Is there any other category for a syllable which is neither an open nor closed syllable?
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English has a rather unusual syllable structure that allows for as many as three leading and five trailing consonants. As a result, there is arguably some ambiguity as to where the articulation point between two syllables falls. However, there is not universal agreement about this point. Wikipedia offers this reasonably clear analysis:
As far as I can tell, the term 'ambisyllabic' applies specifically to consonants, not to syllables; that is, one could speak of an "ambisyllabic consonant" that seems to belong to two different syllables, but not of an "ambisyllabic syllable." However, this article uses the term 'half-closed' to describe syllables that contain ambisyllabic consonants, and that seems like as good a term as any.
There are actually names for syllables not included in the open and closed syllable categories:
Apparently, there are six types of syllables:
This is backed up by this site: