In linguistics, a theoretical entity sometimes postulated when a certain position in a word neutralizes the distinction between two phonemes (sounds). That is, normally there are two different sounds, and they can be in contrast, but in some circumstances they don't both occur. Examples make this clearer.

The /e/ (as in 'bet') and /æ/ (as in 'bat') vowels contrast in English words like 'bet' vs 'bat', 'mess' vs 'mass'. In almost all forms of English the same contrast occurs in 'merry' vs 'marry'. But in (some varieties of) American English there is no such pairwise difference when the following consonant is /r/. In these accents 'merry' and 'marry' are pronounced the same. There is then the vexed question, is this sound both words have, the /e/ of 'bet', or the /æ/ of 'bat'? Postulating an archiphoneme avoids having to decide. You say that in such words there is a special sound, given some ad-hoc symbol such as /E/, that is neither one nor the other: /mEri/.

Note that this does not imply that in phonetic terms the /E/ sound is strictly intermediate betwen /e/ and /æ/.

Australian English has a normal contrast between 'merry' and 'marry', but loses the distinction before /l/. So in (some varieties of) this accent, 'fellow' and 'fallow' are pronounced the same. In this case one might again notate this with an archiphoneme /E/ and call them both /fElou/.

In these dialects of English there is no good reason to insist that the vowel is one thing or the other. The hypothetical archiphoneme is a neutral compromise.

The idea may be extended where it is not appropriate. In German, you do not get voiced stops, such as [d], at the end of a word. So [ra:t] means both 'advice' and 'wheel'. There cannot exist a German word [ra:d]. But this is not a simple phonetic rule, because when you add case or plural endings to the two meanings, they behave differently: 'of the advice' is [ra:tes] but 'of the wheel' is [ra:des]. A solution that put an archiphoneme /T/ in the singular of both words would have to explain how /T/ became [t] in one genitive and [d] in the other. The correct solution is not to use archiphonemes, but to say that [ra:t] 'wheel' is in fact /ra:d/, and add a rule that /d/ is pronounced [t] at the end of a word. (This is how German spelling treats it: Rad and Rat.)

evilrooster reminds me of an omission. In dialects where 'merry' and 'marry' contrast, they have the short vowels of 'met' and 'mat' respectively. The name 'Mary' also contrasts, having a long vowel or diphthong distinct from both the other sounds. But in general American, words like 'Mary', 'hairy, 'fairy' have the same archiphoneme as 'merry', 'ferry', and as 'marry', 'Harry'.

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