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An exercise from English pronunciation in use (Elementary), J. Marks:

Seven of these numbers have /e/. Which are they?

Three, seven, eight, ten, eleven, twelve, thirteen, seventeen, eighteen, twenty, seventy, eighty, a hundred.

The correct answer: seven, ten, eleven, twelve, seventeen, twenty, seventy.

But what about: eight /eɪt/, eighteen /ˌeɪ'tiːn/ and eighty /'eɪti/?

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    /e/ (written thus) is a phoneme. It is a different phoneme from the diphthong /eɪ/. It would be possible to have an analysis of English where /eɪ/ was treated as a sequence of two phonemes, but that is certainly not common.
    – Colin Fine
    Aug 15 '19 at 13:08
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    1. Eight is transcribed phonemically /eɪt/, not /eit/ (phonetically [eɪ̯t], not [eit]). 2. Diphthong /eɪ/ is different from /e/ and /ɪ/ together, which can be denoted more explicitly phonetically by writing it as [eɪ̯] (as I did) or with the ɪ in the superscript (like the 2 in ²). For [eɪ̯eɪ] you'd know there's a diphthong there ([eɪ̯]), together with two other vowels: [e] and [ɪ]. 3. Nobody bothers with this in phonemic transcriptions in English as it probably doesn't occur. 4. If we trust CMUdict, there are no words (pronunciations) matching EH\d? IH\d?.
    – user3395
    Aug 15 '19 at 14:01
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As pointed out in the comments, the /e/ is a different phoneme from the diphthong /eɪ/.

A vowel can either be a monopthong or a diphthong. A monopthong is a vowel sound in which the speech organs remain in the same position throughout the production of the vowel sound (i.e. the vowel quality doesn't change). The /e/ in words like met, set, let etc., is a monopthong

A diphthong, on the other hand, involves a change in the position of the speech organs (i.e. the vowel quality changes). For example, the vowel in mate, sate, late is a diphthong /eɪ/.

In the OP, eight, eighty, eighteen have the diphthong /eɪ/ not the monopthong /e/, so they're excluded from the words having /e/.

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