead.

Finally, while "ä" could theoretically be used for diaeresis even in English (as in coöperation; see the node for the full scoop), this turns out to be quite rare in practice. According to Gritchka, our resident master of obscure linguistic trivia, some poetry by Barnes and Tennyson in the English West Country dialects does use the diaeresis over a, as does Tolkien's Elvish for the initial combinations and .

1: Yup, the same æ that was used for "e" in the last paragraph; the Danish æ is not the same as the German and happens to match the Finnish ä. Confused yet?

References

http://www.unicode.org
http://www2.arts.gla.ac.uk/IPA/vowels.html
http://www.lysator.liu.se/language/Languages/Swedish/Grammar.html
http://www.internetix.ofw.fi/opinnot/opintojaksot/8kieletkirjallisuus/aidinkieli/kirjasuomi/konsvok.html

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