However, I still do not know how many syllables are in "fire."
I understand an r-colored vowel might be at play.
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I know nothing about Haiku, but I can tell you some general things to think about in terms of the syllable in general.
Unfortunately, the syllable is one of those concepts that is difficult to define precisely and uncontroversially in terms of its details, depsite it being one of the few phonological phenomena that your "average" speaker has a good degree of intiution about. What we can say is that speech appears to be organised into "syllables" which are defined by some combination of the following:a syllable generally corresponds to a peak in sonority;a syllable generally corresponds to a unit that speakers intuitively make use of in metalinguistic activities (e.g. singing or clapping one note per syllable);a syllable is an organisational unit: it generally corresponds to a vowel at its nucleus, wth which consonants at the "edges" of the same syllable are associated in some way (e.g. changes in duration can occur across the syllable as a whole unit), and in a given language you can find a relatively small number of patterns that all syllables conform to.
When considering the above factors, there are a few cases where ambiguity arises. For example, in the word "strengths", there is a peak of sonority on the "s", and it"s unusual for such a complex cluster to occur, suggesting that the final "s" may constitute its own syllable. But on the other hand, few if any speakers would make two claps/taps to accompany the word "strengths", or sing it on two notes.
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The word "fire" is another example where there is ambiguity, and probably speaker-to-speaker variation. On the one hand, we may conclude that it is composed of 2 syllables: one with a diphthong followed by one with a single schwa vowel. Or we may conclude that it comprises a single syllable with a triphthong ("single vowel" with three targets). One motivating argument for it being a single syllable might be the existence of alternative pronunciatons in which a single diphthong is present; a motivating argument for two syllables would be where speakers mark the word with two claps/notes, or pronounce a distinct yod ("y" sound) between the diphthong and schwa.