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I taught Singers' Diction in grad school for several years.
Technically, "fire" contains a triphthong, three vowels in a row, and is thus one syllable. In International Phonetic Alphabet, it's [a] then [I] then a schwa, which I can't type in ASCII.
The only other triphthong in English is the one in the word "our" or "hour", which is [a], then [U], then a schwa.
The IPA represenation is not [ə], but rather [ɹ̩]. (There's a tick under that ɹ).
And, to be fair, your statements are accurate for British dialects of English that don't have syllable final rhotics, but the original article was about American English.
The interpretation of the [aIə] as triphthong or diphthong varies according to one's source. For singers (I stated that my background was from diction as taught to singers), we tend to be taught that it's a triphthong, because for singing or stage diction, it's often desirable to not be saying an American "r" for seconds at a time...not a pleasing sound. The IPA symbol used varies as well, for the same reasons.
Of course, in some places in the South, "fire" has the same pronunciation as "far".
Depending on how much emphasis you put on the word, it can range from one to two syllables. If you use it in a sentence ("The owl ate the mouse") spoken at a conversational speed, the sounds that come from your mouth are likely to be indistinguishable from saying the name Al. If you say the word slowly, you'll have an [æ] (the vowel in cat) or an [a] (the first vowel in father), followed by [wl̩] (sounds like a sheep's hair).
This is just a straight up [aU] diphthong, one syllable, in virtually every dictionary you can find. Certainly all the ones that use IPA. Same in my compact OED (the tiny-print one that comes with a magnifying glass).
Pacific Northwest here. I pronounce "squirrel" as "squirl".
It gets worse. There's a town, "Sequim". The way you know someone isn't from around here is if they pronounce it "See-quim". It's actually pronounced "Squim".
>CMU's Pronouncing Dictionry (http://www.speech.cs.cmu.edu/cgi-bin/cmudict?in=fire), however, indicates two possible pronunciations (F AY ER | F AY R), the first of which seems to indicate a two-syllable word.
