I'm constantly struck by how many people seemingly refuse to believe in syllabic consonants. For instance, in my dialect (Standard American, or very close to it) there is absolutely no hint of a vowel in words like "word" or "bird". Yet many phonologists transcribe these words either as having a sequence of schwa plus /r/, or as the "r-colored schwa". I see no reason to posit any difference between the /r/ in the nucleus of "bird" and the /r/ in the onset of "rib". There surely must be a slight phonetic difference, but this is to be expected, because one is in onset position, while the other is in nucleus position. This is analogous to the slight difference between /u/ and /w/, or /i/ and /j/. One clue that it really is an /r/, and not a schwa plus /r/ sequence, or even an r-colored schwa: we get orthographic minimal pairs like "fur" and "fir" that are pronounced identically. This would be fine if they were clitics or unstressed syllables, where vowel reduction could neutralize both to schwa, but stressed "fur" and "fir" even in immaculate careful speech, are to my knowledge phonetically identical (I welcome any evidence to the contrary).
One reason people cling to the belief that syllables must have vowels is doubtless English orthography. Except for "rhythm", I can't off the top of my head think of any words that orthographically have a supposedly syllabic consonant (unless you want to count words like "icicle" that end with an orthographic vowel; feel free to post other examples in the comments if you find them). So in words like "butter", "bottle", and "button", where the second syllables contain a syllabic consonant, we still see a vowel in the written form (and presumably there was a vowel in the historical pronunciation).
Another reason is that we are taught, either explicitly or implicitly, that vowels are in some way the defining characteristic of a syllable. Many people are taught this in school, and become so dependent on orthography that some (native) speakers will even claim that the "th" sound in English (an interdental fricative) is a sequence of /t/ and /h/ (I swear I'm not making this up). However, orthography is always an imperfect clue to pronunciation, and English orthography is far from perfect, since its focus is on preserving the historical source of a word rather than transparently showing the pronunciation (NB: unlike many, I don't necessarily think that makes English orthography "worse" than a phonetically transparent orthography).
In English we have a limited number of syllabic consonants, viz., /n/, /m/, /l/, and /r/, i.e., sonorants. However, many other languages even use obstruents as syllabic. Berber and Bella Coola both utilize almost any consonant as a syllable nucleus; Bella Coola has entire vowelless sentences. What it comes down to is that there is no binary distinction between consonants and vowels; there is only the gradient sonority hierarchy, where sounds higher on the scale are more likely to be syllable nuclei, and sounds lower down are more likely to be syllable margins. For instance, within the five most common vowel quality distinctions, /i/, /e/, /a/, /o/, and /u/, /i/ and /u/ are classified as least sonorous, and are widely used (as /j/ and /w/) in syllabe margins, whereas /e/ and /o/ are more sonorous and rarely used as glides (as far as I know only in a few Papuan languages, and /o/ probably in Blackfoot), and /a/ is never a glide (unless perhaps /h/ is the consonantal version, an intriguing but questionable claim). For some languages, only the most sonorous sounds (vowels) are used as syllable nuclei, but other languages allow also the most sonorous consonants (sonorants) and others consonants lower yet (fricatives or even stops).
The 5th Annual Clarion Write-a-thon
10 months ago