Do all semivowels have vowel equivalents, and vice versa?

Question

I realise this is likely to be highly theoretical, as in “there could be such sounds but they aren’t phonemic in any language”. But I have a burning curiosity, and I’m hoping that there’s a concrete answer: “they do/could exist, and here’s what they do/would sound like”. (Or, pessimistically, “no, they couldn’t exist”.)

These seem to be the four canonical semivowels, listed everywhere and accepted without controversy:

• [j] corresponds to [i]
• [ɥ] corresponds to [y]
• [ɰ] corresponds to [ɯ]
• [w] corresponds to [u]

All of these are close (or high) vowels. Filling out the close vowels, Wikipedia also suggests these, which are uncited but seem entirely plausible:

• [j̈] corresponds to [ɨ]
• [ẅ] corresponds to [ʉ]

And I’ve seen these other semivowels put forward with corresponding vowels:

• [ʕ] (or [ʕ̞]) corresponds to [ɑ]
• [ɹ] and [ɻ] correspond to “vowels such as [ɚ]”

But I’ve seen one putative semivowel, [ʋ], with no counterpart suggested. Does it need to have a particular vowel counterpart? (I would have thought yes, but I could be wrong.) If not, are there other semivowels without associated vowels?

What vowel (if any) does [ʋ] correspond to? Are the above semivowel–vowel pairs correct? Are there others?

(It occurs to me that syllabic [ɹ̩], [ɻ̍] could be used instead of [ɚ]… in which case [ʋ] might correspond to a syllabic [ʋ̩]. But Wikipedia says that [ɹ̩] is another way to write [ɚ], placing it as a central mid vowel. What kind of vowel would [ʋ̩] be?!)

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Conversely, what about the vowels that haven’t been mentioned yet?

• Close vowels across the range have been mentioned. (This makes intuitive sense to me; with the jaw more nearly closed, you can make that not-quite-a-consonant articulation.)
• One mid vowel, [ɚ], has come up. I guess it makes sense that you can still sort-of-articulate with a half-open jaw. So what about the other mid vowels? Is there a semivowel for [e], or [ɛ], or [o]?
• And we have one open vowel in the list, [ɑ]. If you’d told me that open vowels couldn’t have semivowels (no way to sort-of-articulate them), I’d have believed you. Does [ʕ] only happen because it’s pharyngeal, not reliant on the mouth? (In which case, wouldn’t it be the semivowel for all open vowels? Why is [ɑ] special?) What about the other open vowels? Is there a semivowel for [a]?

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Reading before asking this:

• Wikipedia, Semivowel (Lists all of the above.)
• Wikipedia, R-colored vowel (Gives [ɹ̩] as equivalent to [ɚ].)
• Eugenio Martínez-Celdrán, Problems in the classification of approximants (Cited by Wikipedia. Lists the canonical semivowels with vowel counterparts. Names [ɹ], [ɻ], and [ʋ] as “closely related to vowels” without specifying which vowels.)
• Carinus, “Is there a vowel equivalent to the bilabial approximant?” (Lists the canonical semivowels, followed by the enigmatic “and so on”. Also adds [ɹ] corresponding to [ɚ].)
• Draconis, answer to “Are semivowels pronounced differently than vowels?” (Lists [w] and [u], [ɥ] and [y], and [ʕ] and [ɑ]. Explains how IPA diacritics could give a range of additional semivowels, but that doesn’t mean they’re all possible or tell me what they’d sound like.)

Not read, because I don’t have access to it: Ladefoged & Maddieson, The Sounds of the World’s Languages (also cited by Wikipedia). If this has the answers to all my questions, please tell me!

Answer 1

It really depends on what kind of theory (of what?) you are using. In phonology, feature theory (generic theory, there are many specific theories) provides one kind of answer and makes it somewhat meaningful to talk about "corresponding". In phonetics, the IPA sort of constitutes a theory, but it doesn't even have "semivowels" and I don't know what would constitute "corresponding". I'll give you an answer based on generative, generally autosegmental theories of features.

In general: there are high vowels [i u y ɯ], which are "syllabic", or can be syllable peaks. That property is expressed in many ways such as with a feature [syllabic] where vowels are [+syllabic], or by saying that "the segment is in the syllable peak", or "is dominated by V". On occasion, the feature "consonantal" is pressed into service (Hyman and Hayes do this in versions of moraic theory) saying that glides are "+consonantal" and vowels are "–consonantal", but this is contrary to the standard definition – it's essentially reviving [syllabic] which fell out of favor in autosegmental phonology. And then, there are "corresponding" glides [j w ɥ ɰ] which are [–syllabic], dominated by C, or whatever makes a segment be "non-syllabic": the other non-prosodic features are the same. I should also point out that [j w ɥ] are uncontroversial and well-enough attested but the status of [ɰ] as a glide is not at all established.

Next, we have a problem of nomenclature, dealing with semivowels, glides, and approximants. Since these are not technical concepts in phonology, you can call [ɰ] a "glide", or an "approximant", or a "semivowel", its feature properties are still the same (high back unrounded oral sonorant). The term "approximant" is least well suited for describing the traditional concept "semivowel", because there are lots of things that are not "semivowels" that are approximants (Danish ð, ʋ in many languages claimed to have it, ɹ in various dialects of English; ʕ as articulated in many languages). The main problem with the term "glide" is that the laryngeals [h ɦ ʔ] are also referred to as glides, but they are not semivowels.

The existence of non-high glides is controversial, but there are sounds that "sound" like glides which are phonetically not canonical high glides, for example in Kuria, "to eat" is [okorĕa], where [ĕa] sounds like [ja] except that the glide is lower than actual [j] with which it contrasts. The simple solution is to say that this is a short diphthong /ea/ where the first member has little of the entire time assigned to the diphthong. North Saami similarly has (uncontroversial) diphthongs like [ea, oa, ie, uo] plus short versions where the high-vocoid initials sound just like [je, wo]. A phonological analysis shows, however, that these really are diphthongs and not glide-vowel sequences. The first question that a skeptic would ask if you propose a non-high glide is, "how do you know it's not a diphthong?".

As far as "ʋ" is concerned, you have to first sort of what claim is made in invoking "ʋ": what is it different from, what does it contrast with. Often, ʋ is used to indicate a labial continuant like [β], but with some subtle phonetic difference. The wiki page lists numerous cases of that letter being used, and as you can see, it is often a dialect~idiosyncratic or allophonic variant of /v, w/ or a lenited version of /b/ – it apparently never contrasts with all of /v β w/. In those instances where [ʋ] is the phonetic realization of the glide /w/, you expect the "corresponding" vowel to be /u/; but /v,β/ don't fall into the classical vowel/glide correspondence relation. (We do of course say that the glide "corresponding" to /k/ might be [j], in case a language has such a vocalization process, but it might also be [w]).