My friends have a variety of ways of pronouncing "Tuesday":
Which one is considered correct?
There are really only two phonemic variants here:
The variation between these two forms is an instance of the widespread idiosyncratic insertion of /j/ before /u:/ in many English dialects. Another common word with this variation is tune, which is pronounced both as /tu:n/ and /tju:n/. Generally both variants are accepted as correct, though your dialect will vary in which one it prefers.
The variant pronunciation with "choose-day" is a matter of dialects in which the cluster /tj/ is regularly pronounced as [tʃ] ("ch"). This is another common dialectical feature which is generally accepted as correct in the regions where it occurs. When you have someone who begins "Tuesday" with /tju:/ and produces /tj/ as [tʃ], you'll hear them pronounce the word with [tʃu:] as the first syllable.
It depends on your dialect. Dictionary.com lists your first two choices, as well as another pronunciation of day: /ˈtuzdeɪ, -di, ˈtyuz-/ (four possibilities - tooz-day, tyooz-day, tooz-dee, and tyooz-dee).
"Correct" is relative. If your dialect pronounced Tuesday choose-day, it would be considered correct by those used to the dialect. The only thing dictionaries are useful for (when you're looking up pronunciations) is to find which is the most popular/preferred.
Note that there are boundaries, though, to correctness - it's not all mush. For instance, it would be unequivocably incorrect to pronounce Tuesday Sunday.
In Australian English, I hear /tʃuz/ (chooze). In British, /tyuz/. In American dialects, thanks to yod-dropping, you hear /tuz/. In some dialects, you can hear a diphthong /iu/, which sounds similar to /yu/.
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